Hodge cohomology with a ramification filtration, I

We consider a filtration on the cohomology of the structure sheaf indexed by (not necessarily reduced) divisors “at infinity”. We show that the filtered pieces are functorial with respect to transfers, have fpqc descent, and are so called cube invariant. In the presence of resolution of singularities and weak factorisation they are invariant under blowup “at infinity”. As such, they lead to a realisation functor from Kahn, Miyazaki, Saito and Yamazaki’s category of motives with modulus over a characteristic zero base field.


Introduction
In his celebrated work [Voe00], Voevodsky constructed the triangulated category of mixed motives DM eff k over a field k.In the series of papers [KMSY21a], [KMSY21b], [KMSY20], Kahn, Miyazaki, Saito and Yamazaki define and study a triangulated category MDM eff k which contains Voevodsky's category DM eff k as a full subcategory.One of their motivations is to obtain a motivic framework where one can study various non-A 1 -invariant cohomology.An example of such a cohomology is the coherent cohomology H i Zar (−, O) of the structure sheaf O. Indeed, O is represented by A 1 which is contractible in DM eff k by definition.Somewhat surprisingly, it has be unknown for a long time whether the most obvious non-A 1 -invariant cohomology theory H i Zar (−, O) is representable in MDM eff k or not.In this paper we show that it is, at least over any field of characteristic 0 (see Cor. 2 below).As a consequence, we observe another fact, also surprisingly unknown for a long time, that MDM eff k is strictly larger than DM eff k .In fact, we will represent a suitable filtration of H i Zar (−, O) graded by divisors "at infinity": for any choice of open immersion X ⊆ X and invertible sheaf of ideals I ⊆ O X whose vanishing locus X ∞ = Spec(O X /I) satisfies X = X \ X ∞ , we can consider the image of the morphism 1 H q ((X, X ∞ ), M O) := H q Zar (X, √ I ⊗ I ⊗−1 ) → H q Zar (X, O X ).
The first author was supported by JSPS KAKENHI Grant (19K14498).The second author is supported by JSPS KAKENHI Grant (21K13783).
1 The strange use of √ I is motivated by the indexing of certain filtrations appearing in class field theory.
In this way, we obtain a filtration on the cohomology of X indexed by the multiplicity of effective Cartier divisors.Moreover, one can prove that this filtration is exhaustive As such, instead of the smooth varieties which generate DM eff k , the category MDM eff k is generated by modulus pairs.A modulus pair can be defined as a pair X = (X, X ∞ ) such that X is a variety, X ∞ ⊆ X is a closed subscheme, and X • := X \ X ∞ is smooth, 3 In the same way that M (X) ∈ DM eff k represents the cohomology of X, the object M (X ) ∈ MDM eff k represents a filtered piece of the cohomology of X • , namely the cohomology with ramification bounded by X ∞ .
Our main theorem is the following.
Theorem 1 (Theorem 7.3).Let k be a field of characteristic zero.there exists an object MO ∈ MDM eff k such that for any smooth variety X, any effective Cartier divisor X ∞ with normal crossings support, and any n ∈ Z we have As an immediate corollary of Theorem 1, we obtain Corollary 2. For any X ∈ Sm k , by taking X = (X, ∅) in Theorem 1, we have a representation of the Hodge cohomology groups The strategy can be summarised as follows.We define M O on PSm k (recalled in §2) and show that it is a quasi-coherent étale sheaf, §3, then show that its Nisnevich fibrant replacement is blowup invariant, §4, cube invariant, §5, and has transfers, §6.
We begin in §2 with a recollection of the general theory, and in particular the construction of MDM eff k .In Appendix A, we collect some definitions and facts about resolution of singularities and weak factorisations.In Appendix B, we give a self-contained proof that MZar (resp.Mét) cohomology can be calculated as the colimit of Zariski cohomology (resp.étale cohomology) over abstract admissible blowups.In Appendix C, we make some basic computation of cohomologies on projective spaces generalising classical computations in SGA6 [BGI67].
In future work, this paper's results and techniques will be used to develop the analogue of Corollary 2 for H p Zar (X, Ω q X ), as well as Hochschild homology with modulus satisfying an HKR isomorphism.
Related work.In [BPO22] a Hodge-type realization with log poles is constructed which should compare to the realisation constructed in this paper in case of reduced divisor or in case there is no divisor at infinity (that is, the case of tame ramification or the case of arbitrary ramification with no pole restriction).

