The Galois action on geometric lattices and the mod-$\ell$ I/OM

This paper studies the Galois action on a special lattice of geometric origin, which is related to mod-$\ell$ abelian-by-central quotients of geometric fundamental groups of varieties. As a consequence, we formulate and prove the mod-$\ell$ abelian-by-central variant/strengthening of a conjecture due to Ihara/Oda-Matsumoto.


Introduction
Our story begins with a question of Ihara from the 80's, which asks whether there is a combinatorial/topological description of the absolute Galois group of Q. More precisely, this combinatorial description should be in the spirit of Grothendieck's Esquisse d'un Programme [Gro97], which suggested studying absolute Galois groups via their action on objects of "geometric origin," and specifically the geometric fundamental group of algebraic varieties. Ihara's question was formulated into a precise conjecture by Oda-Matsumoto, and so we refer to this question/conjecture as "the I/OM." The original I/OM conjecture, which deals with the full geometric fundamental group, and which we call "the absolute I/OM" below, was proven by Pop in an unpublished manuscript from the 90's. Nevertheless, Pop recently formulated and proved a strengthening of the absolute I/OM, which instead deals with the maximal pro-ℓ abelian-by-central quotient of the geometric fundamental group. The pro-ℓ abelian-by-central I/OM implies the absolute I/OM, and both contexts are treated by Pop in [Pop14].
In this paper, we formulate and prove a further strengthening of I/OM, which deals with the modℓ abelian-by-central quotient of the geometric fundamental group. This mod-ℓ context strengthens both the pro-ℓ abelian-by-central and the absolute situations. Furthermore, the mod-ℓ abelianby-central quotient is the smallest possible functorial (pro-ℓ) quotient which remains non-abelian. Therefore, in some sense, the mod-ℓ context yields strongest possible results one could hope for.
Most importantly however, the mod-ℓ abelian-by-central context gets much closer to the spirit of Ihara's original question of finding a combinatorial description of absolute Galois groups. Indeed, the geometric fundamental group of a variety and its pro-ℓ abelian-by-central quotients are both finitely-generated profinite resp. pro-ℓ groups, and the topology of such groups plays a crucial role in both situations. In contrast to this, the mod-ℓ abelian-by-central quotient can be seen as a (discrete) finite-dimensional Z/ℓ-vector space endowed with some extra linear structure. I.e. the mod-ℓ abelian-by-central quotient of a geometric fundamental group is an object of purely combinatorial nature, being a finite-dimensional linear object over Z/ℓ.
The precise notation and context of the paper is somewhat involved. For readers' sake, we now give some brief (and mostly unmotivated) definitions in order to state the primary main theorem of the paper, but the rest of the introduction will provide the full detailed notation and motivation. Let ℓ be a fixed prime. Let k 0 be a field of characteristic = ℓ, and let X be a normal, geometricallyintegral k 0 -variety. For such an X, we write (see §1.4 for the precise definitions of (2) and (3)): (1)π 1 (X) := πé t 1 (X,x) for the geometricétale fundamental group of X, i.e. theétale fundamental group of the the base-changeX of X tok 0 , with respect to some geometric point x.
For an essentially small category V of normal geometrically-integral k 0 -varieties, we consider the group Aut c (π a | V ) which consists of systems (φ X ) X∈V , where X varies over the objects of V, and the φ X ∈ Aut c (π a (X)) are compatible with morphisms arising from V. Since π a (X) is a Z/ℓ-vector space for all X ∈ V, and the morphisms π a (X) → π a (Y ) arising from morphisms X → Y in V are Z/ℓ-linear, we obtain a canonical action of (Z/ℓ) × on Aut c (π a | V ) by left-multiplication. We write Aut c (π a | V ) := Aut c (π a | V )/(Z/ℓ) × for the quotient by this canonical action of (Z/ℓ) × .
The action of Gal k 0 := Gal(k 0 |k 0 ) onX for X as above yields a canonical (outer) action on π a (X) and π c (X), hence also a canonical Galois representation ρ c k 0 ,X : Gal k 0 → Aut c (π a (X)). Finally, by collecting the ρ c k 0 ,X for all X in V, we obtain a canonical Galois representation ρ c k 0 ,V : Gal k 0 → Aut c (π a | V ) ։ Aut c (π a | V ). With this notation, the main theorem of the present paper reads as follows.
Main Theorem: Let k 0 be an infinite perfect field of characteristic = ℓ, and let V be a "sufficiently large" category of normal geometrically-integral k 0 -varieties. Then the canonical map ρ c k 0 ,V : Gal k 0 → Aut c (π a | V ) is an isomorphism.
We will define precisely what we mean by "sufficiently large" in §1.7, where the precise assumption/terminology is that V should be "5-connected." However, we note here that both the full category of all normal geometrically-integral k 0 -varieties and the full category of smooth quasiprojective k 0 -varieties are "sufficiently large" in this sense.
1.1. Automorphism groups of functors. We first introduce some general notation and terminology which will help simplify the exposition. Let C be an essentially small category, and let F : C → D be an arbitrary functor to another category D. We consider Aut(F), the automorphism group of the functor F. To be explicit, the elements of Aut(F) are systems (φ X ) X∈C with φ X ∈ Aut(F(X)) parameterized by the objects X of C, such that for every morphism f : X → Y of C, one has a commutative diagram in D F(X) For a subcategory C 0 of C, we write F| C 0 for the restriction of F to C 0 , and we will frequently consider the automorphism group Aut(F| C 0 ) of this restriction. Note that restriction yields a canonical restriction morphism φ → φ| C 0 : Aut(F) → Aut(F| C 0 ).
In explicit terms, this restriction morphism sends a system (φ X ) X∈C as above to the restricted system (φ X ) X∈C 0 .
We will frequently consider several different target categories D. To keep the notations consistent throughout, if some symbol/notation is used to denote the automorphism groups of objects in D, then we will use the same symbol/notation to denote the automorphism group of a functor whose values are in D.
1.2. The Absolute I/OM. Throughout the paper, we will work with a fixed infinite perfect field k 0 , and k =k 0 will denote the algebraic closure of k 0 . Furthermore, Gal k 0 := Gal(k|k 0 ) will denote the absolute Galois group of k 0 . We will only consider normal geometrically-integral k 0 -varieties, and we denote the category of all such varieties by Var k 0 . Namely, the objects of Var k 0 are schemes which are normal, geometrically-integral, separated and of finite-type over k 0 . The morphisms in Var k 0 are just morphisms of k 0 -schemes.
For every X ∈ Var k 0 , we writeX = X ⊗ k 0 k = X × Spec k 0 Spec k for the base-change of X to k. Since k 0 is perfect, it follows thatX is a normal k-variety for every X ∈ Var k 0 . Furthermore, recall that Gal k 0 acts onX in the usual canonical way. Namely, σ ∈ Gal k 0 acts onX via the automorphism 1 × Spec(σ −1 ) of We let Prof Out denote the category of profinite groups with outer-morphisms. Namely, the objects of Prof Out are just the profinite groups, but the set of morphisms G → H in Prof Out is given by Hom Out (G, H) := Hom cont (G, H)/ Inn(H).
In particular, the automorphism group of a profinite group G in Prof Out is precisely Out(G), the outer-automorphism group of G.
For every X ∈ Var k 0 , consider the geometric (étale) fundamental group of X, π 1 (X) := πé t 1 (X,x) 3 with respect to some geometric pointx. We will considerπ 1 (X) as an object of Prof Out , and because of this, the choice of geometric point becomes irrelevant. We therefore omit the geometric pointx from the notation. To summarize, we obtain a canonical functor π 1 : Var k 0 → Prof Out .
Observe that the action of Gal k 0 onX and the functoriality of πé t 1 with values in Prof Out , yields a canonical outer Galois representation ρ k 0 ,X =: ρ k 0 : Gal k 0 → Out(π 1 (X)).
Naturally, this outer representation agrees with the one arising from the fundamental short exact sequence 1 →π 1 (X) → πé t 1 (X,x) → Gal k 0 → 1. Moreover, it is clear that if X → Y is a morphism in Var k 0 , then the induced morphismπ 1 (X) → π 1 (Y ) in Prof Out is compatible with the action of Gal k 0 . In particular, Gal k 0 acts on the functor π 1 itself.
Similarly, if V is any (essentially small) subcategory of Var k 0 , then Gal k 0 acts onπ 1 | V . Following our notational convention mentioned in §1.1, we denote the automorphism group ofπ 1 | V by Out(π 1 | V ). We therefore obtain a canonical outer Galois representation ρ k 0 ,V : Gal k 0 → Out(π 1 | V ).
With this notation, the Absolute I/OM refers to the following general question.
The Aboslute I/OM: For which fields k 0 and subcategories V of Var k 0 as above, is the Galois representation ρ k 0 ,V : Gal k 0 → Out(π 1 | V ) an isomorphism?
The nature of the map ρ k 0 ,V in general is still quite mysterious. Nevertheless, for k 0 = Q, the injectivity of this morphism has been extensively studied. For instance Drinfel'd first observed that Belyi's theorem implies ρ Q,V is injective as soon as V contains the tripod, P 1 Q {0, 1, ∞}. More generally, it follows from the work of Voevodsky, Matsumoto and Hoshi-Mochizuki that ρ Q,V is injective as soon as V contains a (possibly affine) hyperbolic curve.
The surjectivity of ρ k 0 ,V is much less understood, even in the case k 0 = Q. For instance, if one takes V = {M 0,n } n with the "connecting morphisms" (or certain smaller subcategories) then Out(π 1 | V ) is the intensively studied Grothendieck-Teichmüller group. The surjectivity of ρ Q,V in this case (and for other subcategories of the Teichmüller modular tower) is still a major open question in modern Galois theory.
In any case, Ihara asked in the 1980's whether ρ Q,V is an isomorphism in the case where V = Var Q , and it was Oda-Matsumoto in the 90's who finally conjectured that the answer to this question should be "yes." In 1999, Pop proved in an unpublished manuscript that ρ k 0 ,V is an isomorphism for more general fields k 0 in the case where V = Var k 0 . Pop's proof was finally released in [Pop14], along with a stronger pro-ℓ abelian-by-central variant which we now summarize.
1.3. The Pro-ℓ Abelian-by-Central I/OM. In order to get closer to the spirit of Ihara's original question of finding a combinatorial description of the absolute Galois group of Q, it makes sense to replace the geometric fundamental group by certain smaller functorial quotients. The first such strengthening was formulated and proved by Pop [Pop14] who uses the maximal pro-ℓ abelianby-central quotient ofπ 1 . We will give only a very brief summary of the pro-ℓ abelian-by-central context, since the purpose of this paper is to develop a mod-ℓ variant/strengthening of loc.cit.
Note that both Π A (X) and Π C (X) are functorial in X. Moreover, note that Z × ℓ acts on Π A (X) by left multiplication. It turns out (by general group-theoretical facts) that this action lifts to Π C (X). We consider the set of liftable automorphisms of Π A (X), defined by Aut C (Π A (X)) := Image(Aut(Π C (X)) → Aut(Π A (X))), as well as the quotient Aut C (Π A (X)) := Aut C (Π A (X))/Z × ℓ by the canonical action of Z × ℓ . Similarly to the absolute context, for every X ∈ Var k 0 one has canonical representations ρ C k 0 ,X : Gal k 0 → Aut C (Π A (X)) → Aut C (Π A (X)). Given a subcategory V of Var k 0 , one defines Aut C (Π A | V ) as the set of systems (φ X ) X∈V , φ X ∈ Aut C (Π A (X)) which are compatible with morphisms from V, similarly to the absolute context. We may also consider the quotient by the canonical action of Z × ℓ : similarly to our definition of Aut C (Π A (X)). As before, we obtain canonical Galois representations . With this notation, the pro-ℓ abelian-by-central I/OM is completely analogous to the absolute I/OM, as it refers to the following general question.
The Pro-ℓ Abelian-by-Central I/OM: For which fields k 0 and subcategories V of Var k 0 as above, is the Galois representation Pop shows in [Pop14] that ρ C k 0 ,V is an isomorphism for so-called "connected" subcategories V of Var k 0 . This condition holds, in particular, for Var k 0 itself, as well as for the category of smooth quasi-projective k 0 -varieties. Although Pop's notion of connectedness is somewhat technical, we note that it is similar to what we call "2-connected" below.
Pop's pro-ℓ abelian-by-central context gets closer to a truly combinatorial description of absolute Galois groups than the absolute context. However, the groups considered in this context, Π A (X) and Π C (X), are still quite large as they still have a non-trivial profinite topology which plays a very significant role.
In this paper, we will consider a further strengthening of the I/OM which considers the modℓ abelian-by-central quotient ofπ 1 (X). As mentioned before, this quotient ofπ 1 which we consider is the smallest functorial pro-ℓ quotient ofπ 1 which remains non-abelian; in particular, it is a quotient of Π C (X), and it can be seen as a purely combinatorial (i.e. finite and discrete) object. Thus, the mod-ℓ abelian-by-central context is essentially the best one could hope for. We will prove that the analogous mod-ℓ abelian-by-central I/OM holds for so-called "5-connected" subcategories V of Var k 0 . Similarly to Pop's notion of connectedness, our notion of 5-connectedness applies to Var k 0 itself, as well as for the full category of smooth quasi-projective k 0 -varieties.
It is particularly important to note that Pop [Pop14] uses ideas related to Bogomolov's Program [Bog91] in birational anabelian geometry, which considers pro-ℓ abelian-by-central Galois groups of higher-dimensional function fields over k. In a few words, Pop's proof of the pro-ℓ abelian-bycentral I/OM first reduces to a birational context, and eventually uses both the local theory [BT02], [Pop10] and global theory [Pop12c] from pro-ℓ abelian-by-central birational anabelian geometry.
The initial step in our proof of the mod-ℓ abelian-by-central I/OM is more-or-less the same as the pro-ℓ context, in the sense that we will reduce the mod-ℓ I/OM to a birational context. We will then use techniques from the mod-ℓ abelian-by-central variant of Bogomolov's Program, including both the mod-ℓ local theory [Pop12a], [Top15a], [Top15c] and the mod-ℓ global theory [Top15d]. However, because of this strategy, we run into precisely the same problems/difficulties that arise when one passes from the pro-ℓ to the mod-ℓ abelian-by-central variants of Bogomolov's Program.
These fundamental differences between the pro-ℓ and mod-ℓ context were described in detail in the introduction of [Top15d], and we refer the reader there for these details. Nevertheless, we mention here that, in the pro-ℓ context, one eventually uses the Fundamental Theorem of Projective Geometry applied to an infinite-dimensional k-projective space embedded in H 1 (K, Z ℓ (1)), where K is a function field over k. The main difficulty in the mod-ℓ context is that H 1 (K, Z/ℓ(1)) contains no such k-projective space. Therefore, our proof of the mod-ℓ I/OM is fundamentally different than the proof of the pro-ℓ variant. See §2 for a detailed summary of the proof of the mod-ℓ I/OM, and see the introduction of [Top15d] for more on the comparison between the pro-ℓ and mod-ℓ contexts. We now introduce the mod-ℓ abelian-by-central context in detail.
1.4. The Mod-ℓ Abelian-by-Central Quotients. For a profinite group G, we let G (i) denote the i-th term of the mod-ℓ Zassenhauss filtration of G. We will only need to consider the first two non-trivial terms of this filtration, which are defined explicitly as follows: ( We will consistently denote the quotients G/G ( * ) for * = 2, 3 as follows: We call G a resp. G c the mod-ℓ abelian resp. mod-ℓ abelian-by-central quotient of G. Indeed, note that G a is an ℓ-elementary abelian pro-ℓ group and G c is a central extension of G a by an ℓ-elementary abelian pro-ℓ group. Suppose now that σ, τ ∈ G a are given, and choose liftsσ,τ ∈ G c of σ, τ . Since G c is a central extension of G a , it follows that the commutator [σ, τ ] :=σ −1τ −1στ depends only on σ, τ ∈ G a and not on the choice of liftsσ,τ ∈ G c of σ, τ . Next, define the completed wedge product of G a with itself as where H varies over the open subgroup of G a , and where ∧ 2 (M ) = M ⊗ M/ x ⊗ x : x ∈ M for a discrete Z/ℓ-module M . Then the commutator defined above extends linearly to define a canonical morphism and we define R(G) as the kernel of this canonical map. Note that one has G a = (G c ) a and R(G) = R(G c ), so the datum (G a , R(G)) is completely determined by the quotient G c . Suppose now that G 1 , G 2 are two profinite groups, and that f : G a 1 → G a 2 is a morphism. In this context, we say that f is compatible with R if the induced map restricts to a map R(G 1 ) → R(G 2 ). We write Hom c (G a 1 , G a 2 ) for the collection of morphisms f : G a 1 → G a 2 which are compatible with R. Similarly, for a profinite group G, we write Aut c (G a ) for the collection of automorphisms of G a which are compatible with R and whose inverse is also compatible with R.
