Para-abelian varieties and Albanese maps

We construct for every proper algebraic space over a ground field an Albanese map to a para-abelian variety, which is unique up to unique isomorphism. This holds in the absence of rational points or ample sheaves, and also for reducible or non-reduced spaces, under the mere assumption that the structure morphism is in Stein factorization. It also works under suitable assumptions in families. In fact the treatment of the relative setting is crucial, even to understand the situation over ground fields. This also ensures that Albanese maps are equivariant with respect to actions of group schemes. Our approach depends on the notion of families of para-abelian varieties, where each geometric fiber admits the structure of an abelian variety, and representability of tau-parts in relative Picard groups, together with structure results on algebraic groups.


Introduction
The Albanese variety and the Albanese map are truly fundamental constructions in algebraic geometry.The terms were coined by André Weil ([53], page 438 and commentary on page 570, compare also [32], page 328), in reference to work of Giacomo Albanese on zero-cycles and correspondences for surfaces ( [2] and [3]).Originally, it was a purely transcendental construction, depending on path integrals over closed holomorphic one-forms.A general formulation for compact complex spaces X was given by Blanchard [7]: The Albanese variety is the complex torus V * /∆, constructed with the dual of the vector space V ⊂ H 0 (X, Ω 1 X/C ) of closed forms, and the Lie group closure ∆ of the subgroup given by integrals over loops.The resulting Albanese map is defined by integrals over paths, and depends on the choice of a base point x 0 .For compact connected Riemann surfaces, this simplifies to J = H 0 (X, Ω 1 X ) * /H 1 (X, Z).Together with the principal polarization stemming from the intersection form this is called the jacobian variety.For genus g ≥ 1 the Albanese map is a closed embedding X ⊂ J, and for genus g ≥ 2 one obtains an embedding of the Deligne-Mumford stacks M g ⊂ A g , which comprises families of curves and principally polarized abelian varieties, of genus and dimension g, respectively.
It is natural to ask for algebraic constructions of Albanese varieties and Albanese maps, say for projective varieties, that extend to general proper schemes X over arbitrary ground fields k, and beyond.It quickly became clear that such a generalization is possible: Matsusaka [36] and Serre [49] constructed the Albanese map in the classical language of algebraic varieties, by regarding them as universal maps to abelian varieties, compare also the discussion of Esnault, Srinivas and Viehweg [16].Grothendieck used Picard schemes and Poincaré sheaves to obtain Albanese maps, as outlined in [19], and [20], Section 3. The representability of the Picard functor was extended to arbitrary proper schemes by Murre [39].
In the absence of rational points, however, notorious complications arise and Albanese maps take values in principal homogeneous spaces rather than abelian varieties.These problems become even more pronounced over imperfect fields of characteristic p > 0: To our best knowledge, the existence of Albanese map, and its base-change behavior, is only established for geometrically integral proper schemes, or more generally for geometrically reduced and geometrically connected proper schemes, compare the discussions of Conrad [11] and Wittenberg [54].Also see Achter, Casalaina-Martin and Vial [1] for non-proper X.
The goal of this paper is to settle such issues, by systematically working in the relative setting over some base scheme S, and also to use algebraic spaces rather than schemes.Recall that algebraic spaces are important generalization of schemes introduced by Michael Artin.Roughly speaking, they take over the role of Moishezon spaces from complex geometry.However, their definition is much more indirect: algebraic spaces are contravariant functors X : (Aff/S) → (Set) that satisfy the sheaf axiom with respect to the étale topology, and are also otherwise closely related to schemes.Notions like the underlying topological space |X| or the structure sheaf O X still exist but lose much of their immediate significance, in comparison to the schematic situation.
Let X be an algebraic space over S such that the structure morphism is proper, flat, of finite presentation and cohomologically flat in degree d = 0, with h 0 (O Xs ) = 1 for all points s ∈ S. In particular, the structure morphism X → S is in Stein factorization.Our main result is: Theorem.(see Thm. 10.2) Assumptions as above.If Pic τ X/S admits a family of maximal abelian subvarieties, then there is an Albanese map f : X → Alb X/S .Moreover, it is universal for morphisms into families of para-abelian varieties, equivariant with respect to actions of group spaces, and commutes with base-change.
If the base scheme S is integral, and the generic fiber of Pic τ X/S → S is proper, all assumptions are satisfied at least on some dense open set U , over which the Albanese map thus exists.If S is the spectrum of a field k we get for each proper algebraic space X with h 0 (O X ) = 1 unconditional results.Note that X may be reducible, non-reduced, or non-schematic, and the ground field k is arbitrary.It would be possible to develop the theory where the base S itself is an algebraic space rather then a scheme, but for the sake of exposition we refrain from doing so.
To establish the above result we develop a theory of para-abelian varieties, which is of independent interest.Apparently, the name was coined by Grothendieck ([20], Theorem 3.3), but did not gain widespread use.Again it is crucial to work in the relative setting: A family of para-abelian varieties is a smooth proper morphism P → S such that for each point s ∈ S, there is a field extension k of the residue field κ(s) such that P ⊗ k admits the structure of an abelian variety.Note that our definition does not involve any a priori torsor structure.However, we show in Section 5 that a certain inertia subsheaf in Aut P/S is a family of abelian varieties, which a posteriori yields a torsor structure.
The second main ingredient is a systematic study for the tau-part Pic τ X/S of the relative Picard functor.Roughly speaking, it parameterizes invertible sheaves that are fiberwise numerically trivial.We provide a new point of view, by directly using Artin's representability criteria [4].Here the representation is via algebraic spaces rather than schemes, even if X is schematic, which highlights that the correct framework for our goals is given by algebraic spaces.As already pointed out by Grothendieck [20], the tau-parts are much better behaved than the connected components Pic 0 Xs/κ(s) for the fiberwise Picard groups: The latter form an abelian sheaf that is usually not representable by algebraic spaces, and we give in Proposition 2.2 an explicit example with families of Enriques surfaces.
An important technical issue are the separation properties of the ensuing quotient N = Num X/S of the Picard space by its tau-part, which we call the numerical sheaf.These are algebraic spaces endowed with a group law whose fibers are étale group schemes.In general, they are neither schematic nor separated, but at least locally separated, which means that the diagonal monomorphism N → N × N is an embedding.This very weak separation axiom is enough to carry out some crucial constructions that lead to Albanese maps.
An important insight of this paper is to re-define the notion of Albanese maps for algebraic spaces X over S: Here it denotes pairs (P, f ) where P is an S-family of para-abelian varieties, and f : X → P is an S-morphism such that for each point s ∈ S, the induced homomorphism f * identifies the abelian variety Pic τ Ps/κ(s) with the maximal abelian subvariety for the group scheme Pic τ Xs/κ(s) .In some sense, this definition is rather close to the original approach with path integrals, and its basechange properties follow very easily.We establish a posteriori that such Albanese maps have a universal property, which ensures uniqueness and equivariance.
In this approach, the maximal abelian subvariety plays a crucial role.Extending some results of Brion [10] on the structure of group schemes G of finite type over a base field k, we introduce a three-step filtration G 3 ⊂ G 2 ⊂ G 1 and describe the maximal abelian subvariety G ab ⊂ G in terms of the filtration and extensions of abelian varieties by multiplicative groups, which appears to be of independent interest.
The paper is organized as follows: In Section 1 we briefly review algebraic spaces and local separatedness, and give an existence result for quotients in a relative setting.In Section 2 we thoroughly study the tau-part Pic τ X/S for the relative Picard functor, and collect criteria for separatedness, properness, flatness and smoothness.This leads to the numerical sheaf Num X/S , which is studied in Section 3. In Section 4 we develop a theory of para-abelian varieties and their families.Using actions of automorphism group schemes, we uncover in Section 5 the relation between families of para-abelian varieties and families of abelian varieties.In Section 6 and 7 we work over ground fields k and obtain general results on the structure of group schemes and the maximal abelian subvariety.We then combine our results so far and state the definition of Albanese maps in Section 8.Here we already establish some partial results concerning its uniqueness and existence.To proceed, we have to analyze the notion of Poincaré sheaves in Section 9.In the final Section 10, we obtain existence and uniqueness for Albanese maps and Albanese varieties in full generality.In the Appendix, we collect some facts concerning embeddings of algebraic spaces.
Acknowledgement.We wish to thank Jeff Achter, Brian Conrad, Laurent Moret-Bailly, Olivier Wittenberg and the referees for valuable comments that helped to improve the paper and correct mistakes.This research was conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology, which is funded by the Deutsche Forschungsgemeinschaft.