Review of the general theory
In this section, we recall basic definitions concerning the category of modulus pairs, Rec.2.1, modulus topologies, Rec.2.3, finite correspondences in the modulus setting, Rec.2.5, and the construction of MDM eff k , Rec.2.6.One can find more details in many places: We fix a perfect base field k with the case of interest being char(k) = 0. We restrict our attention to modulus pairs over k with smooth interior so as not to frighten the reader, but a large part of what we write holds over general bases, cf.[KelMiy21].

Recollection 2.1 (Modulus pairs).
(1) A modulus pair4 over k is a pair such that (a) X (called the total space) is a separated k-scheme of finite type, (b) X ∞ ⊆ X (called the modulus) is an effective Cartier divisor, and (c) studied in [KelMiy21] where X is qc separated, and X • = X \ X ∞ is Noetherian.
5 So for example, there is a tower of morphisms So it is something like a "global" version of Raynaud's approach to rigid analytic spaces.
In particular every morphism in MSm k can be written in the form f • s −1 where s ∈ Σ and f is ambient.(5) The category PSm k 7 has categorical fibre products Y = T × S X in the case f : T → S is minimal, i.e., in the case Y is an integral closed subscheme, and Z → X is finite and dominates an irreducible component of X .
There is a canonical functor Sm k → Cor k which sends a morphism f : (2) The structure presheaf O on Sm k which send X to Γ(X, O X ) has a structure of transfers in the sense that there exists O tr : (3) We write MCor k for the category of modulus correspondences. 9Objects are the same as MSm k and morphism groups can be defined as the intersections 10 where the colimit is over ambient minimal morphisms W → X such that if there exists a proper surjective morphism W → X with W • → X • finite and W integral, and a finite sum Taking graphs induces a covariant functor MSm k → MCor k .
Recollection 2.6. ( where Z tr (X ) = hom MCor k (−, X ).(2) In analogy with DM eff k from [Voe00] the category MDM eff k is defined to be the Verdier quotient Since the generators Z tr (X ) are compact, 12 The category MSch k of not necessarily smooth modulus pairs is defined in the analogous way: objects are pairs (X, X ∞ ) with X separated and finite type over k, and the modulus X ∞ is an effective Cartier divisor.Ambient morphisms (X, [KelMiy21, Thm.4.47], the localisation functor admits a right adjoint, and MDM eff k can be identified with the full subcategory of objects with cube invariant hypercohomology, cf Eq.(2.3).