The definitions above can be summarized by defining the mod-ℓ abelian-by-central category, denoted AbC ℓ , to be the category whose objects consist of pairs (G a , R) where G a is a profinite group such that (G a ) (2) = 1, and R is a closed subgroup of ∧ 2 (G a ). A morphism from (G a 1 , R 1 ) to (G a 2 , R 2 ) in AbC ℓ is simply a morphism f : G a 1 → G a 2 of profinite groups such that the induced map ∧ 2 (f ) : ∧ 2 (G a 1 ) → ∧ 2 (G a 2 ) restricts to a map R 1 → R 2 . Finally, we have a canonical functor (•) ac : Prof Out → AbC ℓ defined on objects by G ac := (G a , R(G)). The R-compatible morphisms/automorphisms are then given by the morphisms/automorphism in AbC ℓ : Finally, as noted above, for any profinite group, one has G ac = (G c ) ac as objects of AbC ℓ . In other words, the functor (•) ac factors through the endofunctor G → G c of Prof Out .
Similarly to the pro-ℓ context, we will consistently use underlines to denote the process of modding out by the left-multiplication action of (Z/ℓ) × . For instance, note that (Z/ℓ) × acts on G a by multiplication on the left, and that this action is always compatible with R. Thus, (Z/ℓ) × acts on Hom c (G a 1 , G a 2 ) and we write Hom c (G a 1 , G a 2 ) = Hom c (G a 1 , G a 2 )/(Z/ℓ) × . Similarly, (Z/ℓ) × acts on Aut c (G a ) and we write Aut c (G a ) := Aut c (G a )/(Z/ℓ) × .
Note that for every X ∈ Var k 0 , the Galois group Gal k 0 acts on π c (X) by outer-automorphisms, since π c (X) is a characteristic quotient ofπ 1 (X). Since π a (X) is a further characteristic quotient of π c (X), we see that Gal k 0 acts on (π c (X)) ac = (π a (X), R(π c (X))) as an object of AbC ℓ , so that we obtain canonical Galois representations ρ c k 0 ,X : Gal k 0 → Aut c (π a (X)) → Aut c (π a (X)). Suppose now that V is an essentially small subcategory of Var k 0 . In this context, we define Aut c (π a | V ) to be the automorphism group of the functor In other words, Aut c (π a | V ) consists of system (φ X ) X∈V ∈ Aut(π a | V ) with φ X ∈ Aut c (π a (X)) for all X ∈ V, such that the φ X are compatible with morphisms arising from V. As before, (Z/ℓ) × acts on Aut c (π a | V ) in a canonical way by left multiplication, and will denote the quotient by this action with an underline as Aut c (π a | V ) := Aut c (π a | V )/(Z/ℓ) × . Finally, we can combine all of the ρ c k 0 ,X as before to obtain canonical Galois representations ρ c k 0 ,V : Gal k 0 → Aut c (π a | V ) ։ Aut c (π a | V ). With this notation, the mod-ℓ abelian-by-central I/OM is completely analogous to the absolute and pro-ℓ contexts, and it refers to the following general question.
The Mod-ℓ Abelian-by-Central I/OM: For which fields k 0 and subcategories V of Var k 0 as above, is the Galois representation ρ c k 0 ,V : Gal k 0 → Aut c (π a | V ) an isomorphism?
1.6. The Main Result -Birational Systems. With the preparation above, we will now introduce the subcategories of Var k 0 which we consider in this paper. Let X ∈ Var k 0 have dimension ≥ 1, and let U X be a basis of open neighborhoods of the generic point of X. We will always consider U X as a subcategory of Var k 0 whose objects are the elements of U X and whose morphisms are the inclusions among them as open subsets of X. Moreover, we write U + X = U X ∪ {X} for the basis of open neighborhoods of the generic point of X which also includes X as a terminal object.
Let X ∈ Var k 0 be an object. To simplify the exposition, we will say that a subcategory U X of Var k 0 is a birational system of X if U X is a basis of open neighborhoods of the generic point of X. We will use the notation U + X as above to denote the existence of X as a terminal object. In other words, while a birational system U X of X need not have a terminal object, the birational system U + X always has X as a terminal object. We say that U is a birational system if U is a birational system of X for some X ∈ Var k 0 . The dimension of a birational system U , denoted dim U , is defined to be the dimension of one (hence all) of the objects in U . In particular, dim U X = dim X. Now let r ≥ 0 be given, and let a = (a 1 , . . . , a r ) be a (possibly empty) finite tuple of elements a i ∈ k × 0 . We denote the complement of a in G m (over k 0 ) as U a := G m,k 0 {a 1 , . . . , a r }.
For instance, U a = G m if a = ∅ is empty, and U a = P 1 {0, 1, ∞} is the tripod if a = (1). We will furthermore consider a small category U a which is constructed from a positive-dimensional birational system U and U a as follows: (1) The objects of U a are given by U ∪ {U a }.
(2) The morphisms in U a are the inclusions among the objects in U , and all of the dominant morphisms U → U a for U ∈ U . Our first main theorem concerning the mod-ℓ I/OM as stated above shows the bijectivity of the Galois representation for such categories, if the birational system has sufficiently large dimension.
Theorem A. Let k 0 be in infinite perfect field of characteristic = ℓ. Let U be a birational system of dimension ≥ 5, and let a = (a 1 , . . . , a r ) be a (possibly empty) finite tuple of elements of k × 0 . Then the canonical Galois representation ρ c k 0 ,Ua : Gal k 0 → Aut c (π a | Ua ) is an isomorphism.
1.7. The Main Result -Connected Categories. We will use the theorem above to prove the mod-ℓ I/OM for many more subcategories V, including the full category Var k 0 and the category of smooth quasi-projective k 0 -varieties. To state such a general result, we first need some more terminology.
Suppose that V an essentially small subcategory of Var k 0 which contains a positive-dimensional object. Let U 1 , U 2 be two positive-dimensional birational systems. In this context, we say that U 1 dominates U 2 in V provided that V contains U 1 and U 2 , and that the following holds: • If dim U 2 > 1: For all V ∈ U 2 , there exists some U ∈ U 1 such that V contains a dominant morphism U → V . • If dim U 2 = 1: For all V ∈ U 2 , there exists some U ∈ U 1 such that V contains a dominant morphism U → V with geometrically integral fibers. Next suppose that U , U 1 , U 2 are three positive-dimensional birational systems. In this context, we say that U attaches U 1 to U 2 in V if the following hold: (1) The category V contains U a for some finite tuple a of elements of k × 0 .
(2) The birational system U dominates both U 1 and U 2 in V. Now let d ≥ 1 be given, and let U 0 and U 2r be two birational systems. We will say that U 0 and U 2r are d-connected in V if there exist birational systems U 1 , . . . , U 2r−1 such that, for all i = 0, . . . , r − 1, the following conditions hold: (1) One has dim U 2i+1 ≥ d.
(2) The birational system U 2i+1 attaches U 2i to U 2i+2 in V. Finally, we say that V is d-connected if the following conditions hold: (1) V is essentially small and it contains a positive-dimensional object.
(2) For every object X of V, there exists some birational system U + X of X which contains X as the terminal object, such that U + X is contained in V.
Although the precise definition of a d-connected category is somewhat complicated, we note that, for example, both the full category Var k 0 and the category of smooth quasi-projective k 0 -varieties are d-connected for all d ≥ 1. Furthermore, if d ′ ≥ d ≥ 1, we note that V being d ′ -connected implies that V is d-connected. Our next main theorem concerns the mod-ℓ I/OM for 5-connected varieties.
Theorem B. Let k 0 be in infinite perfect field of characteristic = ℓ, and let V be a subcategory of Var k 0 which is 5-connected. Then the canonical Galois representation ρ c k 0 ,V : Gal k 0 → Aut c (π a | V ) is an isomorphism.
1.8. Birational-Galois variant. Let X ∈ Var k 0 be given and let U = U X be a birational system for X, as defined above. Recall that elements of Aut c (π a | U ) consist of systems of elements (φ U ) U ∈U , where φ U ∈ Aut c (π a (U )) are compatible with the morphisms arising from U . By taking the projective limit over U , one obtains an element of Aut c ((Gal k(X) ) a ). Moreover, since X is geometrically normal, it follows that the induced canonical map Aut c (π a (X)) → Aut c ((Gal k(X) ) a ) is injective. Therefore, in order to prove Theorem A, it makes sense to first develop a birational variant of that theorem, which deals with quotients of absolute Galois groups of function fields as opposed to quotients of fundamental groups of varieties. Therefore, the main focus of this paper is to develop and prove birational variants of our main theorems, and we now introduce the appropriate notation and terminology.
Suppose that K 0 is a regular function field over k 0 , and let K = K 0 · k = K 0 ⊗ k 0 k denote the base-change of K 0 to k. Recall that Gal k 0 acts on K = K 0 ⊗ k 0 k in the obvious way, and that one has a canonical isomorphism We denote by G K := Gal(K(ℓ)|K) the maximal pro-ℓ Galois group of K, and we consider its modℓ abelian resp. mod-ℓ abelian-by-central quotients G a K resp. G c K . We also consider the associated 9 object (G K ) ac = (G c K ) ac = (G a K , R(G c K )) of AbC ℓ , as introduced in §1.4. Following the notation above, we denote the automorphism group of G ac K by Aut c (G a K ), and we write Aut c (G a K ) := Aut c (G a K )/(Z/ℓ) × for its quotient by the canonical action of (Z/ℓ) × . Since the projection G c K → G a K is functorial in K, it follows that Gal k 0 acts on G ac K as an object of AbC ℓ . In other words, we obtain canonical Galois representations ρ c k 0 ,K 0 : . Suppose now that a is a (possibly empty) finite tuple of elements of k × 0 , and recall that we write U a := G m a. Note that every non-constant t ∈ K × 0 induces a dominant morphism U → U a for some U ∈ Var k 0 such that k 0 (U ) = K 0 . Thus, for every non-constant t ∈ K 0 , we obtain a canonical morphism of ℓ-elementary abelian pro-ℓ groups . Let H t denote the kernel of π t : G a K → π a (U a ). We will write Aut c a (G a K ) for the subgroup of Aut c (G a K ) consisting of elements φ ∈ Aut c (G a K ) such that φH t = H t for all non-constant t ∈ K × 0 . Note that the canonical action of (Z/ℓ) × on Aut c (G a K ) restricts to an action on the subgroup Aut c a (G a K ), and we will write Aut c a (G a K ) := Aut c a (G a K )/(Z/ℓ × ) for the quotient of this action.
Since the Galois action is clearly compatible with the morphisms π t for non-constant t ∈ K × 0 , we see that any element of Aut c (G a K ) resp. Aut c (G a K ) which arises from Gal k 0 must actually be contained in Aut c a (G a K ) resp. Aut c a (G a K ). In other words, we obtain canonical Galois representations ρ k 0 : Gal k 0 → Aut c a (G a K ) → Aut c a (G a K ). Our Birational-Galois variant of Theorem A is about this canonical morphism.
Theorem C. Let k 0 be an infinite perfect field of characteristic = ℓ. Let K 0 be a regular function field over k 0 of transcendence degree ≥ 5, and put K = K 0 ·k. Let a be an arbitrary (possibly-empty) finite tuple of elements of k × 0 . Then the canonical map ρ k 0 : Gal k 0 → Aut c a (G a K ) is an isomorphism.
1.9. Birational-Milnor variant. It turns out that it will be more convenient to work with the Kummer Dual of Theorem C. While the Kummer dual of G a K is K × /ℓ, it will be a consequence of the Merkurjev-Suslin Theorem [MS82] that the "dual" of the object G ac K can be considered as the mod-ℓ Milnor K-ring of K, which we denote by k M * (K). Thus, the primary focus of this paper will be to prove a Milnor variant of Theorem C, which deals with the mod-ℓ Milnor K-ring of the function field K.
We will recall the definition of k M * (K) in §3, but we note here that one has k M 1 (K) = K × /ℓ and that one has a canonical surjective morphism of Z/ℓ-algebras where T * (K × /ℓ) denotes the tensor algebra of K × /ℓ considered as a vector space over Z/ℓ which is concentrated in degree 1. We denote by Aut M (k M 1 (K)) the collection of automorphisms of k M 1 (K) which extend to an automorphism of k M * (K). Similarly to the above, we have a canonical action of (Z/ℓ) × on k M 1 (K) by left multiplication, and we put Aut M (k M 1 (K)) = Aut M (k M 1 (K))/(Z/ℓ) × . Let a = (a 1 , . . . , a r ) be a possibly empty finite tuple of elements of k × 0 as above. For x ∈ K × , we write {x} K for the image of x in k M 1 (K) = K × /ℓ. We write Aut M a (k M 1 (K)) for the subgroup of all automorphisms φ ∈ Aut M (k M 1 (K)) such that for all non-constant t ∈ K × 0 , the automorphism φ restricts to an automorphism of the subgroup . Finally, we define Aut M a (k M 1 (K)) := Aut M a (k M 1 (K))/(Z/ℓ) × similarly to the above. It turns out that the group Aut M a (k M 1 (K)) can be viewed as the "Kummer Dual" of the group Aut c a (G a K ) considered in §1.8. As before, we have a canonical action of Gal k 0 on k M * (K). Moreover, this action is compatible with subgroups of k M 1 (K) of the form for all non-constant t ∈ K × 0 , and all a 1 , . . . , a r ∈ k × 0 . To summarize, for a (possibly empty) finite tuple a of elements of k × 0 , we obtain canonical Galois representations ) which are the primary focus of the following "Milnor-Variant" of Theorem C.
Theorem D. Let k 0 be an infinite perfect field of characteristic = ℓ. Let K 0 be a regular function field over k 0 of transcendence degree ≥ 5, and put K = K 0 ·k. Let a be an arbitrary (possibly-empty) finite tuple of elements of k × 0 . Then the canonical map ρ M k 0 ,K 0 : Gal k 0 → Aut M a (k M 1 (K)) is an isomorphism.
1.10. A guide through the paper. This paper contains a total of 11 sections, including §1 which is the introduction, and §2 which introduces some notation, and includes a summary of the proof of the main theorems.
Sections 3, 4 and 5 contain mostly generalities, appropriately translated to our context. More specifically, in §3, we recall some basic facts about the Mod-ℓ Milnor K-ring of fields. In §4, we recall the cohomological framework which allows us to translate back and forth between mod-ℓ abelian-by-central Galois groups and mod-ℓ Milnor K-rings -this can be seen as a group-theoretical formulation of the Merkurjev-Suslin theorem [MS82]. Such cohomological results have seen a recent resurgence in [CEM12], [EM11], [EM15], [Top15b], especially in connection with the Merkurjev-Suslin Theorem [MS82] and/or the Bloch-Kato conjecture, which is now a highly-celebrated theorem due to Voevodsky-Rost et al. [Voe11], [Ros98], [Wei09]. Nevertheless, the Merkurjev-Suslin theorem is sufficient for the considerations in §4, as we summarize the appropriate results for our context in Theorem 4.2.
The core of the paper begins in §6, where we discuss the mod-ℓ Milnor K-theory of function fields. The ideas in this section are similar to [Top15d, §3], although the results themselves refine loc.cit. somewhat. Perhaps the most important result in §6 is Corollary 6.4 which shows how to reconstruct a geometric subgroup given sufficiently many of its elements.
In §7, we summarize (see Theorem 7.1) the main results from Evans-Hrushovski [EH91], [EH95] and Gismatullin [Gis08], translated appropriately to the context of the present paper. In §7 we also prove Corollary 7.4, which shows that the absolute Galois group Gal k 0 can be canonically identified with a Galois group of certain geometric lattices associated to K|k and K 0 |k 0 ; this corollary will be used in a fundamental way in the proof of Theorem D.
In §8, we introduce the so-called essential branch locus, and the notion of an essentially unramified point. We use this concept of essential ramification in several technical results in coordination with the local theory, in order to ensure that certain divisorial valuations can be "detected" in the mod-ℓ setting.
In §9, we recall the notion of a general element, and introduce the notion of a strongly-general element. In this section we also recall the so-called Birational-Bertini theorem for general elements. We also prove a Birational-Bertini theorem for strongly-general elements, which uses the "yoga" of essential ramification in a fundamental way.
In §10, we give the detailed proof of Theorem D, and note that Theorem C follows from this by applying Theorem 4.2 from §4. Finally, in §11, we conclude the proofs of Theorems A and B by using Theorem C. Scanlon for numerous technical discussions concerning the topics in this paper. The author also thanks all who expressed interest in this paper, and especially Aaron Silberstein, Ján Mináč and Andrew Obus.