Algebraic spaces
Throughout the paper, S usually denotes a base scheme.We write (Aff/S) for the category of affine schemes T = Spec(R) endowed with a structure morphism T → S. Recall that an algebraic space is a contravariant functor X : (Aff/S) → (Set) satisfying the sheaf axiom with respect to the étale topology, such that the diagonal X → X × X is relatively representable by schemes, and that there is an étale surjection U → X from some scheme U .These are important generalizations of schemes, because modifications, quotients, families, or moduli spaces of schemes are frequently algebraic spaces rather than schemes.We refer to the monographs of Olsson [41], Laumon and Moret-Bailly [34], Artin [5], Knutson [33], and to the stacks project [51], Part 0ELT.
Note that there is an important novel separation axiom for algebraic spaces, namely local separatedness, which means that the diagonal X → X ×X is an embedding (compare the Appendix for more details).This is automatic for schemes, but may fail for algebraic spaces (see the end of this section for examples).Throughout we use the term embedding in the sense of [21], Definition 4.2.1,where the word immersion is used instead.Note that it is not necessarily closed, open, or quasicompact.Also note that some authors impose an additional condition, besides representability, on diagonals for algebraic spaces, but we make no separation assumptions whatsoever.
By definition, an algebraic space X : (Aff/S) → (Set) satisfies the sheaf axiom with respect to the étale topology.According to [51], Lemma 076M it then also satisfies the sheaf axiom with respect to the fppf topology.Throughout, we shall regard (Aff/S) as a site with this Grothendieck topology, if not stated otherwise.It is obtained by the procedure explained in [13], Exposé IV, Section 6 from the families (U i → T ) i∈I , where U i → T are open embeddings, the index set I is arbitrary, and U i → T is surjective, together with the families (V j → T ) j∈J , where the index set J is finite, the V j → T are flat and of finite presentation, and V j → T is surjective.Note that by dropping the assumption that T and U i belong to (Aff/S), the procedure allows to pass to the larger category (Sch/S).
Let G be an algebraic space such that the G(R), with T = Spec(R) from (Aff/S), are endowed with functorial group structures.For simplicity, we say that G is an algebraic space with group structure.Let X be an algebraic space endowed with a Gaction.One may form the quotient X/G as a sheaf on (Aff/S).Note that since the action is from the left, it would be more appropriate to write G\X.This common inconsistency should not cause confusion, because in most of our applications G will be commutative.
We want to understand when the quotient X/G is representable by an algebraic space.Let us say that the action µ : is a monomorphism.In other words, the groups G(R) act freely on the sets X(R), for all T = Spec(R).Note that this deviates from other terminology that is sometimes used, where (µ, pr 2 ) is assumed to be a closed embedding (for example [38], Definition 0.8).For convenience of the reader, we state the following well-known result: Lemma 1.1.Suppose the structure morphism G → S is flat and locally of finite presentation, and that the action on X is free.Then the sheaf quotient X/G is representable by an algebraic space.Moreover, the quotient map X → X/G is flat and locally of finite presentation, and the formation of X/G commutes with basechange.
Proof.To verify this we first consider the quotient as a stack [X/G] over (Aff/S).The fiber categories over T = Spec(R) comprise torsors T with respect to G T , together with equivariant morphisms T → X T .To proceed we check that [X/G] is an algebraic stack.By assumption on G, the projection pr 2 : G × X → X is flat and locally of finite presentation.Obviously, (σ, x) → (σ, σx) defines an automorphism of G × X, so µ : G × X → X is also flat and locally of finite presentation.Since the action is free, the morphism (µ, pr 2 ) : G × X → X × X is a monomorphism.Now [51], Theorem 06FI ensures that the stack [X/G] is algebraic.All its objects have trivial automorphism groups, because the G-action on X is free, and it follows from [34], Corollary 8.1.1 or [51], Theorem 04S6 that [X/G] is actually an algebraic space.By construction, it represents the quotient sheaf X/G.
Obviously, the formation of the quotient stack [X/G] commutes with base-change.Moreover, the quotient map X → X/G is a torsor with respect to G X/G , because the analogous statement holds for the base-change along X → X/G, and it follows that it is flat and locally of finite presentation.Since G → S is flat and locally of finite presentation, the same holds for X → X/G by descent.
In the situation of the Lemma, the quotient morphism q : The vertical map on the right is a torsor with respect to (G × G) Y ×Y , hence flat and locally of finite presentation.Let us close this section with two examples of quotients that are not locally separated, and in particular not schematic: First suppose that S = Spec(k) is the spectrum of a field, that H is a group scheme of finite type, and Λ ⊂ H(k) is an infinite subgroup.Regard the latter as a constant group scheme G = σ∈Λ Spec(k).The canonical homomorphism G → H is a monomorphism, hence the translation action is free, and the quotient X = H/G exists as an algebraic space.It is not locally separated, because the monomorphism G → H is not an embedding.Note that the section S → X coming from the neutral section in H is not an embedding, because this does not hold for the base-change G → H, so Corollary 11.3 below also ensures that X is not locally separated.
The second example lives over the spectrum S = Spec(R) of a discrete valuation ring.Write U for the open set comprising the generic point, and Z for the complementary closed set.Suppose H is a relative group scheme, and N is a closed subgroup scheme.Assume that the structure morphisms of H → S and N → S are fppf.Assume that N U ⊂ H U is also open, and that N Z ⊂ H Z is bijective.Then the disjoint union G = N (H U N U ) is a group scheme, and the resulting G → H is a monomorphism.Obviously, the translation action of G on H is free, and the structure morphism G → S is fppf.It follows that the quotient X = H/G exists as an algebraic space.Now suppose there are a, b ∈ H(S) such that ab −1 does not lie in G, but ab −1 |Z is contained in N (Z).Then the restrictions of a, b to both Z and U are congruent modulo G. Now suppose that the diagonal ∆ : X → X × X would be an embedding.Then the cartesian diagram defines a bijective embedding Z ⊂ S, whence Z = S.This implies that ab −1 ∈ G, contradiction.To make the situation concrete, one may choose H = µ p,R and N = {e} R over the ring R = Z p [e 2πi/p ].

Pic and Pic-tau for families
Let S be a base scheme, X be an algebraic space, and suppose that the structure morphism X → S is flat, proper, of finite presentation, and cohomologically flat in degree d = 0.The latter ensures that the direct image of the structure sheaf O X is locally free on S, and its formation commutes with base-change.According to [4], Theorem 7.3 the sheafification of R → Pic(X ⊗ R) with respect to the fppf topology is representable by an algebraic space Pic X/S , which is locally of finite presentation, quasiseparated and locally separated.By abuse of notation we here write R → Pic(X ⊗ R) instead of the contravariant functor T → Pic(X × S T ).Note that in Artin's formulation of Theorem 7.3, local separatedness is not stated explicitly because only algebraic spaces satisfying this separation axiom were considered.Also note that Pic X/S frequently fails to be schematic, and this already happens for certain families of curves over discrete valuation rings ( [20], Section 0).
If S is the spectrum of a field k, the component of the origin Pic 0 X/k is quasicompact, and the resulting quotient NS X/k is a local system of finitely generated abelian groups, called the Néron-Severi group scheme.Likewise, the preimage Pic τ X/k of the torsion part in NS X/k is quasicompact.According to [6], Exposé XIII, Theorem 4.6 the geometric points on Pic τ X/k correspond to invertible sheaves that are numerically trivial, in other words the intersection number (L • C) = χ(L ) − χ(O C ) vanishes for every curve C ⊂ X.Note that the argument depends on Chow's Lemma, which holds in our context ( [33], Theorem 3.1 or [45], Theorem 8.8), and the proof immediately carries over from schemes to algebraic spaces.
For a general base S, we now define Pic τ (X ⊗ R) as the subgroup of the Picard group comprising all invertible sheaves that are fiberwise numerically trivial.Clearly, this is functorial in R. We need the following result, which is already implicit in Artin's paper [4]: Theorem 2.1.The sheafification of R → Pic τ (X ⊗ R) is representable by an algebraic space Pic τ X/S whose structure morphism is of finite presentation (hence quasiseparated) and locally separated.Moreover, the inclusion into Pic X/S is an open embedding.
Proof.The problems are local in the base, so we may assume that S = Spec(A) is affine.Let A λ ⊂ A be the direct system of subrings that are finitely generated over the ring Z.Using the results from [26], §8 there is an index µ such that X = X µ ⊗ Aµ A for some proper flat X µ over A µ ; compare [45] Proposition B.2 and B.3 for statements entirely in the realm of algebraic spaces.We also have to ensure cohomological flatness: Forming the Čech complex for some étale surjection U µ → X µ with an affine scheme U µ and arguing as in [37], Section 5 we find a bounded complex K • of finitely generated projective A µ -modules giving an identification is finitely generated and its formation commutes with base-change.With the long exact Tor sequence for 0 → H 0 → K 0 → M → 0, we infer that the formation of H 0 (X µ , O Xµ ) commutes with base-change if and only if M is locally free.So by assumption, M ⊗ Aµ A is locally free.Hence there is some λ ≥ µ such that M ⊗ Aµ A λ is locally free.Summing up, we may replace X → Spec(A) with the base change of X µ → Spec(A µ ) to A λ , and assume that A is finitely generated over the excellent Dedekind ring Z.
Artin established the representability for the sheafification of R → Pic(X ⊗ R) in [4], Theorem 7.3 by an application of his Theorem 5.3, which involves checking certain conditions [0']- [5'].The reasoning for Pic τ (X⊗R) is virtually the same.Only condition [1'] requires additional arguments: Let R be a complete local noetherian ring, with residue field k = R/m R .We have to verify that the canonical map is bijective.It is injective, by Grothendieck's Existence Theorem ( [23], Theorem 5.1.4for schemes and [51], Theorem 08BE for algebraic spaces).Moreover, given invertible sheaves L n on X⊗R/m n+1 R such that the restriction of L n+1 to X⊗R/m n+1 R is isomorphic to L n , and that (L 0 •C 0 ) = 0 for every integral curve C 0 in X 0 = X ⊗k, the isomorphism classes come from some invertible sheaf L on X.Our task is to check that the restriction to X ⊗ κ(p) is numerically trivial, for each prime ideal p ⊂ R. By [22], Proposition 7.1.4it suffices to treat the case that R is a discrete valuation ring, p is the zero ideal, and that S = Spec(R).Let η ∈ S be the generic point, and suppose that there is some integral curve The closure C ⊂ X of this curve is flat, whence the closed fiber C 0 ⊂ X 0 is a curve.Note that by [27], Proposition 21.9.11 the total space C carries an ample sheaf, hence the algebraic space C is a scheme.Since Euler characteristics are constant in families ( [24], Theorem 7.9.4),we see This shows that Pic τ X/S is an algebraic space.According to [6], Exposé XIII, Theorem 4.7 the monomorphism to Pic X/S is an open embedding, and the structure morphism Pic τ X/S → S is quasicompact.The proof relies on approximation and Chow's Lemma as above, and carries over from schemes to algebraic spaces.Since Pic X/S is locally of finite presentation, quasiseparated and locally separated, the same holds for the open subspace Pic τ X/S .
In [20], Grothendieck defined Pic τ X/S as the subsheaf comprising those R-valued points of Pic X/S that are torsion in the fiberwise Néron-Severi groups, and showed in his Theorem 1.1 that it is open, under the assumption that Pic X/S is a scheme.The above approach seems to be more adequate in the realm of algebraic spaces.
Similarly, one defines Pic 0 X/S as the abelian subsheaf comprising the R-valued points of Pic X/S that are trivial in the fiberwise Néron-Severi groups.This indeed seems the only possible approach, because the subsheaf is not representable in general, as the following counterexample shows: Suppose S is the spectrum of F 2 [[t]], and let X → S be a family of Enriques surfaces whose generic fiber is classical but whose closed fiber is ordinary or supersingular (for details see [8], Section 3).Then G = Pic τ X/S is finite and flat of degree n = 2.It is the union G = G ∪ G of two sections intersecting in the closed fiber, which is a singleton.Proposition 2.2.In the above situation, the subsheaf Pic 0 X/S inside Pic τ X/S is not representable by an algebraic space.
Proof.Suppose H = Pic 0 X/S is representable.By definition, the canonical morphism to G = Pic τ X/S is a monomorphism, and the set of image points is the zero-section.It follows that the structure morphism H → S is a universal homeomorphism.In particular, it is universally closed, separated, and has affine fibers.According to [45], Theorem 8.5 it must be integral, and in particular affine.Write H = Spec(A) and G = Spec(R), and consider the canonical homomorphism R → A. The induced map R/t n R → A/t n A is bijective for each n ≥ 0. We have n≥0 t n A = 0 by Krull's Intersection Theorem, thus R → A is injective.It follows that the image of H → G contains both generic points, contradiction.
Let us finally collect the basic properties of the tau-part: Proposition 2.3.The algebraic space Pic τ X/S has the following properties: (i) If the geometric fibers of f : X → S are integral, then the structure morphism Pic τ X/S → S is separated.(ii) If the geometric fibers of f : X → S are integral and locally factorial, then Pic τ X/S → S is proper.