The presheaf M O
Definition 3.1.If A is any ring and f ∈ A a nonzero divisor we write Of course, A ⊆ M O(A, f ) with equality if f is invertible, and on the other side, In particular, M O(A, f ) is free of rank one in this case.
Lemma 3.3.Suppose φ : A → B is a homomorphism of rings equipped with nonzero divisors f, g respectively, such that φ(f for some h, the question is whether f hA which is also clear since if a n = f j for some j and n ≥ 1 then (ha) n = f hj ′ for some j ′ .We get the general case by factoring the given morphism as (A, f ) → (B, φ(f )) → (B, g).Proposition 3.4.Suppose that A is a ring, f ∈ A is a nonzero divisor and A → B a faithfully flat morphism (in particular, the images of f in B and B ⊗ A B are again nonzero divisors).Then is exact, where we write f also for the images in B and B ⊗ A B to lighten the notation.
Proof.We are studying the diagram: ] is injective, so the cocycle condition for a implies that f a and (f a) n a are cocycles, so we see that f a and (f a) n a are in the subgroup A ⊆ B as desired.
Lemma 3.5.Let φ : A → B be an étale homomorphism and f ∈ A a nonzero divisor.Then the canonical isomorphism Example 3.6.Even though M O is an fpqc sheaf, Prop.3.4, the statement of Lemma 3.5 does not generalise to flat morphisms.
Of course, this does not preclude the possibility that the comparisons H n Zar (X, M O) → H n fppf (X, M O) be isomorphisms.
Proof.First note that we have φ( already reduced because it is étale over the reduced ring (A/f A) red , so it suffices that Spec(ψ) be a surjective closed immersion.This happens because (A/f A) Theorem 3.7.There is a unique fppf-sheaf M O on PSm k such that for affine modulus pairs with principal modulus (Spec(A), (f )) we have M O(Spec(A), (f )) = M O(A, f ).Furthermore, this is quasi-coherent as an étale sheaf.In particular, its Zariski, Nisnevich, and étale cohomologies agree, and vanish for affines.In the previous section, we have constructed a Zariski (in fact fpqc) sheaf M O of abelian groups on PSm k which is quasi-coherent as an étale sheaf.Our next goal is to prove that M O and its cohomology presheaves are invariant under suitable blow-ups.For global sections M O we need a normality assumption, Prop.4.3.For the cohomology we assume normal crossings, Prop.4.6.
To begin with we characterise of elements of M O(A, f ) in Lemma 4.1, and show Lemma 4.2.Let A be a Noetherian normal domain and f ∈ A a nonzero divisor.Then there is an equality Proof.The inclusion M O(A, f ) ⊆ height p=1 M O(A p , f ) comes from Lemma 4.1.Suppose we have an element a on the right.Then for all height one primes p, we have a ∈ A p [f −1 ] = A[f −1 ] p and there exists n p such that f a, (f a) np a ∈ A p by Lemma 4.1.Since ∈ A is an open condition, for each p, there exists an open neighborhood U p of p on which f a and (f a) np a are still regular functions.Since Spec A is quasi-compact, there exists a finite family p 1 , . . ., p m such that Spec A = ∪ m i=1 U pi .Set n := max(n p1 , . . ., n pm ).Then we have f a ∈ A and (f a) n ∈ A, and hence belongs to the left hand side, as desired.
Proof.Since the étale and Zariski cohomologies of M O agree, Thm.3.7, it suffices to show that the morphism α : M O X ét → Rf * M O Y ét is an isomorphism, where now Rf * is the direct image between the small étale sites.Since this question is étale local, it suffices to find for each point x ∈ X an étale morphism U → X whose image contains x, and such that By the definition of normal crossings with X ∞ (Def.A.1), there exists a diagram of étale morphisms such that x ∈ p(U ), p * X ∞ = q * H and Z × X U = Z 0 × A n U , where H = { a∈A t ra a = 0} and Z 0 = {t b = 0, ∀b ∈ B} for some r a > 0 and A, B ⊂ {1, . . ., n}.Therefore, replacing f by f | U , we may assume that there exists an étale morphism q : where f 0 is the blow-up of A n at Z 0 and q ′ is the morphism induced by the universal property of blow-up.Suppose that we know that the assertion of Prop.4.6 holds for f 0 .That is, suppose that, setting A := (A n , H) and Y 0 := (Y 0 , f * 0 H), we have By applying q * = Rq * to this isomorphism and by using the flat base change q * Rf 0 * ∼ = Rf * q ′ * [Sta18, 02KH], we obtain On the other hand, we have q * M O A = M O X and q ′ * M O Y 0 = M O Y by quasicoherence since q and q ′ are étale by construction.Assembling all these isomorphisms leads to the isomorphism This is done by direct calculation in Proposition 4.8 below, using Lemma 4.7 to reduce to the case Z 0 = {0}.Lemma 4.7.For any ring A and for any non-zero divisor f ∈ A, we have  Proof.First we reduce the assertion to the case that d = 0. Note that

These identifications induce an isomorphism
since strict transform along a flat morphism is a pullback [Sta18,0805].Combining this with Lemma 4.7, we are reduced to the case E, where E is the exceptional divisor of the blowup.So Since we are dealing with vector bundles, we can apply the projection formula [Sta18, 01E8] to find