Notation and a Summary
Throughout the whole paper, we will work with a fixed prime ℓ and a fixed base field k 0 , such that k 0 is infinite, perfect, and of characteristic = ℓ. We denote by Gal k 0 the absolute Galois group of k 0 . We will also fix a function field K 0 over k 0 which is regular, which means that K 0 has a separating transcendence base and that k 0 is relatively algebraically closed in K 0 . We denote by k :=k 0 the algebraic closure of k 0 , and we write When we discuss other fields which are potentially unrelated to K 0 |k 0 and/or K|k and which might have characteristic ℓ, we will use letters such as F, L, M , etc. The perfect closure of a field F will be denoted by F i . If Char F = p > 0, then we will write Frob F for the usual Frobenius map on F , and we note that Frob F i is an automorphism of F i . In order to keep the notation consistent, if Char F = 0, then Frob F is defined to be the identity on F . Finally, the absolute Galois group of a field F will be denoted by Gal F , and the maximal pro-ℓ Galois group of F will be denoted by G F .
For a valuation v of F , we will use the following standard notation associated with v. We denote the valuation ring of v by O v and the maximal ideal of O v is denoted by m v . We will also write U v := O × v for the v-units and U 1 v := (1 + m v ) for the principal v-units. Finally, we write vF for the value group of v and F v for the residue field of v. The residue map O v ։ F v will usually be denoted by t →t. If L is a subfield of F , we will abuse the notation and write vL resp. Lv for the value group resp. residue field of the restriction v| L of v to L.
The majority of this paper deals with k M * (K), the mod-ℓ Milnor K-ring of K, the definition of which is recalled in §3. We note now that for a field F , one has k M 1 (F ) = F × /ℓ, and that the canonical projection F × ։ F × /ℓ = k M 1 (F ) is denoted by x → {x} F . We now introduce some important notation which will be used consistently throughout the whole paper. In particular, we introduce the notion of a geometric subgroup of k M 1 (K), which is the primary object we study in this paper. For a subset S of K, we may consider k(S), the subextension of K|k generated by S, and we write: (1) K(S) := k(S) ∩ K for the relative algebraic-closure of k(S) in K. ( is injective, and its image is K(S). In particular, the map k M 1 (K(S)) → k M 1 (K) induces a canonical isomorphism k M 1 (K(S)) ∼ = K(S). We will also frequently work with valuations of K via the images of their (principal) units in k M 1 (K), and so we must introduce some more important notation here. For a valuation v of K, we will consistently write: (1) . We will frequently consider affine and projective spaces over k, which are given by a certain set of algebraically independent parameters. More precisely, let t = (t 1 , . . . , t r ) be a collection of r algebraically independent indeterminants over k. In this case, we write In other words, one has a canonical open embedding A r t ֒→ P r t , and the function field of A r t and/or P r t can be canonically identified with the rational function field k(t) generated by t.
We will also identify the closed points of A r t resp. P r t with the set A r t (k) = k r resp. P r t (k) = (k r+1 {0})/k × of k-rational points. We will use affine coordinates (a 1 , . . . , a r ) to denote elements of A r t (k) = k r , and we will use homogeneous coordinates (a 0 : · · · : a r ) to denote elements of P r t (k) = (k r+1 {0})/k × . In particular, we identify A r t (k) = k r with the elements of the form (1 : a 1 : · · · : a r ) in P r t (k) = (k r+1 {0})/k × . Since we will consider various representations of Gal k 0 , in order to simplify the notation, we will denote all such representations by ρ k 0 if no confusion is possible. This convention holds in particular for the representations ρ c k 0 , * ( * = X, V, K 0 ) and ρ M k 0 ,K 0 , which were defined in §1. We will also use this implicit terminology when describing the compatibility of certain morphisms with ρ k 0 . To be precise, if •, • are two objects endowed with two Gal k 0 -representations ρ k 0 : Gal k 0 → Aut(•) resp. ρ k 0 : Gal k 0 → Aut(•), we say that a morphism f : Aut(•) → Aut(•) of automorphism groups is compatible with ρ k 0 provided that the following diagram commutes: Furthermore, we will say that an element σ ∈ Aut(•) arises from Gal k 0 if σ is in the image of ρ k 0 . If we wish to make precise the element τ ∈ Gal k 0 which maps to this σ, we will say that σ is defined by τ .
Some of the proofs in this paper are fairly technical, although the overall idea is quite natural, and can be briefly described using the following three key steps:

13
(1) First, reduce all the main theorems to Theorem D.
(2) Second, prove that any element of Aut M a (k M 1 (K)) induces an automorphism of a certain lattice G * (K|k) which is of geometric origin. This step is the most difficult and takes up the majority of the paper.
(3) Finally, we use an analogue of the Fundamental Theorem of Projective Geometry for this lattice G * (K|k), to deduce that any element of Aut M a (k M 1 (K)) arises in a unique way from Gal k 0 . This analogue of the fundamental theorem of projective geometry comes from the work of Evans-Hrushovski [EH91], [EH95] and Gismatullin [Gis08], and it relies on the so-called group-configuration theorem from geometric stability theory. For readers' sake, we now provide a fairly detailed summary of the proofs of the main theorems, to act as a guide for reading the details which are found in the body of the paper.
2.1. Reduction to Theorem A. In the terminology introduced above, suppose that U X and U Y are birational systems for X resp. Y . Furthermore, suppose that U X dominates U Y in V. Note that any element φ of Aut c (π a | V ) defines an element of φ| U X ∈ Aut c (π a | U X ) and an element of φ| U Y ∈ Aut c (π a | U Y ) by restriction. The condition that U X dominates U Y implies the following property: If φ| U X is defined by τ ∈ Gal k 0 , then φ| U Y is defined by τ as well. The "5-connectedness" assumption on V is then used to reduce Theorem B to Theorem A.

2.2.
Reduction to Theorem C. Let X be a normal k 0 -variety with K = k(X), and let U be a birational system of X. By passing to the projective limit over U , one obtains a canonical injective map . Given a finite tuple a of elements of k × 0 , this injective map above induces a map Aut c (π a | Ua ) → Aut c a (G a K ), by first restricting to U , then taking projective limits over U . This induced map turns out to be injective as well, as long as dim U ≥ 2. Since this injection is compatible with ρ k 0 , we see that Theorem A follows from Theorem C.
2.3. Reduction to Theorem D. Kummer theory yields a canonical perfect pairing Moreover, using the well-known duality between H 2 (G c K , Z/ℓ) and the relations in a minimal free presentation of G c K (in the category of pro-ℓ groups), along with the fact that cup-products correspond to commutators in this duality, it is then a consequence of the Merkurjev-Suslin theorem [MS82] that one has a canonical isomorphism . This isomorphism is obtained by dualizing an automorphism of G a K via the Kummer pairing above to obtain an automorphism of k M 1 (K). Although the isomorphism above is not compatible with ρ k 0 on the nose (since we didn't introduce the appropriate cyclotomic twist), the induced isomorphism ) is actually compatible with ρ k 0 . By Kummer theory, it follows that the isomorphism above restricts to an isomorphism Aut c a (G a K ) ∼ = Aut M a (k M 2.4. The Mod-ℓ geometric lattice. The primary focus of the proof of Theorem D is to show that an element φ ∈ Aut M a (k M 1 (K)) induces an automorphism of a certain graded lattice which is contained in the lattice of subgroups of k M 1 (K). The elements of G * (K|k) are the geometric subgroups of k M 1 (K), as introduced above, and the grading is induced by the so-called Milnor-dimension of subsets of k M 1 (K). Moreover, as a consequence of the construction, it will also follow that such an induced automorphism of G * (K|k) fixes all geometric subgroups which arise from K 0 |k 0 . We denote the collection of all such automorphisms of G * (K|k) by Aut * (G * (K|k)|K 0 ).
We show in Proposition 7.3 that the map K(S) → K(S) (see the notations introduced above) induces an isomorphism of graded lattices G * (K|k) ∼ = G * (K|k), where G * (K|k) is the lattice of relatively-algebraically closed subextensions of K|k graded by transcendence degree. Thus, any element of Aut M a (k M 1 (K)) will define an automorphism of G * (K|k) which fixes subextensions arising from K 0 |k 0 . We then use the results of Evans-Hrushovski [EH91], [EH95] and Gismatullin [Gis08] to show that any such automorphism of G * (K|k) arises from some element of Gal k 0 . See Theorem 7.1, Proposition 7.2 and Corollary 7.4 for more details.
2.5. Generic generators of G * (K|k). The idea of the proof is to "produce" elements of G * (K|k), i.e. geometric subgroups of k M 1 (K), using the "given" data of the mod-ℓ Milnor K-ring k M * (K) endowed with some extra structure which is compatible with all automorphisms in Aut M a (k M 1 (K)), and also to ensure that this process is compatible with such automorphisms. The reconstruction process of G * (K|k) relies on a certain "closure operation" called the Milnor Supremum, which takes place entirely in the ring k M * (K), and which takes in a set of geometric subgroups as an input and returns a geometric subgroup as an output.
The fact that this closure operation produces geometric subgroups follows from some explicit vanishing and non-vanishing results in k M * (K). The vanishing results say that k M * (K) = 0 for * > tr. deg(K|k), and this follows from well-known cohomological dimension calculations of K and the Bloch-Kato conjecture, which is now a theorem of Voevodsky-Rost et. al. [Voe11], [Ros98], [Wei09]. The non-vanishing results say that there are "many" elements of k M 1 (K) which have nontrivial products. The "many" above refers to the fact that these non-vanishing results all involve some open condition on some model of K|k (or some subextension of K|k) which is usually the complement of some branch locus.
2.6. Fixing elements of G * (K|k) which arise from K 0 . The fact that an automorphism σ ∈ Aut M a (k M 1 (K)) is compatible with the tuple a implies that σ fixes the elements of G 1 (K|k) which come from K 0 . Applying the "closure operation" described above shows that σ fixes all of the elements of G * (K|k) which come from K 0 . However, this is still very far from what we need, because at this point there is aboslutely nothing we can "construct/produce" which is moved around by the action of Gal k 0 .
To show this, we introduce the concept of a strongly-general element of K|k, which is related to the concept of a general element from [Pop12c], [Pop12b] but has further assumptions. Another key input comes from the local theory in abelian-by-central birational anabelian geometry for function fields over algebraically closed fields. In this context, the local theory says that σ is compatible with quasi-divisorial valuations. But using the previous step, one can show that σ is actually compatible with divisorial valuations. The literature concerning the local theory in abelian-by-central birational anabelian geometry is quite rich, and it includes the following works among others [BT02], [MMS04], [Pop10], [Pop12a], [Top15a], [Top15c]. See the introduction of [Top15a] for a detailed overview of the history of the local theory.
To show that σ ∈ Aut M a (k M 1 (K)) fixes elements from K × 0 , we first show this for elements of K × 0 which are strongly-general in K|k, and this uses the local theory in an essential way. To deduce that σ fixes all elements arising from K × 0 , we prove a Birational-Bertini type result for strongly-general elements, which shows that there are "sufficiently many" strongly-general elements in higher-dimensional function fields.
2.8. The Base Case. By using our "closure operation" described above, in order to show that σ ∈ Aut M a (k M 1 (K)) induces an automorphism of the lattice G * (K|k), it suffices to show that σ induces a permutation of G 1 (K|k), the set of 1-dimensional geometric subgroups. Using the notation introduced above, a one-dimensional geometric subgroups is a subgroup of k M 1 (K) which of the form K(t) for some t ∈ K × k × . Note that every element t ∈ K = K 0 ⊗ k 0 k can be written as a sum a 0 x 0 + · · · + a r x r for some x i ∈ K 0 and a i ∈ k. The proof now proceeds by induction on the length r of such an expression. The case r = 0 was discussed above, and so the base case for the induction is r = 1.
The base case works as follows. Using the concept of essential ramification, we show that there are "many" elements of the form x 0 + a 1 x 1 with x i ∈ K 0 and a i ∈ k which are "acceptable" with respect to σ. The term "acceptable" means that there exists some t ∈ K such that σ sends the geometric subgroup associated to x 0 + a 1 x 1 to the geometric subgroup associated to t. As before, the term "many" is related to a precise open condition on a certain model of a subextension of K|k, with the condition being related to the essential branch locus.
Once we have "many" acceptable elements of the form x 0 + a 1 x 1 , we use the yoga of "generic generators" mentioned above to show that all pairs (t 0 , t 1 ), with t 0 ∈ K 0 and t 1 = a 0 x 0 + a 1 x 1 , x i ∈ K 0 , a i ∈ k, are acceptable with respect to σ (acceptability is defined similarly for pairs as it was for elements of K). We then take appropriate intersections of certain two-dimensional geometric subgroups to deduce that every element of the form a 0 x 0 + a 1 x 1 is acceptable.
2.9. Inductive Case. To conclude the proof, one proceeds by induction on r as above, with the inductive hypothesis being that every element of K of the form a 0 x 0 + · · · + a s x s with s < r, x i ∈ K 0 , a i ∈ k, is acceptable with respect to σ. The proof is now similar in nature to the proof of the base case. Indeed, first we show that pairs of certain elements are acceptable, then take intersections of certain two-dimensional geometric subgroups to deduce that all all elements of the form above are acceptable.
2.10. Concluding the Proof. The argument outlined above shows that σ ∈ Aut M a (k M 1 (K)) induces a permutation of G 1 (K|k). Since σ is compatible with the "closure operation" described above, it follows that σ induces an automorphism of the lattice G * (K|k). Namely, one obtains a On the other hand, we prove that the map Gal k 0 → Aut * (G * (K|k)|K 0 ) is an isomorphism by using the results of Evans-Hrushovski [EH91], [EH95] and Gismatullin [Gis08], as noted above. Thus, to conclude the proof of Theorem D, it remains to show that the map The argument here again uses the theory of strongly-general elements.
Indeed, any element σ in the kernel of the map above must fix all geometric subgroups of k M 1 (K). First, we show this implies that the restriction of σ to any strongly-general geometric subgroup looks like some element of (Z/ℓ) × · 1. Finally, one uses a Birational-Bertini type argument again to deduce that σ is indeed an element of (Z/ℓ) × · 1 k M 1 (K) . This thereby proves the injectivity of the map above, hence concluding the proof of Theorem D.