and each
Spec(B) → S, there is an algebraic space Z, some proper B-morphisms h 1 , . . ., h r : Z → X ⊗ B and integers n 1 , . . ., n r such that the induced map Proof.It suffices to treat the case that S = Spec(R) is the spectrum of a noetherian ring.Set G = Pic τ X/S .We start with assertion (i) and (ii), which follow from Valuative Criteria, similar to [20], Theorem 2.1.Suppose the fibers are geometrically integral.Then the map O S → f * (O X ) is bijective.Let V be a discrete valuation ring with field of fractions F = Frac(V ), such that the residue field k = V /m V is algebraically closed and that V is complete.
For (i) we have to verify that the map Without restriction, we may assume V = R.The Leray-Serre spectral sequence for the structure morphism X → S shows that the canonical map Pic(X) → G(R) is bijective and that the canonical map Pic(X ⊗F ) → G(F ) is injective.So the sections l i come from invertible sheaves L i that are isomorphic on the complement U of the integral Cartier divisor D = X ⊗ k.Choose an isomorphism ϕ : L 1 |U → L 2 |U .It extends to a homomorphism ϕ : L 1 (−nD) → L 2 for some integer n ≥ 0, as in [21], Theorem 6.8.1.This gives a short exact sequence 0 The cokernel is an invertible sheaf on some effective Cartier divisor E ⊂ X supported by D. Since the latter is integral, we have E = mD for some integer m ≥ 0. Setting Computing the determinant of F with the second sequence and using O X (D) O X gives det(F ) O X .Using the first sequence then reveals L 1 L 2 .This shows (i).Note that this generalizes a result of Ischebeck [30], who considered affine schemes.
For assertion (ii), suppose we have a point l ∈ G(F ).Choose a finite separable extension F ⊂ F so that the image l ∈ G(F ) comes from an invertible sheaf on X ⊗ F , and let V be the integral closure of V .Our task is to check that l lies in the subset G(V ) ⊂ G(F ) ( [34], Theorem 7.3 or [51], Lemma 0A3Z).Without loss of generality, we may assume that R = V = V .Let L F be the invertible sheaf on X ⊗ F corresponding to the point l, and F be any coherent extension to X.We now check that its bi-dual L is invertible.The total space X is integral, because this holds for the closed fiber X ⊗ k.Since X ⊗ F is locally factorial, the same holds for X, by Nagata's result ( [40], Lemma 1).In particular, X is normal, and L is reflexive of rank one, hence invertible.This settles (ii).
Assertion (iii) is due to Ekedahl, Hyland and Shepherd-Barron ( [15], Proposition 4.2).Note the Betti numbers b i (X s ) are vector space dimensions for the étale cohomology groups which by definition are computed over the algebraic closure of the residue field κ(s).
We finally come to statement (iv), which is an abstraction of Mumford's arguments for abelian varieties ( [38], Proposition 6.7).We saw in Theorem 2.1 that G = Pic τ X/S is of finite presentation over S. By definition of smoothness, we have to check that the canonical map G(A) → G(A/J) is surjective, for each R-algebra A and each ideal J with J 2 = 0.In light of [27], Remark 17.5.4 it suffices to consider the case where A is a local Artin ring.We may further assume that the residue field A/m A is algebraically closed, that the ideal J has length one, and that A = R.
Let h : X → S be the structure morphisms and X aff = Spec Γ(X, O X ).The proof for [48], Lemma 1.4 reveals that we have an identification vanishes, because X aff is an Artin scheme with algebraically closed residue fields.Consequently, the Leray-Serre spectral sequence for h gives an exact sequence Here the vertical arrows are given by the linear combination n i h * i .By assumption, the vertical map on the right is injective, but the vertical map in the middle is zero.So by exactness of the upper row, each invertible sheaf on X ⊗ B that is fiberwise numerically trivial extends to X.
Note that assertion (i) actually hold for the structure morphisms Pic X/S → S of the whole Picard scheme, with the same proof.

The numerical sheaf
Let S be a base scheme, and X be an algebraic space whose structure morphism X → S is proper, flat, of finite presentation and cohomologically flat in degree d = 0. We then define an abelian sheaf Num X/S on the site (Aff/S) by the short exact sequence 0 −→ Pic τ X/S −→ Pic X/S −→ Num X/S −→ 0, and call it the numerical sheaf.Note that the translation action of Pic τ X/S on Pic X/S is free, so the formation of the quotient commutes with base-change.In particular, for each point s and each field k containing the separable closure κ(s) sep , the abelian group Num X/S (k ) is free, and its rank is the Picard number ρ ≥ 0 of X ⊗ κ(s) alg , by [6], Exposé XIII, Theorem 5.1.We actually have: Theorem 3.1.Suppose Pic τ X/S → S is flat.Then Num X/S is representable by an algebraic space.The structure morphism Num X/S → S is locally of finite presentation, quasiseparated and locally separated.Moreover, all fibers are separated, schematic and étale.
Proof.The problem is local in S, and we may assume that S is the spectrum of a noetherian ring R. The first assertion follows from Lemma 1.1.Since Pic τ X/S is noetherian, the inclusion into Pic X/S is a quasicompact embedding, and the assertion on the structure morphism follows from Proposition 1.2.To establish the statement on the fibers, we may assume that R = k is a field.Then G = Pic X/k is a scheme, and H = Pic τ X/k is an open subgroup scheme.Hence it is also closed, and the zero element e ∈ G/H is a closed point.In particular, the algebraic space with group structure G/H is separated.It must be étale and hence zero-dimensional, because H ⊂ G is open.Summing up, the algebraic space G/H is locally of finite type over the field k, and zero-dimensional.It the must be schematic by [51], Lemma 06LZ.
In the situation of the above theorem, the diagonal morphism for Num X/S is actually an open embedding, not just an embedding.It apparently gives rise to a further separation condition, which is stronger than local separatedness.We think that this deserves further study.Suppose now that Pic τ X/S is flat over S, so that the numerical sheaf Num X/S is an algebraic space, for example by the criteria in Proposition 2.3.We need the following fact, which is a variant of Mumford's Rigidity Lemma ( [38], Proposition 6.1): Lemma 3.2.Let A be an algebraic space whose structure morphism α : A → S is proper, flat, of finite presentation, cohomologically flat in degree d = 0, and with h 0 (O As ) = 1 for all s ∈ S. Then for each ϕ ∈ Num X/S (A), there is a unique σ ∈ Num X/S (S) making the following diagram commutative: (1) Proof.Write N = Num X/S .In light of the uniqueness assertion, the problem is local, and as usual we may assume that S is the spectrum of a noetherian ring R.
Again by uniqueness, together with fppf descent ( [29], Exposé VIII, Theorem 5.2 for schemes and [51], Lemma 0ADV for algebraic spaces), it suffices to treat the case α : A → S admits a section σ.According to [43], Theorem 2.2.4 the arrow α : A → S is an epimorphism of algebraic spaces.So the diagram (1) reveals that if σ exists, it must coincide with the section ϕ • σ, which thus already settles the uniqueness assertion.It remains to verify that σ = ϕ • σ indeed makes the diagram commutative.
To start with, suppose that R = k is a field, so that N is a separated scheme whose underlying topological space is discrete.Since A is proper, the set-theoretical image ϕ(A) ⊂ N is closed and quasicompact.It carries exactly one scheme structure, because N is étale, and must be the spectrum of an étale k-algebra L = k 1 × . . .× k r .In particular, the schematic image is affine.Using h 0 (O A ) = 1 we conclude L = k, and ϕ = σ • α follows.
Next suppose that R is artinian.Still N is a separated scheme, and the schematic image ϕ(A) ⊂ N is affine.Since A is separated, the section σ is a closed embedding.Write A 0 ⊂ A for the corresponding closed subspace.Both maps ϕ and ϕ|A 0 factor over the affine hulls of A and A 0 , respectively.It thus suffices to check that The first map is also bijective, by the assumptions on cohomological flatness, hence the second map is bijective, too.
We now come to the general case.First note that σ : S → N is not necessarily a closed embedding.However, since N → S is locally separated, the section σ must be an embedding, according to Corollary 11.3 below.It is also quasicompact, because S is noetherian and N → S is quasiseparated.Thus σ factors over some Zariski open U ⊂ N via some closed embedding σ : S → N , by Lemma 11.1 below.We next verify that ϕ : A → N also factors over U .Since the latter is open, this can be checked fiberwise, and was established above.It remains to show that ϕ : A → U factors over the closed embedding σ : S → U .The latter corresponds to some quasicoherent I ⊂ O U , and we have to check that J = ϕ −1 (I )O A is the zero ideal.The problem is local in S, and it suffices to treat the case that the noetherian ring R is complete and local.Let X n = X ⊗ R/m n+1 R , and write X ⊂ X and X n ⊂ X n for the closed subspaces coming from J .We saw above that X n ⊂ X n is an equality.By Grothendieck's Comparison Theorem ( [23], Theorem 5.4.1 for schemes and [51], Lemma 0A4Z for algebraic spaces) the same holds for X ⊂ X, thus J = 0.