Cube invariance of RΓ(−, M O)
The goal of this subsection is to prove that the cohomology presheaves of modulus global sections satisfy cube invariance.First we prepare a general criterion for cohomological cube invariance.We will use it in Proposition 5.4 to show cube invariance on nice modulus pairs for M O.
Lemma 5.1.Let F be an additive presheaf on PSch k , and let τ ∈ {Zar, Nis, ét}.Let X = (X, X ∞ ) be a modulus pair such that X is quasi-compact and F X is a quasi-coherent sheaf of O-modules on the small site X τ .Suppose moreover that for any affine open subscheme U = Spec A ⊂ X with U ∞ := X ∞ ∩ U = Spec A/(f ) a principal Cartier divisor, the sequence of A-modules f ) → 0 is exact.Then, for any i ∈ Z, the first projection X ⊠ → X induces an isomorphism of abelian groups Note that this almost never represents the Cartesian product in MSm k .
Proof.It suffices to treat the case τ = Zar since the étale and Zariski cohomology agree if F is a quasi-coherent étale sheaf, [Mil80, Rem.III.3.8].By Mayer-Vietoris and induction on the minimal size of a finite affine covering of X, we are reduced to the case when X is affine.Let P 1 = U 0 ∪ U 1 be the standard open covering.Then, for any i = 0, 1 and j > 1, we have H j Zar (X × U i , F X ) = 0 since X × U i is affine and F X is quasi-coherent by assumption.Therefore, the Mayer-Vietoris long exact sequence is simplified as Zar (P 1 X , F ) → 0, where U 01 = U 0 ∩ U 1 , and (−) X = (−) × X (we omit the subscripts of F for the simplicity of notation).Therefore, the right exactness of Eq.(5.1) shows and the left exactness of Eq.(5.1) shows Now, we move on to the proof of the cube invariance of the cohomology of M O.We start with the following lemma.
Lemma 5.3.Suppose that A is reduced and f is a nonzero divisor.Then we have Proof.By functoriality, there exists a canonical inclusion for some n ≥ 0 (note this will imply that a ∈ A[t, f −1 ]).We will show that for some n these two elements are in both Write a = t m b where b ∈ A[t, 1 f ] has non-zero constant term and m ∈ Z. Then we have for some n ≥ 0 by Lemma 4.1.Since b ∈ A[t, 1 f ] has non-zero constant term and A is a reduced ring, b n+1 also has non-zero constant term.So we have (m+1 f ] follows from Eq.(5.3).
Proposition 5.4.Suppose that (A, f ) is a modulus pair with A reduced.Then is a short exact sequence.Consequently, for any X ∈ PSm k with X reduced and for τ ∈ {Zar, Nis, ét}, we have Proof.Since A is reduced, by Lem.5.3, we may replace Then the sequence Eq.(5.4) is a subsequence of the exact sequence Exactness of Eq.(5.4) at M O(A, f ) follows from left exactness of Eq.(5.5).Let's show exactness of Eq.(5.4) in the middle.Suppose that we have a cycle (a, b) in the middle of Eq.(5.4).By exactness of Eq.(5.5), A for some n ≥ 0. Hence, it comes from an element of M O(A, f ).Now let's show right exactness of Eq.(5.4).Suppose that it is not surjective.

Choose an element
t ] for some n) so we must have m < 0 and ℓ > 0. We prove that the condition ℓ > 0 leads to a contradiction as follows.Suppose that we have chosen an element such that ℓ is minimal.The highest degree term of (af ) n a is (a ℓ t ℓ f ) n a ℓ t ℓ .But then from af, (af ) n a ∈ A[t, 1 t ] we deduce that a ℓ t ℓ f and (a ℓ t ℓ f ) n a ℓ t ℓ are in A[t], so a ℓ t ℓ is in the image of M O(A[t], f ).Since a is not in the image, a − a ℓ t ℓ is also not in the image, so a did not have minimal ℓ; a contradiction.
Finally, the second assertion in the statement follows from Lem. 5.1 since M O| X ét is quasi-coherent étale sheaf by Prop.3.4.
Remark 5.5.The above exactness is false if A is not reduced, as one sees immediately from the example (A, f ) = (k[ε]/ε 2 , 1).Indeed, in this case we have giving global sections of A + ε t, instead of A.

Transfers on M O
We observe that the structure of presheaf with transfers on O recalled in Recollections 2.5(2) induces a structure of presheaf with transfers on M O.
Remark 6.2.A modulus pair with non-smooth interior will appear in the proof.One checks directly that Lemma 3.3, Proposition 3.4, Lemma 3.5, Theorem 3.7, Lemma 4.1, Lemma 4.2, and Proposition 4.3 all work verbatim for pairs (X, X ∞ ) wth X Noetherian normal, and X ∞ an effective Cartier divisor.In fact these work even more generally than that, cf.Remark 8.2.
Proof.By definition, Rec.2.5(3), there is a morphism of modulus pairs W → X such that W is integral, W → X is proper surjective, W • → X • is finite, and the composition W → X → Y is a finite sum of morphisms of modulus pairs.Normalising, we can assume W is integrally closed in W • .As such, the morphism ( * ) in the diagram certainly exists, and is unique by injectivity of M O(W) ⊆ O(W • ).By Proposition 4.3 the square on the right is Cartesian, so the morphism ( * * ) also exists and is unique.