Milnor K-Theory
Let F be a field. We recall that the r-th Milnor K-group of F is defined as follows: The tensor product makes K M * (F ) := r≥0 K M r (F ) into a graded-commutative algebra over the ring Z = K M 0 (F ), and we call K M * (F ) the Milnor K-ring of F . It is customary to denote the product of a 1 , . . . , a r ∈ K M 1 (F ) = F × in this ring by {a 1 , . . . , a r }. We will use the standard notation k M r (F ) := K M r (F )/ℓ and call k M r (F ) the r-th mod-ℓ Milnor K-group of F . As with K M * (F ), the tensor product makes k M * (F ) := r≥0 k M r (F ) into a graded commutative algebra over k M 0 (F ) = Z/ℓ, and we call k M * (F ) the mod-ℓ Milnor K-ring of F . Given r elements a 1 , . . . , a r of k M 1 (F ) = F × /ℓ, we will denote their product in k M * (F ) by For b 1 , . . . , b r ∈ K M 1 (F ) = F × , we will abuse the notation and write •} F will also be used to indicate this functoriality. Namely, if b 1 , . . . , b r ∈ F × are given, and F ֒→ L is a field extension, then 3.1. Purely Inseparable Extensions. We will frequently reduce some arguments concerning finite field extensions to the case where the extension is separable. This will be possible because a purely-inseparable extension of fields of characteristic = ℓ induces an isomorphism on the mod-ℓ Milnor K-ring which is also compatible with valuations, as the following two lemmas show.
Lemma 3.1. Let L|F be a finite and purely inseparable extension of fields, such that Char F = ℓ.
Proof. Put p = Char F and assume that p > 0. Since L is finite and purely inseparable over F , one has L ⊂ F 1/p n for sufficiently large n. Since p is invertible in Z/ℓ, the canonical map . But such a subextension M |L is purely inseparable, so the argument above shows that η = 0. Thus Lemma 3.2. Let (L, w)|(F, v) be a finite and purely inseparable extension of valued fields, such that Char F = ℓ. Then the canonical map Proof. Put p = Char F and assume that p > 0. Since L|F is purely inseparable, the index [wL : vF ] is a power of p. Therefore, the canonical map vF/ℓ → wL/ℓ is an isomorphism. Next, note that one has a commutative diagram with exact rows As indicated on the diagram, the middle vertical arrow is an isomorphism by Lemma 3.1, and the right vertical map is an isomorphism as noted above. It follows that the left vertical map is also an isomorphism, as required.
3.2. Tame Symbols. Suppose that (F, v) is a discretely valued field of rank 1, so that one has vF ∼ = Z. We recall that the (r-th) tame-symbol associated to v is a morphism We will primarily use tame symbols to prove that the mod-ℓ Milnor K-ring of a function field contains many non-trivial elements. Most such "non-vanishing" results will essentially follow from the following fact concerning the field of Laurent series.
. We proceed to prove this by induction on d, with the base case d = 0 being trivial. For the inductive case, let v be the t 1 -adic valuation on L ′ . Then the residue field of v can be canonically identified with F ((t d )) · · · ((t 2 )) =: M via the obvious map sending t i ∈ U v to t i ∈ M for i ≥ 2. Finally, applying the tame-symbol associated to v, we obtain

Galois Cohomology
In this section, we recall the basic framework which allows us to translate back and forth between the "Galois" context (i.e. mod-ℓ abelian-by-central Galois groups) and the "Milnor" context (i.e. mod-ℓ Milnor K-rings) by using Kummer theory. This theory is more-or-less well known, as it follows from the fact that H 2 of a pro-ℓ group is "dual" to the relations in a minimal free presentation of the group, while the cup product in H * is "dual" to the commutator [•, •] as defined in §1.4.
The essential calculations concerning cup products and commutators were first carried out by Labute [Lab67] (see also [NSW08, §3.9]). These calculations have seen a recent resurgence of interest in [CEM12], [EM15], [EM11], [Top15b], especially in connection with the Merkurjev-Suslin theorem [MS82] and Bloch-Kato conjecture, which is now a theorem of Voevodsky-Rost et al. [Voe11], [Ros98], [Wei09]. The Merkurjev-Suslin theorem is sufficient for our considerations here. In fact, the discussion in this section can be seen as a summary of [Top15d, §8], appropriately translated to the context of this paper.
First we introduce a bit of notation. We let H * = H * cont denote continuous-cochain groupcohomology. For a pro-ℓ group G, we write H * (G) := H * (G, Z/ℓ), but we will specify the coefficient module if it is different from Z/ℓ. We will denote the decomposable part of H * (G) by H * (G) dec . We will only need to consider H 2 (G) dec , which is defined as the image of the cup-product map Also, recall that the Bockstein morphism, denoted β : H 1 (G) → H 2 (G), is the connecting morphism in the cohomological long exact sequence associated to the short exact sequence of Finally, recall that for a field F , we let G F denote the maximal pro-ℓ quotient of Gal F . Most of the general results we say in this section hold true for general fields F such that Char F = ℓ and µ ℓ ⊂ F . However, in order to simplify the discussion, we will restrict our attention to the function field K|k which is the main focus of the paper.
4.1. Generalities on Cohomology of Pro-ℓ Groups. In this subsection we summarize the basic connection between the group-theoretical structure of a pro-ℓ group G and the mod-ℓ cohomology of its various quotients. We will use the notation introduced in §1.4. First of all, recall that the inflation map induces canonical isomorphisms We will tacitly identify these three cohomology groups. Since H 1 (G a ) = Hom(G a , Z/ℓ), the definition of G (2) implies that G a and H 1 (G a ) are in perfect Z/ℓ-duality. This perfect duality thereby induces a perfect pairing in the usual way. Next, recall that the Lyndon-Hochschild-Serre (LHS) spectral sequence where d 2 is the differential on the E 2 -page of the spectral sequence and ) with its isomorphic image in H 1 (G (2) ) G . Using the discussion and notation introduced above, the following fact summarizes the general group-theoretical properties which we will need.
Fact 4.1. The following hold: (2) The cup product induces a canonical isomorphism ) using the isomorphism from (2) above. Then the map is Z/ℓ-dual (hence Pontryagin-dual) to the map Proof. See [Top15d, Lemma 8.2] for assertion (1). Assertion (2) follows from the Künneth formula along with the fact that G a is isomorphic to a direct power of Z/ℓ; see [Top15d, Fact 8.1] and the surrounding discussion for more details. Assertion (3) is the standard "duality" between the commutator and the cup-product. This "duality" has been well-known for some time (see e.g. [ For an automorphism φ of k M 1 (K), we denote by φ * the automorphism of G a K which is dual to φ via the pairing above. This perfect duality yields a canonical isomorphism of automorphism groups , which is given by mapping φ ∈ Aut(G a K ) to (φ −1 ) * . This isomorphism further induces a canonical isomorphism on (Z/ℓ) × -classes of automorphism groups ). These isomorphisms are actually compatible with the Galois action, once we introduce the appropriate twist. Hence the last isomorphism is compatible with ρ k 0 on the nose. To make things precise, let χ ℓ : Gal k 0 → Z × ℓ denote the ℓ-adic cyclotomic character of the base field k 0 . We can then define the i-th cyclotomic twist ρ k 0 (i) of the representation ρ k 0 : Gal k 0 → Aut(k M 1 (K)) in the obvious way, as ρ k 0 (i)(τ ) = χ ℓ (τ ) i · ρ k 0 (τ ). Next, recall that Kummer theory produces a canonical isomorphism k M 1 (K) = H 1 (G a K , µ ℓ ) = H 1 (G a K ) ⊗ µ ℓ which is compatible with the action of Gal k 0 . It therefore follows that the following diagram commutes K Aut(k M 1 (K)) because we introduced the appropriate cyclotomic twist. But the definition of ρ k 0 ( * ) ensures that the induced isomorphism ) is compatible with ρ k 0 since the cyclotomic twist becomes completely irrelevant after modding out by (Z/ℓ) × . 4.3. Galois vs. Milnor. With this discussion, we may now present the main theorem of this section which allows to pass back and forth between the Galois-setting and the Milnor-setting. The following theorem is essentially a group-theoretical interpretation of the Merkurjev-Suslin theorem [MS82].

20
(2) The canonical isomorphism K : Aut(G a K ) ∼ = − → Aut(k M 1 (K)) restricts to an isomorphism K : ) which is compatible with ρ k 0 . Concerning the proof of Theorem 4.2, the implication (1) ⇒ (2) is clear, while the implication (2) ⇒ (3) follows from the observations about cyclotomic twists made above. It therefore remains to prove assertion (1), which was essentially already proven in [Top15d, Theorem 8.6]; alternatively, one can deduce (1) from the main results of [Top15b]. We give a summary of the proof of Theorem 4.2 below. First, we need to recall some calculations in Galois cohomology. (1) The Bockstein morphism β : is an isomorphism for * = 0, 1, 2.
Proof. Assertion (2) is clear for * = 0, 1, and for * = 2 it follows by considering the LHS spectral sequence associated to the extension Proof of Theorem 4.2. As noted above, it suffices to prove assertion (1) as assertions (2) and (3) follow from this. First, note that the Kummer pairing canonically induces a perfect pairing K . Since k M * (K) is a quadratic Z/ℓ-algebra, in order to prove assertion (1), it suffices to prove that the dual of the inclusion R(G K ) ֒→ ∧ 2 (G a K ) with respect to the pairing above is the (surjective) multiplication map ∧ 2 (k M 1 (K)) ։ k M 2 (K) in the mod-ℓ Milnor K-ring. This follows easily by combining Fact 4.1(3) with the Merkurjev-Suslin Theorem [MS82] and Fact 4.3, as follows.
First, since µ ℓ ⊂ K, we may simplify the notation and choose a fixed isomorphism µ ℓ ∼ = Z/ℓ of Gal K -modules. In particular, we may identify k M 1 (K) with H 1 (G a K ) = H 1 (G K ) via the Kummer pairing, and k M 2 (K) with H 2 (G K ) via the Merkurjev-Suslin Theorem [MS82] and Fact 4.3(2). Next, recall from Fact 4.3 that the Bockstein morphism β : , which is isomorphic to ∧ 2 (k M 1 (K)) by Fact 4.1 and our identification H 1 (G a K ) = k M 1 (K) above. Finally, using the LHS spectral sequence and the fact that H 2 (G K ) dec = H 2 (G K ) by the Merkurjev-Suslin theorem and Fact 4.3(2), we have a canonical short exact sequence by Fact 4.1. Once we identify H 1 (G a K ) with k M 1 (K) and H 2 (G K ) with k M 2 (K), the right-hand map of this sequence is precisely the multiplication map. Finally, the dual of this sequence is precisely by Fact 4.1(3).

The Local Theory
In this section, we recall the required results from the Local Theory in "almost-abelian" anabelian geometry for function fields over algebraically closed fields. All such results have been generally stated for abelian-by-central Galois groups in the literature. But since we work primarily with the "Kummer dual" of this context, i.e. with the mod-ℓ Milnor K-ring, we will need to translate these results to the context of Milnor K-theory via Theorem 4.2.
5.1. Minimized Galois theory and C-pairs. For a field F , we define the ℓ-minimized Galois group of F as g(F ) := Hom(F × , Z/ℓ), endowed with its canonical structure of an abelian pro-ℓ group arising from the point-wise convergence topology. If Char F = ℓ and µ ℓ ⊂ F , then Kummer theory together with a choice of isomorphism of Gal F -modules µ ℓ ∼ = Z/ℓ induces canonically an isomorphism of pro-ℓ groups g(F ) ∼ = G a F . However, the main benefit of working with the minimized context is that it applies also to fields of characteristic ℓ. Since we will need to consider valuations of K which a priori have residue characteristic ℓ, we must work in the minimized context.
A pair of elements σ, τ ∈ g(F ) is called a C-pair provided that for all x ∈ F {0, 1}, one has If F 1 , F 2 are two fields and φ : g(F 1 ) → g(F 2 ) is an isomorphism of pro-ℓ groups, we say that φ is compatible with C-pairs if, for all σ, τ ∈ g(F 1 ), the following are equivalent (1) σ, τ forms a C-pair in g(F 1 ).
(2) φσ, φτ forms a C-pair in g(F 2 ). In the case where Char F = ℓ and µ 2ℓ ⊂ F , it turns out that C-pairs can be determined in a Galois-theoretical manner, as the following fact describes. For a detailed proof of the following fact, see [Top15a, Theorem 12] and [Top15b, Theorem 4].
Fact 5.1. Let F be a field such that Char F = ℓ and µ 2ℓ ⊂ F , and identify g(F ) with G a F via some isomorphism µ ℓ ∼ = Z/ℓ, as above. Let σ, τ ∈ g(F ) = G a F be given. Then the following are equivalent: (1) σ, τ form a C-pair.
(2) One has [σ, τ ] = 0. In particular, if φ is an element of Aut c (G a F ), then φ is compatible with C-pairs. 5.2. Minimized Decomposition Theory. For a valuation v of F , we consider the minimized inertia and decomposition groups of v, which are defined as follows Note that one has I v ⊂ D v ⊂ g(F ), and both I v and D v are closed subgroups of g(F ).
For σ ∈ D v , considered as a homomorphism σ : In particular, the map σ → σ v yields a canonical morphism D v → g(F v). Fact 5.2. Let (F, v) be a valued field, and let w be a valuation of F v. Then the following hold: (1) The canonical map σ → σ v :

22
(2) One has Proof. Assertion (2) is clear, while assertions (1) and (3)  It turns out that the C-pair condition in the minimized decomposition group of a valuation can actually be completely determined from the residue field. We summarize this property in the following fact.
(2) σ v , τ v form a C-pair in g(F v). For r ≤ tr. deg(K|k), we say that a valuation v is an r-quasi-divisorial valuation of K|k if v is the valuation-theoretic composition of r quasi-divisorial valuations as described above.
We will sometimes want to keep track of each term in the composition defining an r-quasidivisorial valuation. In such cases, we will generally consider the flag of quasi-divisorial valuations: v = (v 1 , . . . , v r ) associated to an r-quasi-divisorial valuation v r of K|k. Namely, v i /v i−1 is a quasi-divisorial valuation on Kv i−1 |kv i−1 for all i = 1, . . . , r; here and throughout, we write v 0 for the trivial valuation so that Kv 0 = K. In particular, for all i = 1, . . . , r, the valuation v i is an i-quasi-divisorial valuation of K|k which refines v i−1 .
For any valuation v of K, recall that we write This notation is compatible with the notation/terminology from §5.2, as follows. Under the Kummer pairing G a K × k M 1 (K) → µ ℓ , the subgroup U v resp. U 1 v of k M 1 (K) is precisely the orthogonal to the minimized inertia group I v resp. minimized decomposition group D v of v.
The main results concerning the local theory (in the abelian-by-central setting) state that the minimized inertia and decomposition groups of r-quasi-divisorial valuations are preserved under the action of elements of Aut c (G a K ), and that the partially ordered structure of the associated valuations is preserved as well. We present the following theorem merely as a translation of these results which replaces I v resp. D v with U v resp. U 1 v , and Aut c (G a K ) with Aut M (k M 1 (K)). The fact that elements of Aut c (G a K ) preserve minimized decomposition/inertia groups of quasi-divisorial valuations was first proven by Pop in [Pop12a], which uses the Galois context and the usual inertia/decomposition groups. Nevertheless, the key arguments from loc.cit. can be made to work in the minimized context as well. In any case, in order to keep things precise, we will instead use the reference [Top15c] which uses the minimized context exclusively.

23
(1) Let v be an r-quasi-divisorial valuation of K|k, with r < tr. deg(K|k). Then there exists a unique r-quasi-divisorial valuation v σ of K|k such that (2) Let v be an r-quasi-divisorial valuation of K|k and w an s-quasi-divisorial valuation of K|k, with r, s < tr. deg(K|k), and let v σ , w σ be as in (1) above. Then v is a coarsening of w if and only if v σ is a coarsening of w σ .
Proof. Before we begin the proof, we choose an isomorphism µ ℓ ∼ = Z/ℓ which will be fixed throughout, and which allows us to identify G a K with g(K) by Kummer theory. Proof of (1): Let σ * denote the element of Aut c (G a K ) which is the dual of σ under the Kummer pairing (see Theorem 4.2(1)). Recall that U v resp. U 1 v is the orthogonal of I v resp. D v under this Kummer pairing G a K × k M 1 (K) → µ ℓ ∼ = Z/ℓ. Therefore, it suffices to show that there exists an r-quasi-divisorial valuation v σ such that as the uniqueness of this v σ will be discussed at the end of the proof.
Recall from Theorem 4.2(1) that σ * is compatible with R, hence σ * is compatible with C-pairs by Fact 5.1. The proof now proceeds by induction on r. The base case r = 1 follows immediately from [Top15c, Theorem D] and/or [Pop12a, Theorem 1].
Letting v σ := w ′ • w 1 , it follows from Fact 5.2 that σ * I v = I v σ and σ * D w = D v σ , as required.
Proof of (2): Since vK and wK contain no non-trivial ℓ-divisible convex subgroups, it follows from [Top15c, Lemma 3.1] or [Top15a, Lemma 3.4] that v is a coarsening of w if and only if I v ⊂ I w . By assertion (1), this is true if and only if I v σ ⊂ I w σ , which is similarly equivalent to v σ being a coarsening of w σ . This observation also implies that v σ is uniquely determined by v and by σ.
We will continue to use the notation v σ which was implicitly introduced in Theorem 5.4. In other words, for an r-quasi-divisorial valuation v of K|k with r < tr. deg(K|k), and σ ∈ Aut M (k M 1 (K)), we denote by v σ the unique r-quasi-divisorial valuation of K|k such that Since v σ is compatible with coarsening/refinement of v, we may also use this notation for flags of quasi-divisorial valuations. Namely, if (v 1 , . . . , v r ) is a flag of quasi-divisorial valuations of K|k, then (v σ 1 , . . . , v σ r ) is again a flag of quasi-divisorial valuations of K|k.