Para-abelian varieties
Let us call an algebraic space P over some field k a para-abelian variety if there is a field extension k ⊂ k such that the base-change P = P ⊗ k k admits the structure of an abelian variety.The terminology goes back to Grothendieck, who introduced it in [20], Theorem 3.3 by a different condition.A posteriori, we shall see that the notions are equivalent.By fpqc descent, our P is proper and smooth over k, with h 0 (O P ) = 1.Moreover: Lemma 4.1.The algebraic space P is a projective scheme.
Proof.Choose an algebraically closed field extension k ⊂ Ω.According to [37], page 62 there is an ample sheaf L on the base-change P ⊗ k Ω.This defines a morphism Spec(Ω) → Pic P/k , which factors over some connected component Pic l P/k .Fix a closed point a in this component.Then the field extension k ⊂ κ(a) is finite, and there is a finite extension κ(a) ⊂ k and some invertible sheaf A on P ⊗ k mapping to a ∈ Pic P/k .Choose an embedding k ⊂ Ω.By the numerical criterion for ampleness the base-change of A to P ⊗ Ω is ample ( [31], page 343, Theorem 1).It follows that P ⊗ k admits an ample invertible sheaf N .Let N = N (N ) be the norm with respect to the finite locally free morphism P ⊗ k → P .According to [22], Proposition 6.6.1 the base-change of N to P ⊗ k is ample.Thus N is ample on P .
For our purposes it is crucial to work in the relative setting: Definition 4.2.A family of para-abelian varieties over some scheme S is an algebraic space P , together with a morphism P → S that is proper, flat and of finite presentation, such that the fibers P s are para-abelian varieties over the residue field κ(s), for every s ∈ S.
Particular examples are the families of abelian varieties.By this we mean an algebraic space A, together with a proper flat morphism of finite presentation A → S endowed with a group structure, such that all fibers are abelian varieties.These are often called abelian schemes in the literature.According to Raynaud's result (see [17], Theorem 1.9) the total space A indeed must be a scheme.It actually satisfies the AF-property, that is, each finite set of points admits a common affine open neighborhood, provided the base scheme S is affine.Note, however, that there are examples without ample sheaves ( [42], Chapter VII, 4.2), and this happens already over the spectrum of the ring of dual numbers In what follows, we fix a family of para-abelian varieties f : P → S. Note that the structure morphism is proper, surjective and smooth, in particular of finite presentation.Moreover, the canonical map O S → f * (O P ) is bijective ([24], Proposition 7.8.6).In other words, f is cohomologically flat in degree d = 0, and h 0 (O Ps ) = 1 for all points s ∈ S. In contrast to families of abelian varieties, the total space P often fails to be a scheme.This already happens in relative dimension g = 1 over local schemes S of dimension n = 2, see [42], Chapter XIII, Section 3.2 and also [55].
We seek to relate the sheaves Aut P/S and Pic P/S to P → S. Let us start with some useful observations, which generalize [38], Theorem 6.14: For each e ∈ P (S), there is a unique group law µ : P × S P → P that turns P → S into a family of abelian varieties, with e : S → P as the zero section.
Proof.Uniqueness and hence also existence are local problems, so we may assume that S = Spec(R) is affine.Suppose there are two group laws µ 1 and µ 2 with e as zero section.Since P and hence P × S P are of finite presentation, the algebraic space P and the morphisms µ i are already defined over some noetherian subring R 0 ⊂ R. As explained in [38], Corollary 6.6, the Rigidity Lemma applied over R 0 ensures that µ 1 = µ 2 .This settles uniqueness.
It remains to verify existence.For this it also suffices to treat the case that R is noetherian.Suppose first that there is an fpqc extension R ⊂ R such that a group law µ exists for P = P ⊗ R R , with some origin e ∈ P (R ).Using translation by e ⊗ 1 − e , we may assume that e is the base-change of e.Consider the ring R = R ⊗ R R .By fpqc descent, we have to verify that the two pull-backs µ ⊗ 1 and 1 ⊗ µ to R coincide.Both are group laws, and in both cases the origin is the pull-back of e. Uniqueness ensures µ ⊗ 1 = 1 ⊗ µ .Note that this settles the assertion if R = k is a field.
Suppose next that S is the spectrum of some local Artin ring R. Then P red is the closed fiber, which is schematic according to Proposition 4.1.By [45], Corollary 8.2 the total space P is schematic as well.Hence the group law µ exists by [38], Proposition 6.15.Now suppose that R is a general noetherian ring.Fix a closed point a ∈ S, corresponding to a maximal ideal m ⊂ R. The preceding paragraph gives a formal group law over the completion R = lim ← − R/m n a .It comes from a group law over R, according to Grothendieck's Existence Theorem ( [23], Theorem 5.4.1 for schemes and [51], Lemma 0A4Z for algebraic spaces).Since R m ⊂ R is an fpqc extension, the group law already exists over R m .Thus it is already defined over some open neighborhood U ⊂ S of a ( [26], Theorem 8.8.2 for schemes and [45], Proposition B.2 for algebraic spaces).Applying this for all closed points a ∈ S, we obtain an open covering S = U 1 ∪ . . .∪ U r such that the group law exists over each U i .By uniqueness, these local group laws glue and yield the desired global group law.
Corollary 4.4.Suppose S = Spec(R) is henselian, with closed point a ∈ S. Then P → S admits the structure of a family of abelian varieties if and only if the closed fiber P a contains a rational point.
Proof.The condition is obviously necessary.Conversely, suppose there is a rational point e a ∈ P a .It follows from [27], Corollary 17.16.3 that there is a subscheme Z ⊂ P containing e a , and such that Z → S is étale and quasi-finite.In turn, the singleton {e a } is a connected component of the closed fiber Z a .It corresponds to a connected component U ⊂ Z, because R is henselian.Moreover, U → S is étale and finite, and thus defines a section e ∈ P (S) extending e a .By the Proposition, P → S becomes a family of abelian varieties.
Corollary 4.5.There is an étale surjection S → S such that the base-change P = P × S S admits the structure of a family of abelian varieties over S .
Proof.According to [27], Corollary 17.16.3there is an étale surjection S → S such that P (S ) is non-empty.The Proposition ensures the existence of a group law.
Recall that B = Pic τ P/S is an algebraic space, and the morphism B → S is of finite presentation.For each integer n ≥ 1, the kernel B[n] for multiplication by n is another algebraic space.Write G n = Aut B[n]/S for the ensuing automorphism sheaf.
Corollary 4.6.In the above situation, the structure morphism B → S is a family of abelian varieties.Moreover, for each integer n ≥ 1 the morphism B[n] → S is finite and locally free, and G n → S is relatively representable by affine schemes.
Proof.We first check that B → S is a family of abelian varieties.The morphism is proper according to Proposition 2.3, part (ii).To check smoothness, it suffices to treat the case that P → S admits a section ( [27], Proposition 17.7.1).So P → S becomes a family of abelian varieties, by our Proposition.Mumford then showed that that B = Pic τ P/S is smooth ( [38], Proposition 6.7).He actually assumed that P → S is projective to have the existence of Pic τ B/S as projective scheme.However, his arguments carry over without change to our situation (relying on Theorem 2.1 and Proposition 2.3, part (iv)).It remains to verify that B → S has geometrically connected fibers.For this it suffices to treat the case that S is the spectrum of an algebraically closed field k, and that P is an abelian variety.Then the Néron-Severi group NS(P ) is torsion free, according to [37], Corollary 2 on page 178.Thus B → S is a family of abelian varieties.
For the remaining assertions, it suffices to treat the case that S is the spectrum of a noetherian ring R, and P → S is a family of abelian varieties.Since B → S is separated, the inclusion B[n] ⊂ B is closed, hence the structure morphism B[n] → S is proper.It is also quasi-finite ( [37], Appendix to §6).By [27], Corollary 18.12.4 it must be finite.For each point s ∈ S, the fiberwise multiplication n : B s → B s is finite and surjective, and hence flat ( [50], Proposition 22 on page IV-37).In light of [26], Proposition 11.3.11this holds true for n : B → B. Consequently B[n] → S is finite and locally free.It follows that G n → S is affine, for example by [35], Lemma 4.1.
Let G be an algebraic space endowed with a group structure, and assume that the structure morphism g : G → S is flat and of finite presentation, and that O S → g * (O G ) is bijective.Suppose that we have a relative G-action on P .By functoriality, it induces a relative action on B = Pic τ P/S .Corollary 4.7.In the above situation, the G-action on B is trivial.
Proof.It suffices to treat the case that S is the spectrum of a noetherian ring R. The induced action on B[n] is trivial, because the latter are affine, whereas R = Γ(G, O G ).According to [26], Theorem 11.10.9 the collection of closed subgroup schemes B[n] ⊂ B, n ≥ 1 is schematically dense, and it follows by from loc. cit., Proposition 11.10.1 that the action on B must be trivial as well.

Equivariance
Fix a base scheme S, and let X be an algebraic space whose structure morphism X → S is locally of finite presentation and separated.According to [4], Theorem 6.1 the Hilbert functor Hilb X/S is representable by an algebraic space that is locally of finite presentation and separated.Recall that its R-valued points are the closed subspaces Z ⊂ X ⊗ R such that the projection Z → Spec(R) is proper, flat and of finite presentation.If furthermore X → S itself is proper and flat, one sees that Aut X/S is an open subspace of Hilb (X×X)/S , by interpreting automorphisms via their graphs.
Now let P → S be a family of para-abelian varieties.The action from the left of the algebraic space Aut P/S on P induces an action from the right on the family of abelian varieties B = Pic τ P/S , via pull-back of invertible sheaves.Write G ⊂ Aut P/S for the ensuing inertia subgroup sheaf ; its group of R-valued points comprises the R-isomorphisms f : P ⊗ R → P ⊗ R where the induced map f * : B ⊗ R → B ⊗ R is the identity.
Proposition 5.1.The inclusion G ⊂ Aut P/S is representable by open-and-closed embeddings, the structure morphism G → S is a family of abelian varieties, and the total space G is a scheme.
Proof.First observe that once we know that G → S is a family of abelian varieties, the total space must be a scheme by Raynaud's result ( [17], Theorem 1.9).To verify the statements on G ⊂ Aut P/S and G → S, it suffices to treat the case that P → S is a family of abelian varieties, and S is the spectrum of a noetherian ring R. Note that then the translation action gives a monomorphism P → Aut P/S .This is a closed embedding, because P is proper and Aut P/S is separated ( [27], Corollary 18.12.6).
By Corollary 4.7 we have P ⊂ G as subsheaves inside Aut P/S .We claim that this inclusion is an equality.This is a statement on R -valued points; by making a basechange it suffices to treat the case R = R .Suppose we have some automorphism f : P → P that lies in G(R).Write f = t a • h, where h respects the chosen group law and t a is the translation map with respect to the section a = f (e).The induced homomorphism h * on Pic τ P/S is the identity, because this holds for the translation t * a by Corollary 4.7, and for f * by assumption.Our task is to verify h = id P .In light of Grothendieck's Comparison Theorem ( [23], Theorem 5.4.1 for schemes and [51], Lemma 0A4Z for algebraic spaces) it suffices to treat the situation when R is artinian, and we also may assume that the residue field k = R/m R is algebraically closed.
Suppose first that R = k is a field.According to [37], Corollary on page 131 the classifying map stemming from a normalized Poincaré sheaf P on P × B is an isomorphism of abelian varieties.Moreover, one easily sees that it is natural with respect to P .In particular, the homomorphism h : P → P coincides with (h * ) * .Since h * = id B , it follows that h = id P .We refer to Section 8 for a detailed discussion of Poincaré sheaves.
In the general situation, choose a composition series m R = a 0 ⊃ . . .⊃ a r = 0 whose quotients have length one, and write R i = R/a i .We now show by induction on i ≥ 0 that h i = h⊗R i are identities.We just checked this for i = 0. Now suppose i > 0, and that h i−1 is the identity.Recall that H 0 (P 0 , Θ P 0 /k ) is the Lie algebra for the group scheme Aut P/S ⊗k, where Θ P 0 /k = Hom(Ω 1 P 0 /k , O P 0 ) is the tangent sheaf.It follows that each g ∈ Aut P/S (k[ ]) with g ⊗ k = e ⊗ k is the translation by some a ∈ P (k[ ]).Now both h i and the identity on P ⊗ R i are extensions of h i−1 .As explained in [52], Corollary 4.4 we may view the difference as an element in H 0 (P 0 , Θ P 0 /k ) ⊗ a i /a i−1 , and conclude that h i is translation by some a i ∈ P (R i ).The latter must be the neutral element e i ∈ P (R i ), because h i is a homomorphism.This shows that G = P as subsheaves of Aut P/S .
Summing up, we have shown that G ⊂ Aut P/S is representable by closed embeddings, and that the structure morphism G → S is a family of abelian varieties.It remains to verify that the inclusion G ⊂ Aut P/S is open.For this it suffices to check that for a given automorphism f : P → P , the set Since this is a closed set in a noetherian topological space, the task is to verify that it is stable under generization.So we may assume that R is local and f s is a translation for the closed point s ∈ S, and have to show that U = S.Using Grothendieck's Comparison Theorem ( [23], Theorem 5.4.1 for schemes and [51], Lemma 0A4Z for algebraic spaces), it suffices to treat the case that R is artinian.We then argue as in the preceding paragraph.
Proposition 5.2.The canonical action of G on P is free and transitive.
Proof.We have to show that the morphism (µ, pr 2 ) : G × S P → P × S P is an isomorphism, where µ : G × P → P denotes the action.It suffices to treat the case that P → S is a family of abelian varieties (Proposition 4.3), and that S is the spectrum of a noetherian ring R. Then we saw in the proof for Proposition 5.1 that the G-action coincides with the P -action via translations.The latter is obviously free and transitive.
Let us sum up the content of the preceeding propositions: Theorem 5.3.The family of para-abelian varieties P → S induces, in the above canonical way, a family of abelian varieties G → S inside Aut P/S , and P becomes a principal homogeneous G-space, in other words a representable G-torsor.
In turn, we get a cohomology class [P ] ∈ H 1 (S, G), where the cohomology is taken with respect to the fppf topology.Since the structure morphism G → S is smooth, the cohomology remains unchanged if computed with the étale topology ( [28], Theorem 11.7).Indeed, we already saw in Corollary 4.5 that P → S admits sections locally in the étale topology.Note, however, that the order of the cohomology class [P ] may be infinite ( [42], Chapter XIII, Section 3.2 and also [55]).Now let f : P 1 → P 2 be a morphisms between families of para-abelian varieties, and G i ⊂ Aut P i /S be the resulting families of abelian varieties as above, such that P i is a principal homogeneous G i -space.Proposition 5.4.In the above situation, there is a unique homomorphism f * : G 1 → G 2 between families of abelian varieties such that f : P 1 → P 2 is equivariant with respect to the action of G 1 .
Proof.Uniqueness is clear: For each R-valued point σ ∈ G 1 (R), there is an fppf extension R ⊂ R such that there is some a ∈ P 1 (R ).Then We now verify existence.In light of the uniqueness and fppf descent, it suffices to treat the case that there is a section e 1 ∈ P 1 (S).Composition with f yields a section e 2 ∈ P 2 (S).In turn, we obtain identifications G i = G i •e i = P i .With respect to these identifications, we can regard f : P 1 → P 2 as a morphism f * : G 1 → G 2 between families of abelian varieties that respect the zero sections.This is a homomorphism by [38], Corollary 6.4.Given R-valued points σ ∈ G 1 (R) and a ∈ P 1 (R), we write a = η • e 1 for some η ∈ G 1 (R) and obtain Each R-valued point a ∈ P (R) can be seen as an isomorphism ξ : G R → P R , which comes with an inverse ϕ = ξ −1 , and each l ∈ Pic G/S (R) yields some ϕ * (l) ∈ Pic P/S .As explained in [42], Chapter XIII, Proposition 1.1 this induces a canonical isomorphism (2) Pic G/S ∧ G P −→ Pic P/S , (l, a) −→ ϕ * (l).
Here the wedge symbol denotes the quotient of Pic G/S ×X by the diagonal left action g • (l, a) = (lg −1 , ga).The G-action on P is free and transitive (Proposition 5.2), whereas the G-action on the invariant open subspace Pic τ G/S is trivial (Corollary 4.7).It follows that the projection (3) pr is an isomorphism.As observed by Raynaud in loc.cit., composing (2) with the inverse of (3) yields: Proposition 5.5.The above maps gives an identification Pic τ G/S = Pic τ P/S of families of abelian varieties.
For each family A → S of abelian varieties, the family of abelian varieties Pic τ A/S is called the dual family.The above observation identifies the family B = Pic τ P/S of abelian varieties coming from our family P → S of para-abelian varieties with the dual family for G → S, where the latter is defined via the inclusion G ⊂ Aut P/S .This has a remarkable consequence: Let P → S and P → S be two families of para-abelian varieties, and consider the canonical map Hom S (P , P ) −→ Hom Gr/S (Pic τ P/S , Pic τ P /S ), f −→ f * .Note that the term on the left is a set, which might be empty, whereas the term on the right is an abelian group.Moreover, the G-action on P induces an action of the group G(S) on the set Hom Sch (P, P ).Lemma 5.6.In the above situation, the action of G(S) on the set Hom S (P , P ) is free, and the fibers of the map f → f * are precisely the orbits.
Proof.Suppose some σ ∈ G(S) fixes a morphism f : P → P .We have to verify that σ = e.By descent, we may replace S with P and assume that there is a section a ∈ P (S).Then f (a) ∈ P (S).The action of G(S) on P (S) is free, and from σ + f (a) = f (a) it follows σ = e.
Corollary 4.7 ensures that the orbits are contained in the fibers.Conversely, suppose that f, g : P → P are two morphisms with f * = g * .We have to produce some σ ∈ G(S) with f = g + σ.It must be unique, if it exists, according to the preceding paragraph.By descent, our problem is local, so we may assume that P → S and hence also P → S is a family of abelian varieties, that S is the spectrum of a noetherian ring R, and that we have an identification G = P .Write f = t a • f 0 and g = t b • g 0 , where f 0 , g 0 : P → P are homomorphisms, and t a , t b are translations by some elements a, b ∈ P (S) = G(S).Our task is to verify that g 0 = f 0 .In light of Grothendieck's Comparison Theorem ( [23], Theorem 5.4.1 for schemes and [51], Lemma 0A4Z for algebraic spaces) it suffices to check this if R is a local Artin ring.From Corollary 4.7 we have f * 0 = g * 0 .We now argue as in Proposition 5.1, by choosing a composition series m R = a 0 ⊃ . . .⊃ a r = 0 whose quotients have length one, and applying induction on i ≥ 0 with R i = R/a i .