Hodge realisation for M O
We now combine the above to prove our main theorem for M O, Theorem 7.3.The idea is that RΓ Nis (−, M O) can be equipped with transfers, and should be blow up invariant and cube invariant.Sadly, Example 8.3 below shows that RΓ Nis (−, M O) is not blowup invariant without some stricter hypotheses.We use normal crossings.Notation 7.1.Write MSm nc k ⊆ MSm k (resp.MCor nc k ⊆ MCor k ) for the full subcategory of quasi-projective normal crossings modulus pairs.Remark 7.2.In general, we have the following.
(1) If Y → X is an abstract admissible blowup, then Y ⊗ → X ⊗ is again an abstract admissible blowup.15So any functor on MCor k which sends X ⊗ → X to an isomorphism, also sends Y ⊗ → Y to an isomorphism.(2) If X ∈ MCor nc k then X ⊗ ∈ MCor nc k .If k satisfies (RoS), then we also have: (1) The inclusions MSm nc k ⊆ MSm k and MCor nc k ⊆ MCor k are equivalences of categories.
(2) Consequently, the canonical comparison functor is an equivalence of categories.
Then there is a unique object MO ∈ MDM eff k such that for X with normal crossings we have where the colimit is over abstract admissible blowups.Consider the case that X ∈ MSm nc k .By Proposition A.7 we can assume all Y are actual blowups of X, and by (RoS) we can assume that all Y are normal crossings.By (WF), such Y → X are zig zags of abstract admissible blowups V → W such that V → W is a blowup in a regular centre that has normal crossings with W ∞ .Since X ∈ MSm nc k by Proposition 4.6 (and Theorem 3.7) the functor Finally, by Proposition 5.4 we deduce that Eq.(7.1) is cube invariant, at least for normal crossings X .But this is sufficient to deduce that it is cube invariant for all X ∈ MSm k , Rem.7.2.So MO lies in the full subcategory MDM eff k ⊆ D(Shv MNis (MCor k )).