Divisorial
Valuations. An r-quasi-divisorial valuation is called r-divisorial provided that the restriction of v to k is trivial. As in the quasi-divisorial setting, we may wish to consider a flag of divisorial valuations v = (v 1 , . . . , v r ) where v i /v i−1 is a divisorial valuation of Kv i−1 |kv i−1 for all i ≥ 1; as before, here v 0 denotes the trivial valuation of K.
It turns out (using the valuative criterion for properness) that any divisorial valuation indeed arises from some Weil-prime-divisor on some normal model of K|k. More generally, any r-divisorial valuation arises from a flag of Weil-prime-divisors on some normal model. See [Pop06,§5] for more details on this.
In general, it is still a major open question to determine which r-quasi-divisorial valuations are actually r-divisorial, using the group-theoretical structure of G c K resp. the ring structure of k M * (K). Nevertheless, in our context, we can use geometric subgroups (as defined in §2) which arise from K 0 to distinguish the r-divisorial valuations among the r-quasi-divisorial ones. See also [Pop12a,Theorem 19] for a related result which uses a similar argument.
Proposition 5.5. Let v be an r-quasi-divisorial valuation of K|k, with r < tr. deg(K|k). Then the following are equivalent: (1) The valuation v is an r-divisorial valuation of K|k.
(2) There exists some t ∈ K 0 k 0 such that the intersection K(t) ∩ U 1 v is finite. Proof. First of all, note that K|K 0 and k|k 0 are both algebraic extensions. Thus, the associated residue extensions, Kv|K 0 v and kv|k 0 v, are also algebraic. Moreover, since k is algebraically closed, the residue field kv is also algebraically closed. In particular, if tr. deg(Kv|kv) ≥ 1, then K 0 v cannot be contained in kv, and therefore there exist (many) elements t in U v ∩ K × 0 whose imaget in Kv is transcendental over kv.
Now assume that v is r-divisorial, hence v is trivial on k. Since r < tr. deg(K|k), it follows that Kv is a function field of transcendence degree ≥ 1 over k. Arguing as above, we see that there exists some element t ∈ (U v ∩ K × 0 ) k 0 such that the imaget of t in Kv is transcendental over k. This implies that K(t) × ⊂ U v and that the image of K(t) in Kv is a function field of transcendence degree 1 over k. Let L ′ denote the image of K(t) in Kv and let L denote the relative algebraic closure of L ′ in Kv. Note that L is a finite extension of L ′ , and that K(t) → L ′ is an isomorphism.
If we consider the canonical map , which is finite by Kummer theory. On the other hand, this kernel is precisely U 1 v ∩ K(t). Conversely, assume that v is not r-divisorial, hence v is non-trivial on k. Since Kv is a function field of transcendence degree ≥ 1 over kv, we can again choose some t ∈ (K × 0 ∩ U v ) k 0 such that the image of t in Kv is transcendental over kv. Now let a ∈ k × be any element such that v(a) < 0. Then one has On the other hand, there are infinitely many such a ∈ k, hence the set is an infinite linearly-independent subset of k M 1 (k(t)). Since the map given by the composition k M 1 (k(t)) → k M 1 (K(t)) ∼ = K(t) has a finite kernel by Kummer theory, it follows that U 1 v ∩ K(t) is infinite, as required.

Milnor K-Theory of Function Fields
In this section, we will prove some important vanishing and non-vanishing results for the mod-ℓ Milnor K-ring of a function field. In particular, we will show that a significant portion of the "algebraic-independence" structure of a function field K over an algebraically closed field k is encoded in the mod-ℓ Milnor K-ring of K.
6.1. Vanishing. We begin by recalling the following fact which follows from some well-known cohomological dimension bounds, combined with the highly celebrated Voevodsky 6.2. Non-Vanishing. The non-vanishing results in mod-ℓ Milnor K-theory of function fields follow the "yoga" that algebraically independent elements should have non-trivial products in Milnor Ktheory. This turns out to be true, but with various exceptions which arise from modding out by ℓ-th powers. Nevertheless, it turns out that these exceptions can be avoided because they are related to some ramification phenomena which is concentrated in codimension one.
Lemma 6.2. Let t 1 , . . . , t r ∈ K be algebraically independent over k. Then there exists a non-empty open subset U of A r k such that, for all (a 1 , . . . , a r ) ∈ U (k), one has {t 1 − a 1 , . . . , t r − a r } K = 0.
Proof. By extending t 1 , . . . , t r to a transcendence base t = (t 1 , . . . , t d ) for K|k, we may assume without loss of generality that r = d = tr. deg(K|k). Let X = A d t denote affine d-space over k with parameters t, so that k(t) is canonically identified with the function field of X. Let K ′ be the maximal separable subextension of K|k(t), and let Y denote the normalization of X in K ′ .
Then Y → X is a finite separable (possibly branched) cover of k-varieties. Since the branch locus of Y → X has codimension one, there exists a non-empty open subset U of X such that Y is unramified over U . Let x ∈ U be a given closed point. Then x corresponds to a k-rational point and in this case (t 1 − a 1 , . . . , t d − a d ) is a system of regular parameters at x. If y is any closed point of Y lying above x, then (t 1 − a 1 , . . . , t d − a d ) is also a system of regular parameters at y, since y is unramified over x. By taking the m y -adic completion at y, we obtain a k-embedding of K ′ into the field of Laurent series k((T 1 , . . . , T d )) =: L which sends t i − a i to T i . But {T 1 , . . . , T d } L = 0 by Fact 3.3, so it follows that Finally, one has {t 1 − a 1 , . . . , t d − a d } K = 0 as well by Lemma 3.1.
The following proposition is our second main "non-vanishing" result, and it will be crucial in describing geometric subgroups of k M 1 (K). This will play a primary role in the reconstruction of the "geometric lattice" in the proof of the main theorems. Proposition 6.3. Let t 1 , . . . , t r ∈ K be algebraically independent over k. Let z ∈ K × be such that {z} K / ∈ K(t 1 , . . . , t r ). Then there exists a non-empty open subset U of A r k such that for all (a 1 , . . . , a r ) ∈ U (k), one has {t 1 − a 1 , . . . , t r − a r , z} K = 0.
Proof. Put d = tr. deg(K|k) and s := d − r. Let t = (t 1 , . . . , t r , y 1 , . . . , y s ) be a transcendence base of K|k extending t 1 , . . . , t r . Let K ′ denote the maximal separable subextension of K|k(t). By Lemma 3.1, there exists some z ′ ∈ K ′ such that {z ′ } K = {z} K . By replacing z with such a z ′ , we may assume without loss of generality that z ∈ K ′ . Let F ′ denote the relative algebraic closure of k(t 1 , . . . , t r ) in K ′ . In particular, note that F ′ is separable over k(t 1 , . . . , t r ) and that F := K(t 1 , . . . , t r ) is purely inseparable over F ′ . Therefore, one has {z} K ′ / ∈ {(F ′ ) × } K ′ by Lemma 3.1. This implies that the field F ′ is relatively algebraically closed also in K ′ ( ℓ √ z) =: L. And since L is separable over k(t), it follows that L is regular over F ′ .
Let B denote the normalization of A r t 1 ,...,tr in F ′ . Moreover, we let X 0 denote A s B;y 1 ,...,ys , affine s-space over B with coordinates (y 1 , . . . , y s ). Finally, let X 1 denote the normalization of X 0 in K ′ , and let X 2 denote the normalization of X 0 in L.
We will now pass to a sufficiently small non-empty open subset V of B which has the six properties listed below. This is possible since each one of these properties holds on a dense open subset of B, as described in each item below.
(1) The fibers of X i → B, i = 0, 1, 2, over points of V are all geometrically integral. This is an open condition on B since k(X i ) is regular over F ′ = k(B), hence X i → B has generically geometrically integral fibers.
(2) Any point of V is unramified over A r t 1 ,...,tr . This is an open condition on B because the extension F ′ |k(t 1 , . . . , t r ) is finite and separable, hence the ramification locus of B → A r t 1 ,...,tr has codimension one.
(3) For all x ∈ V , the function z is regular and non-zero on the generic point of (X 1 ) x , the fiber of X 1 → B over x. This is clearly an open condition on B, since the support of z has codimension one in X 1 , and X 0 is flat over B of relative dimension s. (4) For all x ∈ V , lettingz denote the image of z in k((X 1 ) x ), one has This is an open condition on B because L = K ′ [ ℓ √ z], hence for a sufficiently small non-empty affine open subset of X 1 , say Spec A, one has z ∈ A and the normalization of A in L is precisely x . This is clearly an open condition on B, since the branch locus of X 1 → X 0 is a proper closed subset of X 0 , and X 0 is flat over B. Let U denote the image of V in A r t 1 ,...,tr , and note that U is a non-empty (hence dense) open subset of A r t 1 ,...,tr . Suppose that x ∈ U is a closed point with associated rational point (a 1 , . . . , a r ) ∈ U (k) ⊂ k r , and let y ∈ V be a closed point lying above x. Then (t 1 − a 1 , . . . , t r − a r ) is a system of regular parameters at x. Since y is unramified over x by condition (2) above, the tuple (t 1 − a 1 , . . . , t r − a r ) is also a system of regular parameters at y. Since X 0 = A s B , we see that (t 1 − a 1 , . . . , t r − a r ) is a system of regular parameters for the generic point of the fiber (X 0 ) y . By condition (6), (t 1 − a 1 , . . . , t r − a r ) is also a system of regular parameters for the generic point η y of the fiber (X 1 ) y , which is integral by condition (1). Finally, by (4) and (5), we know that {z} k(ηy) = 0 as an element of k M 1 (k(η y )). Taking the completion with respect to the maximal ideal of O X 1 ,ηy , we obtain an embedding of K ′ into the field of Laurent series M := k(η y )((T 1 , . . . , T r )) which sends t i − a i to T i . Fact 3.3 implies that {T 1 , . . . , T r , z} M = 0, hence Finally, one has {t 1 − a 1 , . . . , t r − a r , z} K = 0 by Lemma 3.1, and this concludes the proof of the proposition.
As a corollary, we obtain our primary criterion for constructing geometric subgroups of k M 1 (K). Since it appears in the statement of the corollary, we briefly note that, given r elements f 1 , . . . , f r of k M 1 (K), one obtains a canonical morphism . The description of geometric subgroups of k M 1 (K) will use the kernels of such morphisms. Corollary 6.4. Let t 1 , . . . , t r ∈ K be algebraically independent over k. Let S 1 , . . . , S r be arbitrary infinite subsets of k. Then one has K(t 1 , . . . , t r ) = (a 1 ,...,ar) where (a 1 , . . . , a r ) varies over the elements of S 1 × · · · × S r .
Proof. It is clear that where (a 1 , . . . , a r ) varies over S 1 × · · · × S r , by Fact 6.1. The converse follows immediately from Proposition 6.3, since the fact that S i are all infinite implies that the set  The following lemma shows how the Milnor-supremum can be used to construct "large" geometric subgroups from smaller geometric subgroups.

28
Lemma 6.6. Let (S i ) i be a collection of subsets of K, and put K i = K(S i ). Moreover, put Then one has sup M (K) = K(S).
Proof. Put d ′ := tr. deg(K(S)|k) and d := dim M (K). First we show that d = d ′ . The inequality d ≤ d ′ follows from Fact 6.1 since K ⊂ K(S). Conversely, we note that there exist s 1 , . . . , s d ′ ∈ S which are algebraically independent over k. Thus, d ≥ d ′ by Lemma 6.2. Since d = tr. deg(K(S)|k) = dim M (K(S)) and K ⊂ K(S), it follows again from Fact 6.1 that Conversely, we again use the fact that there exist t 1 , . . . , t d ∈ S which are algebraically independent over k. Then the inclusion follows from Corollary 6.4.

Geometric Lattices
A graded lattice is a Z ≥0 -graded set L * = i≥0 L i endowed with a partial ordering ≤, such that the following two conditions hold: (1) Every subset S of L * has a greatest lower bound ∧S and a least upper bound ∨S in L * .
Namely, the partially ordered set (L * , ≤) is a complete lattice.
(2) If A ∈ L s and B ∈ L t are such that A < B, then one has s < t. Namely, the partial ordering ≤ is strictly compatible with the grading of L * .
An isomorphism between two graded lattices L * 1 and L * 2 is a graded bijection which respects the partial ordering. We will denote the automorphism group of a graded lattice L * by Aut * (L * ).
7.1. Relative Algebraic Closure. Assume now that F is a perfect field and that L is an extension of finite transcendence degree over F , such that F is relatively algebraically closed in L. We denote by G * (L|F ) the graded lattice of relatively algebraically closed subextensions of L|F , graded by transcendence degree. More precisely, we let G s (L|F ) denote the collection of relatively algebraically closed subextensions of L|F of transcendence degree s over F . Then one has G * (L|F ) = s≥0 G s (L|F ), and the ordering on G * (L|F ) is the partial ordering given by inclusion of subextensions of L|F .
Note that one has a canonical isomorphism of graded lattices defined by M → M i , with inverse given by M i → M i ∩L. In particular, this canonical isomorphism induces an isomorphism of automorphism groups: . 29 We will denote by Aut {F } (L i ) the set of automorphisms φ of L i which satisfy φF = F . Note that any element of Aut {F } (L i ) induces an automorphism of G * (L i |F ) in the obvious way. Thus we obtain a canonical homomorphism The work of Evans-Hrushovski [EH91] [EH95] and Gismatullin [Gis08] considers the surjectivity/injectivity of this map. We summarize their main results in the following theorem.
Theorem 7.1 (see [Gis08,Theorem 2.4]). Let F be a perfect field and let L be an extension of finite transcendence degree over F , such that F is relatively algebraically closed in L. Assume furthermore that tr. deg(L|F ) ≥ 5. Then the canonical map is surjective, and its kernel is the subgroup generated by Frob L i .
Proof. The only part of this theorem which doesn't follow directly from [Gis08, Theorem 2.4] is that loc.cit. uses the combinatorial geometry associated to G * (L|F ) as its basic structure whereas we use the whole lattice G * (L|F ). These two formulations are easily seen to be equivalent, as follows.
First, note that the combinatorial geometry G(L|F ) associated to L|F is defined to be the set G 1 (L|F ) with a closure operation cl on subsets S ⊂ G 1 (L|F ). This closure operation is easily interpretable in the lattice G * (L|F ) as Conversely, given the set G 1 (L|F ) endowed with the closure operation cl defined above, it is easy to see that G * (L|F ) is canonically isomorphic to the lattice of flats associated to the combinatorial geometry (G 1 (L|F ), cl). In other words, the lattice G * (L|F ) can be identified with the lattice of closed subsets of G 1 (L|F ) with respect to the closure operation cl described above, and the grading of this lattice is given by the combinatorial dimension of closed subsets.
With these observations, it is easy to see that restricting φ ∈ Aut * (G * (L|F )) to G 1 (L|F ) induces an isomorphism of automorphism groups φ → φ| G 1 (L|F ) : Aut * (G * (L|F )) ∼ = Aut cl (G 1 (L|F )) where Aut cl (G 1 (L|F )) denotes the set of permutations of G 1 (L|F ) which are compatible with the closure operation cl. This isomorphism of automorphism groups is clearly compatible with the canonical morphisms originating from Aut {F } (L i ). Hence, the assertion of the theorem follows immediately from [Gis08, Theorem 4.2]. 7.2. The Galois Action on the Geometric Lattice. We now consider the canonical Galois action of Gal k 0 on G * (K|k). First, recall that Gal k 0 can be canonically identified with the Galois group Gal(K|K 0 ), since K 0 |k 0 is a regular function field. In particular, Gal k 0 acts on the (relatively algebraically closed) subextensions of K|k in the obvious way, and this action preserves transcendence degree. In other words, Gal k 0 acts on the graded lattice G * (K|k).
Using the notation introduced above, we have and G r (K|k) consists of all those K(S) such that tr. deg(K(S)|k) = r. We will consider the subgroup Aut * (G * (K|k)|K 0 ) of Aut * (G * (K|k)) consisting of all automorphisms Φ ∈ Aut * (G * (K|k)) such that ΦK(S) = K(S) for all subsets S ⊂ K 0 . We can also describe Aut * (G * (K|k)|K 0 ) as a "Galois Group," as follows. First, observe that one has a canonical injective map G * (K 0 |k 0 ) → G * (K|k) 30 which sends F 0 ∈ G * (K 0 |k 0 ) to F := K(F 0 ). Therefore, we see that Aut * (G * (K|k)|K 0 ) is the Galois group of G * (K|k) over G * (K 0 |k 0 ), i.e. the set of automorphisms of G * (K|k) which fix G * (K 0 |k 0 ) pointwise. Note that Gal k 0 fixes K(S) setwise (as a subset of K) for any subset S ⊂ K 0 . Therefore, the action of Gal k 0 on G * (K|k) induces a canonical Galois representation We now use Theorem 7.1 to prove that this map is an isomorphism.
Proposition 7.2. Assume that tr. deg(K|k) ≥ 5. Then the canonical map is an isomorphism.
Proof. We first use Theorem 7.1 to make the following observation: Any automorphism Φ of G * (K|k) arises from an automorphismΦ of K i which is unique up-to composition with some power of Frob K i . Assume moreover that Φ restricts to an automorphism Φ 0 of G * (K 0 |k 0 ). Then Theorem 7.1 implies that Φ 0 arises from an automorphismΦ 0 of K i 0 which is unique up-to composition with some power of Frob K i 0 . It therefore follows from the functoriality of the situation that this automorphismΦ restricts to an automorphism of K i 0 , which is preciselyΦ 0 , up-to composition with a power of Frob K i 0 .