Some extensions of group schemes
Let k be a ground field of characteristic p ≥ 0. Recall that an abelian variety A is a group scheme that is smooth, connected and proper.By abuse of notation, we simply say that A is abelian.A group scheme N is called of multiplicative type if there is a field extension k ⊂ k such that N ⊗ k is isomorphic to the spectrum of the Hopf algebra k [Λ] for some commutative group Λ.We are mainly interested in the case that the scheme N is of finite type; then the group Λ is finitely generated, and one may choose k ⊂ k finite and separable.Moreover, N is a twisted form, already in the étale topology, of Here the summands are the kernels of the multiplicative group G m with respect to multiplication by n i ≥ 0. For brevity, we call such N multiplicative.
Throughout this section, we study extensions of group schemes where A is abelian, N is multiplicative, and the middle term E is commutative, and analyze their splittings.Some of the assertions below are valid over general base schemes, but for the sake of exposition we stick to a ground field k.First note that Hom(A, N ) = 0, because N is affine and h 0 (O A ) = 1.Hence a splitting is unique, if it exists.
Recall that exactness means that for each T = Spec(R), the sequence of groups 0 → N (R) → E(R) → A(R) is exact, and for each a ∈ A(R) there is an fppf extension R ⊂ R such that the base-change a is in the image of E(R ) → A(R ).
For any commutative group scheme G, we can consider the resulting abelian sheaves Ext i (G, G m ) on the category (Aff/k) of affine schemes, endowed with the fppf topology.These are defined with injective resolutions of G m , but can also be seen as sheafification of the presheaf that assigns to each ring R the groups Ext i (G R , G m,R ).
Its elements can also be interpreted as equivalence classes of Yoneda extensions, formed with sheaves of abelian groups.Note that the Ext i (G, G m ) may or may not be representable by group schemes.Proposition 6.1.We have Ext 1 (N, G m ) = 0.Moreover, Hom(A, H) = 0 for any affine group scheme H.
Proof.For the first assertion, suppose we have some ring R and a short exact sequence 0 → G m,R → F → N R → 0, with some abelian sheaf F in the middle.We have to show that the extension splits on some fppf extension.The sheaf F is a G m,R -torsor over the scheme N R , hence representable by some scheme.According to [14], Exposé XVII, Proposition 7.1.1 the relative group scheme F is multiplicative, over the base S = Spec(R).Clearly, the structure morphism F → S is of finite type.According to [14], Exposé X, Corollary 4.5 we may replace R with some étale ring extension such that N R and F become diagonalizable.Then the extension F corresponds to the pull-back to S of some short exact sequence 0 → Λ → Λ → Z → 0 of commutative groups, which of course splits.This gives the first assertion.
For the second assertion, suppose we have a homomorphism f : A R → H R over some ring R. Since the formation of Γ(A, O A ) = k commutes with flat ring extensions, we have R = Γ(A R , O A R ), and conclude that f factors over the zero section.
We infer that the short exact sequence (4) yields an exact sequence of abelian sheaves.The sheaf Ext 1 (A, G m ) is representable by an abelian variety, and the theory of bi-extension gives an identification with the dual abelian variety B = Pic τ A/k , see the discussion in [44], Section 2. Actually, A → B is an antiequivalence of the category of abelian varieties with itself, coming with the biduality identification A = Ext 1 (B, G m ).
Let us call a group scheme L a local system if it is a twisted form of the constant group scheme (Λ) k , where Λ is a finitely generated abelian group.Equivalently, L is étale, and the group L(k sep ) is finitely generated.For each multiplicative group scheme G, the sheaf L = Hom(G, G m ) is a local system.Actually, the functor G → L is an anti-equivalence between the category of multiplicative group schemes and the category of local systems, which follows from [14], Exposé IX, Corollary 1.2.Summing up, our extension (4) gives a coboundary map Proof.The condition is obviously necessary.Suppose now that the coboundary map vanishes.We already remarked that the section is unique, if it exists, so with Galois descent it suffices to treat the case that k is separably closed.Then N is a direct sum of copies of the multiplicative group G m and the kernels µ n = G m [n], hence it is enough to consider the cases N = G m and N = µ n .In the former case, the extension class of ( 4) is the image of the identity map id : N → G m under the coboundary map, whence the extension splits.It remains to treat the case N = µ n .The canonical inclusion µ n ⊂ G m yields a push-out extension On the right we have the dual abelian variety B = Ext 1 (A, G m ).The image of the identity on G m is a rational point b ∈ B, and we have b = 0 by assumption.Thus E splits.The Kummer sequence gives an exact sequence The term on the left vanishes.Thus the map on the right is injective, and E splits as well.
The coboundary map is functorial in the extension (4), by the very definition of delta functors ([18], Section 2.1).Consequently, the pull-back along some homomorphism of abelian varieties A → A splits if and only if the composition L → B → B vanishes.We exploit this as follows: Let Z ⊂ B be the Zariski closure of the settheoretical image for L → B. This is a smooth subgroup scheme, and its formation commutes with ground field extensions ( [14], Exposé VIb, Proposition 7.1).The short exact sequence 0 → Z → B → B → 0 defines an abelian variety B = B/Z, and we now consider the dual abelian variety A .It comes with a homomorphism Theorem 6.3.The homomorphism f : A → A has the following properties: (i) The formation of A and f commutes with ground field extensions.
(ii) The pull-back of our extension (4) along f : A → A splits.
(iii) The induced homomorphism A → E is a closed embedding, and its image contains every abelian subvariety inside E. (iv) For every homomorphism g : A → A from an abelian variety A such that the pullback of (4) along g splits, there is a unique factorization over f : A → A.
Proof.Assertion (i) follows from the fact that the formation of B commutes with ground field extensions, whereas (ii) is a consequence of Proposition 6.2.We next verify (iv).Suppose our extension is split by some g : A → A. Let B → B be the dual homomorphism.Its kernel K ⊂ B is a closed subscheme.We already observed that the composite map L → B → B vanishes.Thus the set-theoretical image of L → B is contained in K, hence the Zariski closure Z is contained in K.The Isomorphism Theorem gives a unique factorization of B → B → B .Dualizing gives the desired factorization A → A → A. This factorization is unique, by biduality.
It remains to establish (iii).The splitting for E = E × A A is unique, as we already observed below (4).In turn, there is a unique lift g : A → E for f : A → A. Consider the schematic image A 0 = g(A ) inside E, and the induced factorization A → A 0 → A, and the dual factorization B → B 0 → B .Obviously, the pull-back of (4) along A 0 → A splits.From (iv) we get a decomposition A = A 0 ⊕ A 1 .This ensures that B 0 → B is a closed embedding.Using that B → B is surjective, we infer that B 0 = B is bijective, and it follows that A → A 0 is an isomorphism.
The sheaf kernel Ker(∂) and the sheaf image Im(∂) = L/ Ker(∂) for the homomorphism ∂ : L → B are group schemes that are locally of finite type, and in fact local systems.According to Lemma 1.1, the sheaf cokernel Coker(∂) = B/ Im(∂) is an algebraic space that is not necessarily locally separated.