Consider the following subgroup of Aut
. By Theorem 7.1 and the observations made above, we see that the canonical map is surjective, and its kernel is the subgroup generated by Frob K i .
On the other hand, it is easy to see that one has a canonical isomorphism Finally, we conclude the proof by recalling that the canonical map ρ k 0 : Gal k 0 → Gal(K|K 0 ) is an isomorphism, since K 0 |k 0 is a regular function field. As the isomorphisms are all compatible with ρ k 0 , the assertion follows.
7.3. The Mod-ℓ Geometric Lattice. As a first step towards the mod-ℓ context, we will prove that there is an isomorphic copy of G * (K|k) inside of the lattice of subgroups of k M 1 (K). We write G * (K|k) for the collection of geometric subgroups of k M 1 (K), ordered by inclusion in k M 1 (K), and graded by dim M . Namely, as a partially-ordered set, one has where the partial-ordering is given by inclusion of subgroups of k M 1 (K). The grading of is defined by the condition that G s (K|K) consists of all geometric subgroups A of k M 1 (K) such that dim M (A) = s. The next proposition shows that we can identify G * (K|k) with G * (K|k), and that the Milnor-supremum defined in §6.4 corresponds to the least-upper-bound in G * (K|k), via this identification.
Proposition 7.3. The following hold: (1) The set G * (K|k), endowed with the ordering by inclusion in k M 1 (K) and the grading by dim M , is a graded lattice.
(3) For a collection (K i ) i of elements of G * (K|k), the least-upper-bound of (K i ) i in the lattice G * (K|k) is precisely the Milnor-supremum sup M ( i K i ).
Proof. Note that the map K(S) → K(S) yields a canonical surjective map G * (K|k) → G * (K|k). By Fact 6.5, this surjective map restricts to a surjective map on the graded pieces, and it is clear from the definition that this map respects the ordering. Since G * (K|k) is a graded lattice, in order to prove (1) and (2) it suffices to prove that this canonical map is injective.
With this in mind, suppose that S, T are two subsets of K. Assume for a contradiction that K(S) = K(T ) and K(S) = K(T ). Then Fact 6.5 implies that tr. deg(K(S)|k) = tr. deg(K(T )|k) =: d. But as K(S) and K(T ) are relatively algebraically closed in K, the assumption that K(S) = K(T ) implies that tr. deg(K(S ∪ T )|k) > d.
Thus, there exist t 0 , . . . , t d ∈ S ∪ T which are algebraically independent over k. By Lemma 6.2, there exist a 0 , . . . , a d ∈ k such that However, since K(S) = K(T ) =: K, we have {t i − a i } K ∈ K. But then (7.1) provides a contradiction to the fact that dim M (K) = d.
This proves that the map G * (K|k) → G * (K|k) is injective, hence bijective, which thereby proves assertions (1) and (2). Assertion (3) follows immediately from assertion (2) and Lemma 6.6. 7.4. Galois Action on the Mod-ℓ Lattice. By Proposition 7.3(1)(2), the partially ordered set G * (K|k) is a graded lattice which is canonically isomorphic to G * (K|k). Moreover, the definition of this isomorphism immediately shows that it is compatible with the action of Gal k 0 . In any case, since G * (K|k) is a graded lattice, we can define Aut * (G * (K|k)|K 0 ) similarly to the way we defined Aut * (G * (K|k)|K 0 ). Namely, Aut * (G * (K|k)|K 0 ) consists of all automorphisms Φ of G * (K|k) (as a graded-lattice) such that ΦK(S) = K(S) for all S ⊂ K 0 . As before, one has a canonical Galois representation On the other hand, Proposition 7.3(2) immediately implies that we have a canonical isomorphism of automorphism groups: and it is easy to see that this isomorphism is compatible with ρ k 0 . Thus, by combining Propositions 7.2 and 7.3, we immediately deduce the following corollary.
Corollary 7.4. Assume that tr. deg(K|k) ≥ 5. Then the canonical map is an isomorphism.

Essentially Unramified Points and Fibers
A key difficulty which arises by working in the mod-ℓ context is that the presence of ramification can make certain valuations "invisible." More precisely, suppose that t is a non-constant element of K. While every divisorial valuation of K(t)|k is the restriction of some divisorial valuation of K|k, there may be some divisorial valuations v of K(t)|k such that {U v } K is not of the form U w ∩ K(t) for any divisorial valuation w of K|k. Dualizing using the Kummer pairing, there may be some minimized inertia subgroups of G a K(t) arising from divisorial valuations of K(t)|k which are not the image of any minimized inertia subgroup of G a K that arise from a divisorial valuation of K|k. This "difficulty" is clearly intimately tied to ramification (specifically, ramification indices which are divisible by ℓ). In this section, we introduce some general conditions which will suffice to prevent such problems.
8.1. The flag associated to regular parameters. We will use a basic construction in algebraic geometry which associates an r-divisorial valuation and/or a flag of divisorial valuations of length r to a system of regular parameters at a point of codimension r in a regular k-variety. More precisely, let X be a regular k-variety, and let x be a (possibly non-closed) regular point of X. Let O X,x be the regular local ring at x. For a subset S ⊂ O X,x , we use the usual notation to denote the closed subscheme of Spec O X,x associated to the ideal (S) of O X,x .
Let (f 1 , . . . , f r ) be a system of regular parameters for the (maximal ideal of the) regular local ring O X,x , and put L := k(X). We will abuse the terminology and also say that (f 1 , . . . , f r ) is a system of regular parameters at x in X.
In any case, to the system (f 1 , . . . , f r ) of regular parameters of O X,x , we associate a flag (v 1 , . . . , v r ) of divisorial valuations of L|k, by letting v i /v i−1 be the divisorial valuation of f 1 , . . . , f i−1 )) associated to the prime Weil-divisor V (f 1 , . . . , f i ) on V (f 1 , . . . , f i−1 ). As before, v 0 stands for the trivial valuation of L by convention. The following is a summary of the basic facts concerning the relationship between (f 1 , . . . , f r ) and (v 1 , . . . , v r ).
Fact 8.1. In the context above, the following hold: (1) For all i = 1, . . . , r, one has Moreover, one has f i ∈ U v i−1 and its image in (2) For all i = 1, . . . , r, one has 8.2. ℓ-unramified prolongations. Suppose now that k(X) = L is a subextension of K|k, and (v 1 , . . . , v r ) is a flag of divisorial valuations of L|k. Let (w 1 , . . . , w r ) be a flag of divisorial valuations of K|k which prolongs (v 1 , . . . , v r ); i.e. w i is a prolongation of v i for i = 1, . . . , r. Note that this condition ensures that the canonical map k M 1 (L) → k M 1 (K) restricts to a morphism {U v i } L → U w i for all i = 1, . . . , r. We will say that (w 1 , . . . , w r ) is an ℓ-unramified prolongation of (v 1 , . . . , v r ), provided that the canonical map k M 1 (L) → k M 1 (K) induces isomorphisms . . , r (recall that v 0 resp. w 0 denote the trivial valuations).
The notion of an ℓ-unramified prolongation and Fact 8.1 lead to the following particularly useful observation. If the flag (v 1 , . . . , v r ) arises from a system of regular parameters (f 1 , . . . , f r ) at a regular point x ∈ X as described in §8.1, and (w 1 , . . . , w r ) is an ℓ-unramified prolongation of (v 1 , . . . , v r ) to K, then {f i } K is a generator of U w i−1 /U w i ∼ = Z/ℓ for all i = 1, . . . , r by Fact 8.1(2). 33 8.3. The Essential Branch Locus. Let X be a regular k-variety, and suppose that K is a finite extension of k(X). Let K ′ denote the maximal separable subextension of K|k(X), and let Y → X denote the normalization of X in K ′ . Thus, Y → X is a finite separable (possibly branched) cover of k-varieties. The branch locus of this finite separable cover Y → X will be called the essential branch locus of X in K. Recall that the essential branch locus of X in K is actually a divisor of X. However, we will only be interested in the support of this divisor. Namely, we will consider the essential branch locus of X in K only as a closed subvariety of X which has codimension one.
Along the same lines, if x is a (scheme-theoretic) point of X, we say that x is essentially unramified in K if x is not contained in the essential branch locus of X in K. Similarly, if Z is an integral closed subvariety of X, we say that Z is essentially unramified in K if the generic point of Z is essentially unramified in K. Since the essential branch locus of X in K is closed in X, we note that Z is essentially unramified in K if and only if Z is not contained in the essential branch locus of X in K.
The concept of essentially unramified points and the essential branch locus allows us to produce "many" ℓ-unramified prolongations of flags of divisorial valuations, as follows. In the context above, suppose that x is an essentially unramified regular point of X, and let y ∈ Y be a point in the fiber of x. Moreover, let (f 1 , . . . , f r ) be a system of regular parameters at x, and let v := (v 1 , . . . , v r ) be the flag of divisorial valuations of k(X)|k associated to (f 1 , . . . , f r ). Since y is unramified over x, it follows that (f 1 , . . . , f r ) is a system of regular parameters at y as well, and we let w ′ := (w ′ 1 , . . . , w ′ r ) be the flag of divisorial valuations of k(Y ) = K ′ associated to (f 1 , . . . , f r ). Note that the canonical . . , r (recall that v 0 resp. w ′ 0 denote the trivial valuations). Finally, let w = (w 1 , . . . , w r ) be any prolongation of w ′ to K. Then w is a flag of divisorial valuations, and it follows from Lemma 3.2 that w is an ℓ-unramified prolongation of v as defined in §8.2. We summarize this discussion for later use in the following fact.
Fact 8.2. Suppose that X is a regular k-variety and that K is a finite extension of k(X). Let K ′ denote the maximal separable subextension of K|k(X), and let Y denote the normalization of X in K ′ . Let x be a regular point of X which is essentially unramified in K. Let (f 1 , . . . , f r ) be a system of regular parameters at x with associated flag v of divisorial valuations, and let y be any point of Y in the fiber above x. Then the following hold: (1) The system (f 1 , . . . , f r ) is a system of regular parameters at y.
(2) Any prolongation w to K of the flag of divisorial valuations of k(Y ) = K ′ associated to (f 1 , . . . , f r ), considered as a system of regular parameters at y, is an ℓ-unramified prolongation of v.
8.4. Essentially Unramified Fibers. We will primarily use Fact 8.2 in the case where x ∈ X is the generic point of a fiber of some smooth morphism. More precisely, suppose that X → S is a dominant smooth morphism of regular k-varieties with geometrically integral fibers, and let s ∈ S be a closed point in the image of X → S. Let η s ∈ X be the generic point of the fiber of X → S over s. With the setup above, if (f 1 , . . . , f r ) is a system of regular parameters at s ∈ S, then (f 1 , . . . , f r ) is also a system of regular parameters at η s ∈ X. Thus, if K is a finite extension of k(X) and η s is essentially unramified in K, then we may apply Fact 8.2 to η s ∈ X endowed with a system of regular parameters (f 1 , . . . , f r ), which arises from s ∈ S. 8.5. Mod-ℓ Divisors. In this subsection, we will use the concept of essentially unramified fibers, as discussed in §8.4, to compare the divisorial valuations on a field of the form K(t) with the divisorial valuations which can be detected in the mod-ℓ setting from K.
For t ∈ K k, we denote by D t the set of divisorial valuations of K(t)|k. Since K(t) is a onedimensional function field over k, the elements of D t are in canonical bijection with the closed points of the unique complete normal model of K(t)|k. As for the mod-ℓ analogue of D t , we define D t to be the set where v varies over the divisorial valuations of K|k. Note that one has K(t)/U v ∼ = Z for all v ∈ D t , and that K(t)/V ∼ = Z/ℓ for all V ∈ D t . Our primary goal in this section is to compare D t and D t . First, we show that D t can be embedded canonically in D t .
Lemma 8.3. Let t ∈ K k be given. For every V ∈ D t , there exists a unique v ∈ D t such that Proof. Let w be a divisorial valuation of K|k such that V = U w ∩ K(t). Since K(t) ⊂ U w , we deduce that w is non-trivial on K(t), hence w| K(t) is a divisorial valuation on K(t). Denote this divisorial valuation of K(t) by v. Then clearly one has Concerning the uniqueness of v, suppose that v ′ is another divisorial valuation of K(t) such that In particular, v and v ′ must be dependent as valuations of K(t), for otherwise U v · U v ′ = K(t) × by the approximation theorem for independent valuations. Since both v and v ′ have value groups isomorphic to Z, it follows that v = v ′ . This proves the uniqueness of v, as required.
A primary difficuly which arises in the mod-ℓ case is that the canonical map D t ֒→ D t described in Lemma 8.3 need not be surjective in general. Nevertheless, we can use the theory of essentiallyunramified fibers to give a sufficient condition for an element of D t to arise from D t . Although we will not need it later, we note that an argument similar to the proof of Lemma 8.4 below shows that all but finitely many of the elements of D t arise from some element of D t via this injection.
Lemma 8.4. Let t ∈ K k be given and let S be the unique proper normal model of K(t)|k. Let v ∈ D t be given, and let s ∈ S be the (unique) center of v. Assume that there exists a regular k-variety X and a smooth dominant morphism X → S with geometrically integral fibers, such that the following hold: (1) K is a finite extension of k(X).
(2) s is in the image of X → S.
(3) The fiber X s of X → S over s is essentially unramified in K. Then there exists some V ∈ D t such that V = {U v } K , i.e. v is in the image of the canonical injective map D t ֒→ D t from Lemma 8.3.
Proof. Let π ∈ K(t) be a uniformizer for v, and let s ∈ S be the (unique) center of v on S. Furthermore, let η ∈ X be the generic point of the fiber of X → S over s. Following the discussion of §8.4 with the morphism X → S, it follows that η is a regular point of codimension one, and that π is a local parameter at η. Let w 0 be the divisorial valuation of k(X) associated to π at η. By Fact 8.2, there exists an ℓ-unramified prolongation w of w 0 to K|k. The lemma follows by taking V = U w ∩ K(t) and noting that the image of {π} K is a generator of k M 1 (K)/U w ∼ = Z/ℓ.

Strongly General Elements and Birational Bertini
In this section, we recall the notion of a general element of a regular function field. We also introduce the notion of a strongly-general element which is related to the notion of a general element, but has further assumptions which are motivated by the discussion of §8. The primary goal of this section is to prove so-called Birational Bertini results for both general and strongly-general elements, which will show that there are "many" such elements in higher-dimensional function fields.
9.1. General Elements. Let L|F be a regular field extension and let x ∈ L F be given. We say that x is separable in L|F if x / ∈ F · L p where p = Char F . If Char F = 0, then every element of L is separable in L|F by convention. We say that x is general in L|F provided that L is a regular extension of F (x). In particular, if x is general in L|F then F (x) is relatively algebraically closed in L.
The following lemma is our so-called Birational Bertini result for general elements. The first assertion of this lemma can be found in [Lan72,Ch. VIII,pg. 213]. The second assertion of this lemma has essentially the same proof as in loc.cit., but since it hasn't explicitly appeared in the literature, we provide a detailed proof below.
Lemma 9.1. Let F be an infinite field, and let L be a regular function field over F . Let x, y ∈ L be algebraically independent over F with y separable in L|F . Then the following hold: (1) For all but finitely many a ∈ F , the element x + ay is general in L|F .
(2) There exists a non-empty open subset U of A 2 F such that for all (a, b) ∈ U (F ), the element Proof. As mentioned above, we only need to prove assertion (2) since assertion (1) can be found in [Lan72,Ch. VIII,pg. 213]. To simplify the notation, for P = (a, b) ∈ F 2 , we write Since y is separable in L|F , there exists an F -derivation D of L such that D(y) = 0. Now we may calculate: Hence D(t a,b ) = 0 if and only if t a,b = D(x)/D(y), so this can only happen for at most one pair P 0 ∈ F 2 . Thus, for any pair (a, b) different from this (possible) exceptional one P 0 , the element t a,b is separable in L|F , which implies that L|F (t a,b ) has a separating transcendence base. For each P ∈ F 2 , write E P := F (t P ) and E ′ P for the relative algebraic closure of E P in L. If P, Q are two different points of F 2 , note that one has E P · E Q = F (x, y). Note that E ′ P and E ′ Q are regular of transcendence degree 1 over F , since they are contained in L which is regular over F . This implies that E ′ P and E ′ Q are linearly disjoint over F , since L|F is regular. Let M denote the relative algebraic closure of F (x, y) in L, and note that there are at most finitely many intermediate subextensions of M |F (x, y). In particular, there are finitely many subextensions of M |F (x, y) of the form E ′ P (x, y). Let P 1 , . . . , P n be finitely many points in F 2 such that E ′ P i (x, y) exhaust all such subextensions. Now suppose that Q is any point of F 2 which is different from P 0 , . . . , P n . Since Q = P 0 , we see that E ′ Q is the separable algebraic closure of E Q in M . Moreover, by the discussion above, E ′ Q (x, y) must be linearly disjoint from E ′ P i (x, y) over F (x, y), for all i = 1, . . . , n. This forces E ′ Q (x, y) to be precisely F (x, y), and therefore E ′ Q = E Q . In other words, E Q = F (t Q ) is algebraically closed in L, and since Q = P 0 , we see that L is regular over E Q = F (t Q ), as required. 36 9.2. Strongly-General Elements. Let t ∈ K be a non-constant element. We say that t is a strongly-general element in K|k if the following two conditions hold: (1) The element t is general in K|k.
(2) The canonical injective map D t → D t from Lemma 8.3 is surjective (hence bijective).
In the spirit of Lemma 9.1, we now prove the following Birational Bertini result for strongly-general elements of K|k.
Note that if (t − ab)/(u − b) is strongly-general in K|k with a, b ∈ k, then so is Thus, using Proposition 9.2, the fact that k 0 is infinite, and the assumption that k 0 ⊂ L, we can find a, b ∈ k 0 such that both are strongly-general in K|k. But then we see that is a product of two elements of L which are strongly-general in K|k. This concludes the proof of the lemma.
9.3. Rational-Like Collections. Let t ∈ K k be given. A particularly useful consequence of Lemma 8.3 is that every element x ∈ K(t) is contained in all but finitely many of the V ∈ D t . In particular, the projection maps K(t) ։ K(t)/V for V ∈ D t together induce a canonical morphism we thereby obtain a map div Φ : . As the notation div Φ suggests, this morphism should be considered as a "mod-ℓ" analogue of the usual divisor map div : albeit with respect to the collection of isomorphisms Φ. Suppose now that t is strongly general in K|k, so that the canonical injective map D t → D t given by Lemma 8.3 is actually a bijection. In this context, we say that a collection of isomorphisms Φ = (Φ V : K(t)/V ∼ = − → Z/ℓ) V∈Dt as above is a rational-like collection provided that the induced map fits into a short exact sequence For every strongly-general element t ∈ K k, there is a canonical rational-like collection Ψ = (Ψ V ) V∈Dt associated to the field K(t) = k(t), which is defined as follows. For each V ∈ D t 38 with associated element v ∈ D t (i.e. V = {U v } K ), the isomorphism Ψ V is the unique one making the following diagram commute: The following Lemma shows that any rational collection is obtained from the canonical one by multiplying by some element of (Z/ℓ) × .
Lemma 9.4. Let t ∈ K k be strongly general in K|k, and let Ψ = (Ψ V ) V∈Dt be the canonical rational-like collection associated to K(t) = k(t). Also, let (Φ V ) V∈Dt be another rational-like collection. Then there exists a unique ǫ ∈ (Z/ℓ) × , such that Proof. For each V ∈ D t , there exists some ǫ V ∈ (Z/ℓ) × such that We must show that ǫ V is independent of V. So let V i , i = 1, 2 be two elements of D t , and put . But the fact that Φ is a rational-like collection implies that ǫ 1 = ǫ 2 , as required.
We now use the notions of strongly-general elements and rational-like collections to prove a proposition which will be useful in several steps of the proof of Theorem D.
Proof. The implication (1) ⇒ (2) is trivial. The proof of the non-trivial direction (2) ⇒ (1) has three main steps: (1) First, we show that for all t ∈ K k and for all V ∈ D t , one has σV = V.
(2) Second, we show that for all t which is strongly-general in K|k, the restriction of σ to K := K(t) is of the form ǫ K · 1 K , for some ǫ K ∈ (Z/ℓ) × , which a priori might depend on K.
(3) Lastly, we show that ǫ K from step (2) doesn't actually depend on K, and then conclude the proof of the proposition by using Lemma 9.3.
Step (1): Let v be a divisorial valuation of K|k. Since Kv is a function field of transcendence degree ≥ 1 over k, it is easy to see that U v is multiplicatively generated by elements x ∈ U v whose image in Kv is transcendental over k. For all such x ∈ U v , one has K(x) × ⊂ U v . Thus, U v is generated by subgroups of the form K(x) such that K(x) ⊂ U v . Since σK(x) = K(x) for all x ∈ K k, it follows that σU v = U v . For any t ∈ K k, it follows that σV = V for all V ∈ D t by the definition of the elements of D t .
Step (2): Let t be strongly-general in K|k and put K = K(t). Consider the canonical rational-like collection Ψ = (Ψ V ) V∈Dt associated to K(t) = k(t). Since σV = V for all V ∈ D t , we obtain an induced rational-like collection Φ = (Φ V ) V∈Dt , where Φ V := Ψ V • σ. By Lemma 9.4, there exists some ǫ K ∈ (Z/ℓ) × depending only on K and σ, such that Φ V = ǫ K · Ψ V for all V ∈ D t . In other words, for all x ∈ K, one has The injectivity of div Ψ in the definition of Ψ being a rational-like collection implies that σ| K = ǫ K ·1 K .
Step (3): Let t 1 , t 2 be strongly-general in K|k, and put K i := K(t i ) and ǫ i := ǫ K i for i = 1, 2. If t 1 , t 2 are algebraically dependent over k, then K(t 1 ) = K(t 2 ), so that ǫ 1 = ǫ 2 . Assume, on the other hand, that t 1 , t 2 are algebraically independent over k. Since t i is general in K|k, we see that {{t i − a} K : a ∈ k} is a linearly-independent subset of k M 1 (K). Thus, by Proposition 9.2, we may choose a, b ∈ k such that {t 1 − a} K and {t 2 − b} K are Z/ℓ-independent in k M 1 (K) and such that t 0 : We can now calculate: Since {t 1 − a} K and {t 2 − b} K are linearly independent in k M 1 (K), it follows that ǫ 1 = ǫ 2 = ǫ 0 . This proves that ǫ K doesn't depend on K = K(t) for t strongly-general in K|k.
Letting ǫ = ǫ K for some (hence any) K = K(t) with t strongly-general, we deduce that σ| K = ǫ·1| K for all K = K(t) with t strongly-general. By Lemma 9.3, we see that k M 1 (K) is generated by its subgroups of the form K(t) for t strongly-general in K|k. Hence σ = ǫ · 1 k M 1 (K) , as required. 9.4. Faithfulness of the Galois Action. We conclude this section by proving that the Galois action of Gal k 0 on the mod-ℓ Milnor K-ring of K is faithful. Although there are many ways to prove this fact, we can use geometric subgroups and the Birational-Bertini results to prove this for function fields of dimension ≥ 2. The result also holds for function fields of dimension 1, but a different argument is needed in that case.
Proof. First if K = k(t) with t ∈ K 0 , then the claim is clear simply because the set {{t − a} k(t) : a ∈ k} is linearly independent in k M 1 (k(t)), and since the action of τ ∈ Gal k 0 on {t − a} k(t) satisfies τ {t − a} k(t) = {t − τ a} k(t) .
Next, if K has transcendence degree ≥ 2, we note that the action of τ ∈ Gal k 0 on k M 1 (K) restricts to an automorphism on any geometric subgroup of the form K(S) for S ⊂ K 0 . By Lemma 9.1, there exists some t ∈ K 0 which is general in K|k. In this case, the map k M 1 (k(t)) → k M 1 (K) is injective with image K(t). This injection is compatible with the action of Gal k 0 , so the assertion follows from the argument above.
Finally, suppose that tr. deg(K|k) = 1 and suppose that τ is in the kernel of the canonical map ρ k 0 : Gal k 0 → Aut M (k M 1 (K)). Note that in order to show τ = 1, it suffices to prove that τ a = a for all but finitely many a ∈ k.
We now choose some t ∈ K 0 such that K is finite and separable over k(t). Let C denote the complete normal model of K|k, and consider the finite (possibly branched) separable cover C → P 1 Now let a ∈ k be given such that the point t = a in P 1 t is unramified in the cover C → P 1 t , and let v be a divisorial valuation of K|k whose center on C lies above this point. By our assumption on τ , it follows that In particular, we see that {t − τ a} K / ∈ U v , which directly implies that τ a = a. Since this condition holds for all but finitely many a ∈ k, we deduce that τ = 1, as required.

The Main Proof
We now turn to the proof of Theorems C and D, which is the main focus of this paper. The primary focus will be on Theorem D since we have been primarily be working with mod-ℓ Milnor K-theory, while Theorem C will follow by applying Theorem 4.2.
Using the notation from Theorem D, recall that a = (a 1 , . . . , a r ) is an arbitrary (possibly empty) finite tuple of elements of k × 0 . We start off the proof by working with a fixed element τ ∈ Aut M a (k M 1 (K)), although we will eventually replace τ by another element σ ∈ Aut M a (k M 1 (K)) of the form ǫ · τ for some ǫ ∈ (Z/ℓ) × . In particular, σ and τ represent the same element of Aut M a (k M 1 (K)), but this σ will have some further special properties which we will need. In any case, if A is any subgroup of k M 1 (K) and τ, σ are as above, then one has σA = τ A. Since the primary goal of the proof is to show that τ induces an automorphism of the lattice of geometric subgroups of k M 1 (K), this observation shows that it doesn't actually matter if we replace τ with σ = ǫ · τ . We now fix this initial element τ ∈ Aut M a (k M 1 (K)). Recall that the definition of Aut M a (k M 1 (K)) says that τ is an automorphism of k M 1 (K) which satisfies the following two properties: (1) τ extends to an automorphism of k M * (K).
(2) For all t ∈ K 0 k 0 , τ restricts to an automorphism of the subgroup The following fact summarizes the formulation of these two conditions which will form the starting point of our proof.
10.1. Acceptable Subsets. As discussed above, the primary goal of the proof is to show that the action of τ on the subgroups of k M 1 (K) induces an automorphism of the lattice G * (K|k) of geometric subgroups. With this in mind, we say that a subset S ⊂ K is τ -acceptable if there exists a subset T of K such that τ K(S) = K(T ). Thus, the primary goal of the proof is to show that every subset of K is τ -acceptable, and we will then conclude the proof by applying Corollary 7.4. The following fact follows immediately from Lemma 6.6 and Fact 10.1(1), and it essentially reduces our primary goal to showing that every element of K is acceptable.
Fact 10.2. Suppose that (S i ) i is a collection of τ -acceptable subsets of K, and for each i let T i be a subset of K such that τ K(S i ) = K(T i ).