Maximal abelian subvarieties
Let k be a ground field of characteristic p ≥ 0, and G be a group scheme of finite type.The goal of this section is to describe the maximal abelian subvariety G ab ⊂ G, making evident that its formation commutes with ground field extensions.We start by defining a three-step filtration Let G aff be the spectrum of the ring Γ(G, O G ). Then the group law on G induces a group law on G aff , and the canonical map G → G aff is a homomorphism.We denote by G 1 its kernel.Then G → G aff is flat and surjective, so that G/G 1 = G aff , and furthermore h 0 (O G 1 ) = 1 ( [12], Chapter III, §3, Theorem 8.2).The latter condition means that G 1 is anti-affine.According to [10], Proposition 3.3.4this ensures that G 1 is smooth and commutative.We then define G 2 ⊂ G 1 as the maximal subgroup scheme that is smooth, connected and affine.Finally, write G 3 ⊂ G 2 for the maximal subgroup scheme that is multiplicative, so that G 2 /G 3 is unipotent.This defines the desired three-step filtration on G.
Proposition 7.1.The three-step filtration on G has the following properties: (i) Each homomorphism f : G → G respects the filtrations.
(ii) The formation of G i ⊂ G commutes with ground field extensions.
(iii) The group scheme G 1 is anti-affine, and the quotient G 1 /G 2 is abelian.
Proof.We start with assertion (iii).We already observed above that G 1 is antiaffine.In characteristic p > 0 it is also semi-abelian, so that there is a short exact sequence (5) 0 for some torus T and some abelian variety A, according to [9], Proposition 2.2.The torus is contained in G 2 , by maximality of the latter.The resulting surjection In characteristic zero we use the smallest subgroup scheme N ⊂ G 1 such that the quotient G 1 /N is proper.This exists in all characteristics, and is affine and connected, according to [10], Theorem 2. For p = 0 the group schemes G 1 /N and N are automatically smooth.Consequently G 1 /N is abelian and N ⊂ G 2 , and we conclude again that also vanishes, because the image is a smooth, connected, affine, and also a closed subgroup scheme, hence affine and abelian.Finally, G 3 → G 2 /G 3 is zero, because the domain is multiplicative and the range is unipotent.
We now come to (iv).The extension in question has at most one splitting, because there are no non-trivial homomorphisms from the unipotent group scheme G 2 /G 3 to the multiplicative group scheme G 2 .If k is perfect, such a splitting indeed exists, by [12], Chapter IV, §3, Theorem 1.1.This also ensures that G 3 is smooth and connected.Suppose now p > 0. Then we have an exact sequence (5).The canonical projection G 3 → A vanishes, and we obtain a commutative diagram The induced map G 2 /G 3 → A is zero, so the Snake Lemma gives an inclusion G 2 /G 3 ⊂ T /G 3 .Consequently, G 2 /G 3 is both unipotent and multiplicative, hence zero.In turn, the extension splits for trivial reasons.This also establishes the first part of (iv).In characteristic zero, the unipotent scheme G 2 /G 3 must be smooth.It follows from [12], Chapter IV, §2 that it is isomorphic to a sum of additive groups G a .It remains to establish (ii).For each quasicompact and quasiseparated scheme X, the formation of H 0 (X, O X ) commutes with ground field extensions k ⊂ k .In turn, the same holds for the kernel G 1 ⊂ G. Suppose N is a smooth connected affine subgroup scheme in the base-change Then the quotient H = N /G 2 is smooth, connected and affine, and comes with an embedding into the base-change of the abelian variety A = G 1 /G 2 .It follows that H = 0. Thus G 2 ⊂ G 1 commutes with base-change.Finally, the inclusion G 3 ⊂ G 2 commutes with base-change by [12], Chapter IV, §3, Proposition 1.3.
Let H ⊂ G be an abelian subvariety of largest dimension.Then H ⊂ G 1 , because G/G 1 is affine and h 0 (O H ) = 1.Since G 1 is commutative, and sums and quotients of abelian varieties remain abelian, every other abelian subvariety H ⊂ G 1 must be contained in H. Therefore, we call G ab = H the maximal abelian subvariety of G. Clearly, this is functorial in G.However, it is not immediately evident that its formation commutes with ground field extensions.Set U = G 2 /G 3 and regard it as a subgroup scheme inside G 1 .We thus have an extension (6) 0 U be the abelian subvariety constructed in Section 6 as the dual of B = B/Z.Note that in characteristic p > 0, we have U = 0, and thus can regard A as a subgroup scheme of G.
Theorem 7.2.The formation of the maximal abelian subvariety G ab ⊂ G commutes with ground field extensions k ⊂ k .In characteristic p > 0, we have G ab = A .
Proof.The most difficult case p > 0 was essentially solved above: In this case we have From Theorem 6.3 we thus get G ab = A , and its formation commutes with ground field extensions.Suppose now p = 0. Let k ⊂ Ω be a field extension and write G Ω = G ⊗ Ω for the base-change.It suffices to check that the maximal abelian subvariety N Ω ⊂ G Ω descends to a closed subscheme N ⊂ G.By replacing G with G 1 we may assume that G is commutative.Suppose for the moment that our ground field k is algebraically closed, and fix a prime > 0. For each n ≥ 0, the finite group schemes are étale, so all its Ω-valued points come from k-valued points in G.The closure of all these k-valued points defines the desired closed subscheme N ⊂ G.
If k is an arbitrary field of characteristic zero, it remains to treat the case that Ω is a finite Galois extension.Then N Ω ⊂ G Ω is stabilized by the elements of the Galois group Γ = Gal(Ω/k), and the result follows by Galois descent.

The notion of Albanese maps
Let S be a base scheme, and X be an algebraic space where the structure morphism X → S is proper, flat, of finite presentation, and cohomologically flat in degree d = 0. Then Pic τ X/S exists as an algebraic space whose structure morphism is of finite presentation (hence quasiseparated) and locally separated, according to Theorem 2.1.We now come to the central topic of this paper, where we re-define and generalize classical notions of Albanese varieties and Albanese maps: Definition 8.1.An Albanese map for X is a pair (P, f ) where P → S is a family of para-abelian varieties and f : X → P is a morphism satisfying the following condition: For each s ∈ S the homomorphism f * identifies Pic τ P/S ⊗κ(s) with the maximal abelian subvariety of Pic τ X/S ⊗κ(s).By abuse of notation, we simply say that f : X → P is an Albanese map.From Lemma 11.4 we see that f * : Pic τ P/S → Pic τ X/S is a monomorphism.It is actually a closed embedding provided that Pic τ X/S is separated, and the latter indeed holds if S is artinian.In particular, the fiberwise inclusions Pic τ P/S ⊗κ(s) ⊂ Pic τ X/S ⊗κ(s) are closed embeddings.
A priori, our notion of Albanese maps has good base-change properties: Let f : X → P be a any morphism to a family P of para-abelian varieties.Given S → S we write X = X × S S and P = P × S S for the base-changes, and f : X → S for the induced morphism.Proposition 8.2.In the above situation, if f : X → P is an Albanese map, the same holds for the base-change f : X → P .The converse remains true if S → S is surjective.
Proof.This follows from the fact that the maximal abelian subvarieties are stable under field extensions (Theorem 7.2).
The next observation is an important intermediate step towards existence, uniqueness and universal property of Albanese maps: Proposition 8.3.Suppose f : X → P is an Albanese map.Let g : X → Q be a morphism to some other family Q of para-abelian varieties.Then the homomorphism g * : Pic τ Q/S → Pic τ X/S admits a unique factorization over f * : Pic τ P/S → Pic τ X/S .Moreover, there is at most one morphism h : P → Q such that the diagram Proof.Since f * is a monomorphism, there is at most one factorization for g * .Consequently, the existence of a factorization is local in S for the fpqc topology, and it suffices to treat the case that S is the spectrum of a ring R. Since X is of finite presentation, we moreover may assume that R is noetherian.Set and N = Pic τ P/S and H = Pic τ Q/S .The structure morphism N → S is fppf, and the translation action of N on G is free, hence the quotient G/N exists as an algebraic space, and its formation commutes with base-change, according to Lemma 1.1.We see that g * : H → G factors over N if and only if the composite homomorphism c : H → G/N is trivial.Note that the structure morphism G/N → S is of finite type, because the same holds for G → S and the projection G → G/N is fppf.Moreover, for each point s ∈ S, the inclusion N s ⊂ G s is the maximal abelian subvariety, so the quotient (G/N ) s = G s /N s does not contain any non-zero abelian subvarieties.Now suppose that R is a local Artin ring, such that S = {s}.The closed fiber (G/N ) s is a separated scheme.Hence also G/N is separated, and it is schematic by [45], Corollary 8.2.The set-theoretic image of H → G/N is the origin e s ∈ (G/N ) s in the closed fiber.By the Rigidity Lemma ( [38], Proposition 6.1), the morphism H → G/N must factor over some section σ : S → G/N .We must have σ = e, because the image of the map H(R) → G/N (R) is a subgroup.Using Grothendieck's Comparison Theorem ( [23], Theorem 5.4.1 for schemes and [51], Lemma 0A4Z for algebraic spaces), we infer that H → G/N is trivial if R is a local noetherian ring that is complete.For general local noetherian rings R, the formal completion R = lim ← − R/m n R is an fpqc extension, and it follows from uniqueness applied over R ⊗ R R and fpqc descent that H → G/N then is trivial as well.For general noetherian rings R, one sees that for each prime p ⊂ R, there is an element f ∈ R p such that H → G/N is trivial over the localization R f .Hence H → G/N is trivial over the whole spectrum S = Spec(R).
Suppose there are two morphisms h 1 , h 2 : P → Q making the above diagram commutative.It remains to show that h 1 = h 2 .We just saw that the induced homomorphisms h * i : Pic τ Q/S → Pic τ P/S coincide.Let G ⊂ Aut Q/S be the inertia subgroup scheme for the numerically trivial sheaves.We showed in Section 5 that this is a family of abelian varieties, and its action on Q is free and transitive.By Lemma 5.6 there is a unique section σ ∈ G(S) with h 2 = σ + h 1 .By fppf descent, we may replace S with X and assume that the structure morphism X → S admits a section τ ∈ X(S).The commutativity of the diagram (7) reveals that the h 1 and h 2 coincide on the section f • τ ∈ P (S), thus σ ∈ G(S) must be trivial.This shows In the situation of the Proposition, let S → S be some morphism of schemes, and write X , P , Q for the base-changes, and f : X → P and g : X → Q for the induced maps.
Corollary 8.4.Suppose that S → S is fpqc, and that there is some h : P → Q with g = h • f .Then there is also a morphism h : P → Q with g = h • f , and h equals the base-change of h.
Proof.Consider the fiber product S = S × S S , which comes with two projections pr i : S → S .By the uniqueness in the proposition applied over S , we have an equality pr * 1 (h ) = pr * 2 (h ).From fpqc descent ( [29], Exposé VIII, Theorem 5.2 for schemes and [51], Lemma 0ADV for algebraic spaces) one deduces that there is a unique S-morphism h : P → Q inducing the S -morphism h : P → Q .The property g = h • f can be checked after base-changing to S , where it holds by assumption.