41
Put S = i S i and T = i T i . Then one has τ K(S) = K(T ). In particular, S is τ -acceptable.
10.2. Fixing K 0 . We begin by showing that every subset of K 0 is τ -acceptable. In fact, since our goal will be to apply Corollary 7.4, we must show the stronger property that τ K(S) = K(S) for all subsets S ⊂ K 0 . This assertion is the starting point of the proof and is accomplished in the following lemma.
Lemma 10.3. Let S ⊂ K 0 be a subset. Then one has τ K(S) = K(S). In particular, every subset of K 0 is τ -acceptable.
Proof. By Fact 10.2, it suffices to prove that τ K(t) = K(t) for all t ∈ K 0 k 0 . By Corollary 6.4, one has By Fact 10.1(1), we see that On the other hand, Fact 10.1(2) implies that τ {t − c} K ∈ K(t) for all c ∈ k 0 . Moreover, for all x ∈ K(t), one has K(t) ⊂ ker{x, •} K by Fact 6.1. In particular, we deduce that K(t) ⊂ τ K(t). Repeating this argument with τ −1 ∈ Aut M a (k M 1 (K)) in place of τ , we deduce that K(t) = τ K(t), as required.
Although Lemma 10.3 will be used in the final steps of the proof, we will need a stronger variant of this result. Namely, we will need to prove that there exists a σ of the form σ = ǫ · τ for some ǫ ∈ (Z/ℓ) × such that σ{t} K = {t} K for all t ∈ K × 0 . This stronger variant appears in Proposition 10.6 below, and proving this proposition is the main goal of this subsection. Naturally, this property can be seen as a crude approximation to our goal of proving that τ arises from some element of Gal k 0 .
Let (v 1 , . . . , v r ) be a flag of divisorial valuations of K|k of length r < tr. deg(K|k). Since τ is an element of Aut M (k M 1 (K)), we recall from §5.3 (specifically Theorem 5.4) that there exists a unique flag (v τ 1 , . . . , v τ r ) of quasi-divisorial valuations of K|k which satisfies for all i = 1, . . . , r. We now use Lemma 10.3 to prove that this induced flag actually consists of divisorial valuations.
Lemma 10.4. Let (v 1 , . . . , v r ) be a flag of divisorial valuations of K|k, with r < tr. deg(K|k). Then (v τ 1 , . . . , v τ r ) is a flag of divisorial valuations of K|k. Proof. By Theorem 5.4, the valuation v τ i is the i-quasi-divisorial valuation of K|k which is uniquely determined by the fact that On the other hand, by Proposition 5.5, there exists some t ∈ K 0 k 0 such that K(t) ∩ U 1 v i is finite, since v i is i-divisorial. But then by Lemma 10.3, we see that is finite as well. Hence v τ i is an i-divisorial valuation by Proposition 5.5. To simplify the exposition for the rest of the proof, we will introduce some notation to label the elements of D t and D t for a strongly-general element t. If we fix an element t ∈ K k which is strongly-general in K|k, then the canonical map D t → D t defined in Lemma 8.3 is a bijection. Since K(t) = k(t) (as t is, in particular, general), the elements of D t are in bijection with the closed points of P 1 t . Thus D t is also parametrized by the closed points of P 1 t , via the bijection D t → D t . By fixing the parameter t, we can label the closed points of P 1 t as k ∪ {∞} in the usual way. Namely, for c ∈ k, the corresponding point of P 1 t is the closed point given by the equation t − c = 0, and the point associated to ∞ is the closed point given by 1/t = 0. We will denote the element of D t associated to c ∈ k ∪ {∞} by V[t; c], and we will denote the element of D t associated to c by v[t; c]. Using this notation, we recall from Lemma 8.3 that for all c ∈ k ∪ {∞}, one has as subgroups of k M 1 (K), and that one has k M 1 (K)/V[t; c] ∼ = Z/ℓ for all c ∈ k ∪ {∞}. A change of the parameter t by some fractional-linear transformation yields a corresponding change in the associated element of k ∪ {∞} = P 1 (k). More precisely, if we let GL 2 (k) act on P 1 (k) and on the generators of k(t)|k by fractional-linear transformations, as usual, and if M ∈ GL 2 (k) is given, then for all c ∈ P 1 (k), one has as subgroups of k M 1 (K). For an element x of K(t), we define the support of x in D t as This is completely analogous to the usual notion of the support of a function f ∈ k(t) × in P 1 (k) = D t . In particular, using the notation introduced above, note that one has The following lemma shows that τ fixes subgroups of the form V[t; c] which arise from K 0 , i.e. such that t ∈ K 0 k 0 and c ∈ P 1 (k 0 ). This is a key step towards proving Proposition 10.6 below. The proof of this lemma essentially follows by considering the D t -supports of various elements associated to t and using Fact 10.1(2).
Lemma 10.5. Let t ∈ K 0 k 0 be strongly general in K|k, and let c ∈ P 1 (k 0 ) be given. Then one has τ V[t; c] = V[t; c].
Proof. By replacing t with another element of K 0 which generates K(t) = k(t) over k, it suffices to prove that τ V[t; ∞] = V[t; ∞]. By Fact 10.1(2), we know that By combining Lemmas 10.3 and 10.4 with the definition of D t , it follows that for all V ∈ D t , one has τ V ∈ D t as well. In other words, V → τ V can be considered as a permutation of D t .
Note that the D t -support of any element of Finally, for c ∈ k 0 , one has V[t − c; 0] = V[t; c]. Repeating the argument above with t − c for c ∈ k 0 , we deduce that As mentioned above, the following proposition is the primary goal of this subsection, and it can be seen as the first major step towards the proof of Theorem D.
(2) For all t ∈ K 0 which is strongly general in K|k and for all b ∈ k, there exists a unique c ∈ k such that τ {t − b} K = ǫ · {t − c} K .
Proof. Extend t 1 , t 2 , t 3 to a transcendence base t = (t 1 , . . . , t r ) for K|F , and let K ′ denote the maximal separable subextension of K|F (t). Note that K ′ is then a regular extension of F . By Lemma 9.1(1), for all but finitely many a ∈ k, the field K ′ is regular over F (t 1 + a · t 3 ); since k 0 is infinite, we may choose such an a which lies in k 0 . Applying Lemma 9.1(1) again, for all but finitely many b ∈ k, the field K ′ is regular over F (t 1 + a · t 3 , t 2 + b · t 3 ); again, we may choose such a b which lies in k 0 . We put x = t 1 + a · t 3 and y = t 2 + b · t 3 for the a, b ∈ k 0 as above. Next, put F 1 = F (x), F 2 = F (y) and F 12 = F (x, y), and note that F * is relatively algebraically closed in K ′ for * = 1, 2, 12. Also, for * = 1, 2, 12, let M * denote the relative algebraic closure of F * in K. Since K|K ′ is purely-inseparable, the extension M * |F * is also purely-inseparable.
Since F 12 = F (x, y) is rational over F and F 1 = F (x), F 2 = F (y), it is easy to see that one has {F × 1 } F 12 ∩ {F × 2 } F 12 = {F × } F 12 as subgroups of k M 1 (F 12 ). Since k M 1 (F 12 ) → k M 1 (K ′ ) is injective, we deduce that Finally, the fact that follows easily from Lemma 3.1 using the fact that the extensions M * |F * , * = 1, 2, 12, and K|K ′ are purely-inseparable. This concludes the proof of the lemma since K(t, x) = {M × 1 } K and K(t, y) = {M × 2 } K .
Lemma 10.9. Let K be a subgroup of k M 1 (K) which is maximal among the subgroups ∆ of k M 1 (K) such that dim M (∆) = 1. Assume furthermore that there exist elements of G * (K|k) denoted as follows 1 , B ′ 2 , D ∈ G 3 (K|k) such that the following conditions hold: (1) B 1 ∪ B 2 ⊂ D, B 1 = B 2 , and A ⊂ D.
Proof. First suppose that t ∈ K k is given and consider the geometric subgroup K(t). Then for all a, b ∈ K(t), one has {a, b} K = 0 by Fact 6.1. Moreover, by Proposition 6.3, if c ∈ k M 1 (K) K(t), then there exist (many) elements d ∈ K(t) such that {c, d} K = 0. In particular, K(t) is maximal among subgroups ∆ of k M 1 (K) such that dim M (∆) = 1. Now suppose that K satisfies the assumptions on the lemma. The goal of this proof will be to show that there exists some t ∈ K k such that K(t) ⊂ K. Then the "maximality" in the observation above would imply that K(t) = K. We will tacitly use Proposition 7.3(1), which says that for E 1 , E 2 ∈ G * (K|k), one has E 1 ⊂ E 2 if and only if K(E 1 ) ⊂ K(E 2 ) as subgroups of k M 1 (K). Let F 1 , F 2 ∈ G 2 (K|k) be such that K(F i ) = B i and put F = F 1 ∩ F 2 . Then condition (1) implies that F 1 = F 2 hence tr. deg(F |k) ≤ 1. Let F 12 ∈ G 3 (K|k) be such that K(F 12 ) = D, then condition (1) implies that F 1 · F 2 ⊂ F 12 . Let x ∈ K k be such that A = K(x). Condition (1) implies that x is transcendental over F 12 hence it is also transcendental over F 1 and F 2 . Thus, one has B ′ i = K(F i , x) by condition (2). On the other hand, x being transcendental over F 12 , and F 1 · F 2 ⊂ F 12 , implies that On the other hand, letting M ∈ G 2 (K|k) be such that K(M ) = C, we deduce from condition (3) that M ⊂ K(F 1 , x) and M ⊂ K(F 2 , x), hence tr. deg(K(F, x)|k) ≥ 2. Since tr. deg(F |k) ≤ 1 holds, we deduce that tr. deg(K(F, x)|k) = 2 and therefore tr. deg(F |k) = 1. Finally, one has K(F ) = K(F 1 ∩ F 2 ) ⊂ K(F 1 ) ∩ K(F 2 ) = B 1 ∩ B 2 . Therefore, K(F ) ⊂ K by condition (3). Letting t ∈ K k be such that K(t) = F , we deduce that K(t) ⊂ K, hence K = K(t) as noted in the beginning of the proof.
10.4. The Base Case. Recall that our primary goal is to show that every element of K is σacceptable. Also, recall that one has K = K 0 ⊗ k 0 k, and therefore every element t of K can be written as t = a 0 x 0 + · · · + a r x r for some x 0 , . . . , x r ∈ K 0 and some a 0 , . . . , a r ∈ k. We have already proved in Lemma 10.3 that elements of K 0 are σ-acceptable, hence elements of the form a 0 x 0 where a 0 ∈ k and x 0 ∈ K 0 are also σ-acceptable. The proof that every element of K is σ-acceptable follows by induction on the length r of the expression a 0 x 0 + · · · + a r x r above. The base case for our induction is the case r = 1, which is the focus of this subsection.
We begin by proving that "many" elements of the form a 0 x 0 + a 1 x 1 are σ-acceptable, and we will then use the "intersection" results proved in the previous subsection to deduce the full base case.
Lemma 10.10. Let t 1 , t 2 , t 3 ∈ K 0 be algebraically independent over k. Then there exists a nonempty open subset U of A 3 Now suppose that e ∈ k 0 is given and put t e = (x − e)/y. Consider the (rational) projection P 2 x,y → P 1 te defined by the inclusion of function fields k(t e ) ֒→ k(x, y). The fibers of this morphism are the lines in P 2 x,y passing through the point (1 : e : 0), and so this map is surjective onto P 1 te . Letting Y e denote the preimage of the point (1 : e : 0) in X, and putting X e := X Y e , we find that the composition X e ֒→ X → P 2 x,y → P 1 te is a (regular) smooth surjective morphism with geometrically integral fibers which are all essentially unramified in K.
In particular, if t e = (x − e)/y is general in K|k, then we see that t e is automatically stronglygeneral in K|k by applying Lemma 8.4 to the morphism X e → P 1 te . Moreover, as 1/y is separable in K|k, it follows from Lemma 9.1(1) that t e is general (hence strongly-general) for all but finitely many e ∈ k 0 . Now let b ∈ k be arbitrary as in the statement of the lemma. By Proposition 10.6(2), for every e ∈ k 0 as above (i.e. such that t e is strongly-general), there exists a unique c (which a priori might depend on e) such that σ x Fortunately, as noted above, we may directly use Proposition 10.7 which implies that this c doesn't depend on the given e ∈ k 0 such that t e is strongly-general. In other words, there exists a single c ∈ k such that, for all e ∈ k 0 as above, one has Finally, by Lemma 10.6(1), we may multiply both sides by σ{y} K = {y} K to deduce that σ{x − e + b · y} K = {x − e + c · y} K .
We conclude that σK(x + b · y) = K(x + c · y) by Corollary 6.4, since the above equality holds true for all but finitely many elements e of k 0 .
The following lemma concludes the base case for our induction.
Lemma 10.11. Assume that tr. deg(K|k) ≥ 5 and let x 0 , y 0 ∈ K 0 be given. Then for all d ∈ k, the element x 0 + d · y 0 is σ-acceptable.
Since z ∈ K 0 is σ-acceptable by Lemma 10. Moreover, recall that if S is σ-acceptable and T is a subset such that σK(S) = K(T ), then one has tr. deg(K(S)|k) = dim M (K(S)) = dim M (K(T )) = tr. deg(K(T )|k) by Fact 6.5 and Fact 10.1. It follows from these observations and Proposition 7.3 that σK(t) satisfies the assumptions of Lemma 10.9, as follows. Indeed, recall that K(t) is maximal among subgroups ∆ of k M 1 (K) such that dim M (∆) = 1 by Fact 6.1 and Proposition 6.3. Thus, σK(t) is also maximal with this property by Fact 10.1(1). Finally, in the notation of Lemma 10.9, we can take (1) A = σK(t 3 ) and C = σK(t, t 3 ).
10.5. The General Case. We are now ready to prove the final main step in our proof, that every element t of K is σ-acceptable. As noted above, this will proceed by induction on the length of the expression t = a 0 x 0 + · · · + a r x r , a i ∈ k, x i ∈ K 0 , with the base case r = 1 taken care of by Lemma 10.11.
Proof. Recall that every element of K is of the form a 0 x 0 + · · · + a r x r for a i ∈ k and x i ∈ K 0 . We proceed by induction on r, with the case r = 0 being Lemma 10.3 and the case r = 1 being Lemma 10.11. So assume that r is fixed and that all elements of K which can be written as a 0 x 0 + · · · + a s x s , a i ∈ k, x i ∈ K 0 , with s < r, are σ-acceptable. Let t ∈ K be an element of the form t = a 0 x 0 + · · · + a r x r with a i ∈ k and x i ∈ K 0 . As a first reduction, divide by a 0 to assume without loss of generality that a 0 = 1, so that t = x 0 + a 1 x 1 + · · · + a r x r . Also, we may assume that a r ∈ k k 0 , for otherwise x 0 + a r x r ∈ K 0 , hence t is σ-acceptable by the inductive hypothesis. We may further assume without loss of generality that x r and t are algebraically independent, for otherwise K(t) = K(x r ) and so t is σ-acceptable by Lemma 10.3.
Choose t 1 , t 2 ∈ K 0 which are algebraically independent over K(t, x r ). By Lemma 10.8, we can choose a, b ∈ k 0 such that K(t) = K(t, at 1 + a r x r ) ∩ K(t, bt 1 + a r x r ).
Put x = at 1 + a r x r and y = bt 1 + a r x r . Then t − x, t − y, t − a r x r , x, y and x r are all σ-acceptable by the inductive hypothesis, and so any subset of {t − x, t − y, t − a r x r , x, y, x r } is also τ -acceptable by Fact 10.2. Now we apply Lemma 10.9 to deduce the claim, as follows. First, σK(t) is maximal among subgroups ∆ of k M 1 (K) such that dim M (∆) = 1, arguing as in the proof of Lemma 10.11. Also, we may consider the following geometric subgroups of k M 1 (K) following the notation of Lemma 10.9:
(4) B ′ 1 = σK(t, x, x r ) = σK(t − x, x, x r ) and B ′ 2 = σK(t, y, x r ) = σK(t − y, y, x r ). By Proposition 7.3 and the observations made above, we see that these subgroups of k M 1 (K) satisfy the assumptions of Lemma 10.9. It therefore follows from Lemma 10.9 that σK(t) ∈ G 1 (K|k) is geometric, hence t is σ-acceptable, as required.
10.6. Concluding the proofs of Theorems C and D. For the rest of this section, we assume that tr. deg(K|k) ≥ 5 so that we can use Corollary 7.4 and Lemma 10.12.
By Lemma 10.12, any element of K is σ-acceptable. Thus any subset of K is σ-acceptable by Fact 10.2. Moreover, by Fact 6.5 and Fact 10.1(1), if S, T are two subsets of K such that σK(S) = K(T ), then one has tr. deg(K(S)|k) = tr. deg(K(T )|k). In particular, the map A → σA induces an automorphism of the graded lattice G * (K|k) of geometric subgroups of k M 1 (K). In other words, we obtain a canonical homomorphism σ → (K → σK) : Aut M a (k M 1 (K)) → Aut * (G * (K|k)) which factors through Aut M a (k M 1 (K)). Furthermore, by Lemma 10.3, it follows that σK(S) = K(S) for subsets S of K 0 . In other words, the image of this canonical map Aut M a (k M 1 (K)) → Aut * (G * (K|k)) actually lands in the subgroup Aut * (G * (K|k)|K 0 ) which was defined in §7.4.
Finally, it is easy to see that the map Aut M a (k M 1 (K)) → Aut * (G * (K|k)|K 0 ) is compatible with ρ k 0 . Namely, the following diagram commutes: ' ' P P P P P P P P P P P P Aut M a (k M 1 (K)) Aut * (G * (K|k)|K 0 ) By Corollary 7.4, the map ρ k 0 : Gal k 0 → Aut * (G * (K|k)|K 0 ) is an isomorphism. On the other hand, it immediately follows from Proposition 9.5 that the map Aut M a (k M 1 (K)) → Aut * (G * (K|k)|K 0 ) ⊂ Aut * (G * (K|k)) is injective. Hence the map ρ k 0 : Gal k 0 → Aut M a (k M 1 (K)) is an isomorphism as well. This concludes the proof of Theorem D.
Finally, it easily follows from Kummer theory that the isomorphism Aut c (G a K ) ∼ = Aut M (k M 1 (K)) of Theorem 4.2 restricts to an isomorphism Aut c a (G a K ) ∼ = Aut M a (k M 1 (K)). Thus, Theorem C follows immediately from Theorem D by applying Theorem 4.2.

Concluding the Proof of the mod-ℓ I/OM
We now turn to the proof of Theorems A and B. As we will see, Theorem A follows rather easily from Theorem C. On the other hand, Theorem B follows from Theorem A more-or-less because of our definition of a 5-connected subcategory of Var k 0 . 11.1. Proof of Theorem A. Let X be a normal k 0 -variety of dimension ≥ 5, and let U be a birational system of X. Put K = k(X). First, since X is geometrically normal, we recall that for every U ∈ U , one has canonical surjective morphisms G c K ։ π c (U ), G a K ։ π a (U ).