Poincaré sheaves
We now review the notion of Poincaré sheaves, which will be used in the next section to construct Albanese maps.Fix a base scheme S, and let X be an algebraic space whose structure morphism ϕ : X → S is proper, flat, of finite presentation, and cohomologically flat in degree d = 0, with h 0 (O Xs ) = 1 for all points s ∈ S. Thus ϕ * (O X ) = O S , and this commutes with base-change.Consequently ϕ * (G m,X ) = G m,S , and this also commutes with base-change.By the very definition, the algebraic space Pic X/S represents the higher direct image R 1 ϕ * (G m,X ).For each T , the Leray-Serre spectral sequence for the projection pr 2 : X T → T , together with Hilbert 90, gives a short exact sequence (8) We are particularly interested in the case that T = Pic X/S , or more generally that we have a morphism f : T → Pic X/S : Definition 9.1.An invertible sheaf P on X × T is called a Poincaré sheaf with respect to f : T → Pic X/S if its class in Pic(X T ) maps to the element f ∈ Pic X/S (T ) in the above sequence.
By abuse of notation, we simply say that P is a Poincaré sheaf on X × T .Such sheaves exist if the map H 2 (T, G m ) → H 2 (X T , G m ) is injective, by exactness of the sequence.This obviously holds if the structure map ϕ : X → S has a section, or if the cohomology group H 2 (T, G m ) vanishes.In any case, Poincaré sheaves are unique up to preimages of invertible sheaves on T .Now let L be any invertible sheaf on the product X × T , for some algebraic space T .This gives an element in Pic(X T ), which induces a T -valued point for Pic X/S .Let f : T → Pic X/S be the resulting classifying map.From the exact sequence (8) we immediately get: Proposition 9.2.Suppose P is a Poincaré sheaf P on X ×T .Up to isomorphism, there is a unique invertible sheaf N on T such that (id In particular, for every field k and every l : Spec(k) → T , the isomorphism class of the pull-back P|X ⊗k corresponds to the induced k-valued point l ∈ Pic X/S (k).We also see that in general, Poincaré sheaves do not exist: For example, a smooth curve C of genus g = 0 over a ground field k has constant Picard scheme Pic C/k = (Z) k .By Riemann-Roch, if there is a Poincaré sheaf on T = Pic C/k , or even on T = {1} then C contains a rational point, and is thus isomorphic to the projective line.This leads to the following criterion: Proposition 9.3.Suppose there are morphisms Z i → X, 1 ≤ i ≤ r such that each structure morphism pr 2 : Z i → S is locally free of degree d i ≥ 1, with gcd(d 1 , . . ., d r ) = 1.Then there is a Poincaré sheaf on X × Pic X/S .
Proof.Set T = Pic X/S .Fix an index 1 ≤ i ≤ r and set Z = Z i .The map on the right in (8) factors over H 2 (Z T , G m ).Write ψ : Z T → T for the projection.One easily checks that the direct image sheaves R i ψ * (G m,Z T ) vanish for all degrees i ≥ 1 (compare [48], proof for Lemma 1.4).Consequently, the Leray-Serre spectral sequence gives an identification We conclude that the kernel for the map on the right in ( 8) is annihilated by gcd(d 1 , . . ., d r ) = 1.Thus Pic(X T ) → Pic X/S (T ) is surjective, hence a Poincaré sheaf exists.
We also have a non-existence result: Proposition 9.4.Suppose S is the spectrum of a field k, and that there a quasicompact scheme Z so that the pull-back map H 2 (Z, G m ) → H 2 (X Z , G m ) is not injective.Then there is some point l ∈ Num X/k and some connected component T ⊂ Pic l X/k such that there is no Poincaré sheaf with respect to T .Proof.Choose some section s Z ∈ H 0 (Z, R 1 pr 2, * (G m,X Z )) mapping to some nontrivial element of the kernel for H 2 (Z, G m ) → H 2 (X Z , G m ).The resulting classifying map h : Z → Pic X/k factors over a finite union of connected components, corresponding to points l 1 , . . ., l r ∈ Num X/k .Moreover, s Z is the pull-back of the universal section.The decomposition of Pic X/k into connected components gives a decomposition of the quasicompact scheme Z into open-and-closed subschemes Z 1 , . . ., Z r .By passing to one of them, we may assume r = 1 and let T ⊂ Pic l 1 X/k be the connected component containing the image.Suppose that there is a Poincaré sheaf P X×T .Then the restriction s T of the universal section maps to zero under the coboundary.In turn, its pull-back s Z also maps to zero under the coboundary, contradiction.

Existence and universal property
In this section we establish existence and uniqueness results for Albanese maps.These are consequences of a general criterion, for which we have to generalize maximal abelian subvarieties to a relative setting.We start by doing this.Let S be a base scheme, and G be an algebraic space endowed with a group structure.Assume that the structure morphism G → S is of finite type.Definition 10.1.A family of maximal abelian subvarieties for G is a family of abelian varieties A → S, together with a homomorphism i : A → G such that for each point s ∈ S, the map identifies the fiber A s with the maximal abelian subvariety inside the group scheme G s .
Note that i : A → G is a monomorphism, provided that G is locally separated and S is noetherian, according to Lemma 11.4.Then the translation action of A on G is free, so the quotient G/A exists as an algebraic space by Lemma 1.1.Now suppose that X is an algebraic space whose structure morphism X → S is proper, flat, of finite presentation and cohomologically flat in degree d = 0, with h 0 (O Xs ) = 1 for all points s ∈ S. Then Pic τ X/S is an algebraic space endowed with a group structure whose structure morphism is of finite presentation (hence quasiseparated) and locally separated.Note that it is neither flat nor separated in general.We can formulate the main result of this paper: Theorem 10.2.Assumptions as above.Then G = Pic τ X/S admits a family of maximal abelian subvarieties if and only if there is an Albanese map f : X → P .Moreover, it is universal for morphisms to families of para-abelian varieties, and commutes with base-change.
Note that in the relative setting, the assumption is restrictive: For example, a Weierstraß equation over a discrete valuation ring R whose discriminant ∆ is nonzero and belongs to m R defines a family X ⊂ P 2 R of cubics that does not admit a family of maximal abelian subvarieties.Also note that for the family X → S of Enriques surfaces considered in Proposition 2.2 the Pic τ X/S admits a family of maximal abelian subvarieties, namely the zero family.
The proof of the theorem is given at the end of this section.The universal property ensures that the Albanese map is unique up to unique isomorphism.This justifies to write Alb X/S = P , and we call it the family of Albanese varieties for X.The Albanese map becomes f : X −→ Alb X/S .This is equivariant with respect to actions of algebraic spaces endowed with a group structure, an observation that seems to be new for infinitesimal actions, even over ground fields: Corollary 10.3.Assumptions as in the theorem.Then there is a unique action of Aut X/S on the family of Albanese varieties Alb X/S that makes the Albanese map f : X → Alb X/S equivariant.
Proof.Let σ ∈ Aut(X) be an S-automorphism.By the universal property in the theorem, there is a unique morphism σ * completing the following diagram: The uniqueness ensures that σ → σ * respects compositions and identities.In turn, there is a unique action of the group Aut(X) making the Albanese map equivariant.
Since the Albanese map commutes with base-change, the same holds for the σ * .Applying the above reasoning with R-valued points of Aut X/S and using the Yoneda Lemma, we obtain the desired action of the group scheme Aut X/S making the Albanese map equivariant.
One may reformulate the theorem in categorical language: Let C be the category of all algebraic spaces X whose structure morphism X → S is proper, flat, of finite presentation, cohomologically flat in degree d = 0, with h 0 (O Xs ) = 1 for all s ∈ S, and such that G = Pic τ X/S admits a family of maximal abelian subvarieties.Write C ⊂ C for the full subcategory comprising all families of para-abelian varieties.The arguments for the following, which are purely formal and analogous to the preceding proof, are left to the reader: Corollary 10.4.For each morphism ϕ : X → X in the category C, there is a unique morphism ϕ * : Alb X/S → Alb X /S making the diagram commutative.Moreover, ϕ → ϕ * respects composition and identities, and the resulting functor C → C given by X → Alb X/S is left adjoint to the inclusion functor C → C.
Over ground fields, the assumption in the theorem is vacuous, which gives: Corollary 10.5.Suppose that S is the spectrum of field k.Then there is an Albanese map f : X → Alb X/k .It is universal for morphisms to para-abelian varieties, commutes with field extensions, is equivariant with respect to group scheme actions, and functorial in X.
This generalizes previous results for perfect ground fields, or for geometrically integral schemes, or geometrically connected and geometrically reduced schemes, compare the appendix in [54].See also [1] when X is non-proper.We can actually formulate an unconditional result in the relative setting, in the spirit of [6], Exposé XII, Section 1: Corollary 10.6.Suppose that the base scheme S is integral, and that the generic fiber of Pic τ X/S → S is proper.Then after replacing S with some dense open set U , there is an Albanese map f : X → Alb X/S .It is universal for morphisms into families of para-abelian varieties, commutes with base-change, and is equivariant with respect to actions of relative group spaces.
Proof.Without restriction we may assume that S is the spectrum of an integral noetherian ring R. We have to find some U such that G = Pic τ X/S admits a family of maximal abelian subvarieties over U .Let η ∈ S be the generic point.Applying Proposition 7.1 to the proper group scheme G η over the field of fractions F = Frac(R), we see that the maximal abelian variety A η ⊂ G η is the kernel for the affinization map.
It suffices to treat the case that the algebraic space G is schematic, and that A η extends to a family of abelian varieties A → S, by shrinking S. In the same way we may assume that the inclusion A η ⊂ P η extends to some homomorphism i : A → G, and that the diagonal embedding G → G × G is closed.Now the kernel N = Ker(i) is a closed subgroup scheme of A, and in particular proper, with N η = 0.By Chevalley's Semicontinuity Theorem ( [26], Corollary 13.1.5)we may shrink S further making the fibers N s finite.By generic flatness ( [25], Theorem 6.9.1), we even achieve N = 0.
It remains to verify that A s ⊂ G s are maximal abelian subvarieties for a dense open set of points s ∈ S. Let f : G → S be the structure morphism, and set g = dim(G η ).Again by Chevalley's Semicontinuity Theorem ( [26], Theorem 13.1.3),the set Z ⊂ G of all points x ∈ G with dim x (G f (x) ) ≥ g + 1 is closed.Its image f (Z) ⊂ S is constructible, and disjoint from η.So after shrinking S, we may assume that all fibers of f : G → S are g-dimensional.For dimension reasons, the inclusion A s ⊂ G s must be the maximal abelian subvariety, for all points s ∈ S.
Moret-Bailly and one referee alerted us that the conclusion does not hold without suitable assumption on the generic fiber, as the following example shows: Let X Q be the denormalization of an elliptic curve E Q that identifies the origin e ∈ E Q with a rational point a ∈ E Q of infinite order.Any morphism X Q → P Q to an abelian variety induces a homomorphism E Q → P Q with a in the kernel.It follows that these morphisms are constant, hence Alb X Q /Q is trivial.On the other hand, an extension X over some suitable R = Z[1/n] yields a Pic τ X/S that is an extension of a family E → S of elliptic curves by the multiplicative group.Over each closed point s ∈ S, the residue field κ(s) = F p is finite, hence the corresponding Ext group is finite, and the class of the extension has finite order.It follows that Alb Xs/Fp is one-dimensional.
Proof of Theorem 10.2.If there is an Albanese map f : X → P , the image of the monomorphism f * : Pic τ P/S → Pic τ X/S is a family maximal abelian subvarieties.Our task is to establish the converse: Suppose there is a family A ⊂ Pic X/S of maximal abelian subvarieties.We already saw in Proposition 8.2 that the Albanese map commutes with base-change, once it exists.Our task here is to establish existence and universal property.We proceed in five intertwined steps, with various temporary assumptions: Step 1: We show existence, assuming that there is a Poincaré sheaf P on the product X × Pic X/S .Since the family of abelian varieties A has a section, there is also a Poincaré sheaf F on Pic A/S ×A.Tensoring with the preimage of some invertible sheaf on Pic τ A/S we may assume that F becomes trivial on both {e} × A and Pic τ A/S ×{e}.Such Poincaré sheaves are called normalized.Now regard the restriction P| X×A as a family of invertible sheaves on A parameterized by X.Let f : X → Pic A/S be the classifying map.Then the sheaves P| X×A and (f ×id A ) * (F ) define the same elements in Pic A/S (X).Like (8) we have an exact sequence 0 −→ Pic(X) −→ Pic(X × A) −→ Pic A/S (X), so P| X×A and (f × id A ) * (F ) differ only by the pull-back of some invertible sheaf on X. Form the relative numerical group Num A/S and consider the composite map f : X → Num A/S .This factors over some section δ : S → Num A/S , by Lemma 3.2.
The preimage P = Pic δ A/S is a family of para-abelian varieties.It comes with the structure of a principal homogeneous space with respect to Pic τ A/S , stemming from tensor product of invertible sheaves.By construction, the classifying map factors as f : X → P .We claim that this is an Albanese map.
To see this, it suffices by our very definition to treat the case that S is the spectrum of an algebraically closed field k.In light of Hilbert's Nullstellensatz, there is a rational point a ∈ X.This induces a rational point e = f (a) on P , and the latter becomes an abelian variety.Moreover, the rational point δ ∈ Num A/S comes from an invertible sheaf N on A. Tensor products with N give an isomorphism Pic τ A/S → Pic δ A/S .Composing f with its inverse, we may assume that δ = e.The normalized Poincaré sheaf F induces identifications A = Pic τ P/k and P = Pic τ A/k .By construction, the abelian varieties P and A have the same dimension, so we merely have to check that the kernel N of f * : Pic τ P/k → Pic τ X/k is trivial.Suppose this is not the case.Then N contains either a non-zero rational point or a tangent vector supported by the origin.In any case, there is a k-algebra R of degree [R : k] = 2 with an embedding Spec(R) ⊂ N ⊂ Pic τ P/k = A and some non-trivial invertible sheaf L on P ⊗ k R such that (f ⊗ id R ) * (L ) becomes trivial on X ⊗ k R. In particular, L is numerically trivial.Let l : Spec(R) → Pic τ P/k = A be the classifying map, such that Now recall that P| X×A and (f × id A ) * (F ) differ by the preimage of some invertible sheaf M on X.We infer that P| X⊗ k R M ⊗ k R. Regarding P| X⊗ k R as a family of invertible sheaves on X parameterized by Spec(R), we conclude that the classifying map Spec(R) → Pic X/k factors over a closed point.On the other hand, since P is the Poincaré sheaf, the classifying map is the composition of the embeddings Spec(R) ⊂ A and A ⊂ Pic X/k , contradiction.
Step 2: We verify the universal property for the particular Albanese map above, assuming that X and all the algebraic spaces Pic δ X/S , δ ∈ Num X/S (S) admit sections.Since X has a section, there is a Poincaré sheaf P on X × Pic X/S .Let f : X → P be the resulting Albanese map constructed in step 1, and g : X → Q be a morphism into another family of para-abelian varieties.We have to verify that g factors over f , via some h : P → Q.Note that we already saw in Proposition 8.3 that such a factorization is unique, once it exists.Fix a section s : S → X.This induces a section for Q → S, which therefore becomes a family of abelian varieties.Choose a normalized Poincaré sheaf G on Pic τ Q/S ×Q.Viewing this as a family of invertible sheaves on Pic τ Q/S parameterized by Q, we see that g : X → Q is the classifying map for each of the sheaves (9) M = (id × g) * (G ) ⊗ pr * X (L ) on Pic τ Q/S ×X, where L is any invertible sheaf on X.Our task is to find L such that M (id×f ) * (N ) for some invertible sheaf N on Pic τ Q/S ×P , which then gives the desired factorization g = h•f .We shall achieve this by successively constructing the dashed arrows in the following commutative diagram, starting with ϕ × id and proceeding clockwise.For increased clarity, we write Pic Q rather than Pic Q/S etc.: Fix any invertible sheaf L on X, regard the resulting sheaf M in (9) as a family of invertible sheaves on X parameterized by Pic τ Q/S , and consider the classifying map ϕ : Pic τ Q/S → Pic X/S .In light of Lemma 3.2, the composite map ϕ : Pic τ Q/S → Num X/S factors over some section δ : S → Num X/S .By assumption P = Pic δ X/S admits a section.The latter comes from an invertible sheaf L 0 on X, because X has a section, too.Replacing L with L ⊗ L ⊗−1 0 , we may assume that δ is the zero section.This yields a factorization ϕ : Pic τ Q/S → Pic τ X/S and gives the vertical dashed arrow in (10).By Proposition 8.3, it comes with a factorization ψ : Pic τ Q/S → Pic τ P/S , yielding the two diagonal dashed arrows.By construction, (9) is the pullback of the normalized Poincaré sheaf F on Pic τ P/S ×P .Hence it is the pullback of N = (ψ × id P ) * (F ), giving the desired factorization g = h • f of classifying maps and the upper dashed arrow in (10).
Step 3: We verify the universal property for the particular Albanese map above, now only assuming that X admits a section.Keep the notation from the previous step.There the arguments only relied on the existence of sections in Pic δ X/S for one particular δ ∈ Num X/S (S) occurring along the way.Choose an fppf morphism S → S so that this Pic δ X/S acquires an S -valued point.Then step 2 gives the desired factorization g = h • f for the base-changes X = X × S S , P = P × S S and Q = Q × S S .The two pullbacks of h to S × S S coincide, by the uniqueness in Proposition 8.3.Now fppf descent ( [29], Exposé VIII, Theorem 5.2 for schemes and [51], Lemma 0ADV for algebraic spaces) gives the desired factorization h : P → Q.
Step 4: We prove the universal property for general Albanese maps f : X → P .Let g : X → Q be another morphism into some family of para-abelian varieties.As above we have to find a factorization g = h • f , and we shall achieve this by descent.Note that uniqueness was already established in Proposition 8.3.Choose some fppf morphism S → S so that the base-change X = X × S S admits a section.Set P = P × S S , and let X → P 1 be the Albanese map over S constructed in step 1 with Poincaré sheaves.We then have a factorization P 1 → P , according to step 3.By our definition of Albanese maps, the induced proper homomorphism Pic τ P 1 /S → Pic τ P /S is fiberwise an isomorphism, thus a monomorphism (Lemma 11.4), hence a bijective closed embedding ( [27], Corollary 18.12.6).With the Nakayama Lemma one infers that Pic τ P 1 /S → Pic τ P /S is an isomorphism.By biduality, the original homomorphism P 1 → P is an isomorphism.Regard it as an identification P = P 1 .Using step 3 again, we get a factorization g = h • f .The two pullbacks of h : P → Q to S = S × S S coincide, by uniqueness.Now [51], Lemma 0ADV gives the desired factorization h : P → Q.
Step 5: We establish existence in general.Choose some fppf morphism S → S so that the base-change X = X × S S acquires a section.This happens, for example, with S = X.Now step 1 gives an Albanese map f : X → P .Set S = S × S S , and consider the two base-changes pr * 1 (P ) and pr * 2 (P ).The two induced morphisms from X = X × S S are related by a morphism ϕ : pr * 1 (P ) → pr * 2 (P ), according to the existence part in step 4, applied over S , and uniqueness ensures that ϕ is an isomorphism.This satisfies the cocycle condition over S × S S × S S , in light of the uniqueness part in step 4. In turn, the scheme P over S descends to an algebraic space P over S .By construction, P → S is a family of abelian varieties, hence P → S is a family of para-abelian varieties.In the same way, the Albanese map f : X → P descends to a morphism f : X → P .The latter is an Albanese map, by Proposition 8.2.

Appendix: Embeddings for algebraic spaces
Here we collect some general facts about embeddings of algebraic spaces that were used throughout, and seem to be of independent interest.The topic indeed requires attention, because schemes are locally separated, whereas not all algebraic spaces share this property.Recall that a morphism of schemes f : Z → Y is an embedding if it factors into an closed embedding A morphism of algebraic spaces f : Z → Y is called an embedding if for each affine étale neighborhood V → Y the fiber product Z × Y V is a scheme, and the projection Z × Y V → V is an embedding.Proof.Let V λ → Y , λ ∈ I be the affine étale neighborhoods, and C λ ⊂ V λ be the set-theoretical image of the projection Z × Y V λ → V λ .This is locally closed, so Cλ C λ is a closed set in V λ .Moreover, the scheme Z × Y V λ is quasicompact, by the corresponding assumption on f .For each morphism of neighborhoods i : V λ → V µ we have i −1 (C µ ) = C λ .Since the morphism is étale and hence flat, and the scheme Z × Y V λ is quasicompact, we also have i −1 ( Cµ ) = Cλ by [29], Exposé VIII, Theorem 4.1, and thus i −1 ( Cµ C µ ) = Cλ C λ .Thus the complementary open sets U λ ⊂ V λ are compatible, and thus define an open embedding U → Y , according to [51], Lemma 0ADV.Note that in the latter result, we indeed can ignore the condition on the cardinality of the index set I by working in a fixed Grothendieck universe, confer the discussion in [46], Section 2. By construction, the morphism Z → Y factors over U , and the resulting Z → U is a closed embedding.
from the local system L to the dual abelian variety B. Proposition 6.2.Our extension (4) splits if and only if ∂ : L → B vanishes.
Z → U followed by an open embedding U → Y .Then the set-theoretical image C = f (Z) is locally closed, and Z is determined, up to unique isomorphism, by an open set U ⊂ Y and a quasicoherent I ⊂ O U .Note that such U are not unique, but there is a maximal one, namely the complement of the closed set C C = C ∩ (Y U ).

Lemma 11 . 1 .
Let f : Z → Y be an embedding of algebraic spaces.Suppose that the morphism is quasicompact.Then it admits a factorization into a closed embedding Z → U followed by an open embedding U → Y .