Post-script
Here we collect some odds and ends.
Here is a proof of the claim in Example 3.2.Lemma 8.1.Let A be a UFD and f a non-zero divisor.Then the , where p i are irreducible elements in A and m i > 0 for all i = 1, . . ., n.Then we have Here is a remark about the more general setting.
Lemma 4.2 is a kind of valuative criterion for global sections.There is a much more general version of this lemma.The more general version is valid for any ring A equipped with a nonzero divisor f , and the local rings A p are replaced with local rings of the relative Riemann-Zariski space, denoted V al Spec A[f −1 ] (Spec(A)) in Temkin's article, [Tem11].As such Proposition 4.3 also holds for general modulus pairs, but with the restriction that O X be integrally closed in j * O X • , where j : X • ⊆ X is the inclusion.
For Lemma 5.3, Proposition 5.4 one must further assume that the total space is reduced in the general statements.
Here is a counter-example showing that in Proposition 4.6 we need to at least assume that X has rational singularities inside the divisor.
Example 8.3 (Gabber).Suppose f : Y → X is a resolution of singularities of some X with non-rational singularities.That is, such that R i f * O Y = 0 for some i > 0. Suppose that X admits a reduced effective Cartier divisor X ∞ containing the singularities such that Y An explicit example can be produced by considering the affine cone over an elliptic curve (for example): Suppose that E ⊆ P 2 k is a smooth curve with H 1 (E, O E ) = 0, e.g., an elliptic curve, and choose a k be the affine cone over E (we recall a construction below).Let BE = Bl CE {0} → CE be the blowup of the singular point of CE.Equip CE and BE with the pullbacks C ∞ , B ∞ of the effective Cartier divisor A 2 k ⊆ A 3 k corresponding to the P 1 k ⊆ P 2 k chosen above, and set B = (BE, B ∞ ), C = (CE, C ∞ ).We claim that (8.1) is not an isomorphism where f : BE → CE is the canonical morphism.Note CE is affine, so H 1 Zar (BE, M O B ) is precisely the space of global sections of the quasi-coherent sheaf We recall a construction of BE, CE.To begin with, recall that the blowup Bl A 3 {0} of A 3 in the origin is canonically identified with the total space of the line bundle O P 2 (1) on P 2 via a retraction π : Bl A 3 {0} → P 2 to the exceptional divisor P 2 ⊆ Bl A 3 {0}. 16Then one can define BE ⊆ Bl A 3 {0} and CE ⊆ A 3 by forming the Cartesian square on the left and the surjection f .We also have the further Cartesian square on the right coming from the inclusion of the exceptional divisor For the inclusion of Eq.8.1, first note that since P 1 k ∩ E is reduced, the effective Cartier divisor B ∞ is reduced.Indeed, π and therefore θ is an A 1 -bundle.So (8.2) Since affine schemes have no higher coherent cohomology, we have R j θ * O BE = 0 for j > 0 and so the spectral sequence Appendix A. Resolution of singularities and weak factorisations Definition A.1.Let X be a modulus pair and Z ⊆ X a closed subscheme.We will say that Z has strict normal crossings with X ∞ if for every point x ∈ X the local ring O X,x is regular, and there exists a regular system of parameters 18 t 1 , ..., t n ∈ O X,x such that 16 Indeed, classically Bl A 3 {0} is the variety of pairs (L, x) such that L ⊆ A 3 is a line through the origin and x ∈ L. The projection Bl A 3 {0} → A 3 sends (L, x) to x, and the retraction Bl A 3 {0} → P 2 sends (L, x) to L. The exceptional divisor P 2 ⊆ Bl A 3 {0} is the set {(L, x) | x = 0}. 17Indeed, we have identified Bl A 3 {0} with the total space of O P 2 (1), i.e., with Spec ⊕ i≥0 O P 2 (i), and BE is the fibre product. 18Cf.[Sta18,00KU].
We will say that Z has normal crossings with X ∞ if there exists an étale covering V → X such that Z × X V has strict normal crossings with V ∞ .
We say that X is a normal crossings modulus pair if ∅ has normal crossings with X ∞ .Remark A.2. Note, A ∩ B = ∅ is allowed; in particular, Z ⊆ X ∞ is allowed.
such that for each i = 1, . . ., l, either s i or s −1 i is an abstract admissible blowup in PSCH whose total space V i−1 → V i (resp.V i−1 ← V i ) is the blowup of a regular closed subscheme which has normal crossings with Definition A.4.Consider the following properties that a field k might satisfy.
(RoS) For every X ∈ PSm k , there exists an abstract admissible blowup Y → X such that Y is normal crossings and Y → X is an actual blowup.(WF) Every abstract admissible blowup f : Y → X in PSm k such that Y , X are smooth and Y → X is an actual blowup, admits a weak factorisation.
Theorem A.5 (Resolution of Singularities, [Tem08, Thm.1.1],[Hir64]).Let X be a Noetherian quasi-excellent integral scheme of characteristic zero.Then X admits a semi-strict embedded resolution of singularities.In particular, for every closed subscheme Z ⊆ X there is a blowup f : X ′ → X with centre disjoint from the regular locus of X, such that X ′ is regular, Z × X X ′ is a normal crossings divisor.
Notice that (RoS) and (WF) deal with actual blowups.To can turn abstract admissible blowups into actual blowups we using the following.
Proposition A.7 (Temkin).Suppose that Y → X is an abstract admissible blowup in PSm k with X quasi-projective (e.g., affine) and integral.Then there exists a second abstract admissible blowup Y ′ → Y such that Y ′ → X is an actual blowup.
Proof.This is essential [Tem11, Cor.3.4.8] which says that any X such that there exists an f -ample O Y ′ -module L equipped with a global section which is invertible on X • .The existence of the ample sheaf L implies that f is projective, [Sta18, 0B45], and therefore it is an actual blow up, [Liu02, 8.1.24].
Proof.First we treat the case n = 0.If U 0 → X is a τ -covering, then U 0 → X is an M τ -covering by definition of M τ .Conversely, suppose that U 0 → X is an M τcovering.Since U 0 → X is étale (resp.locally an open immersion) as an object of X ét (resp.X Zar ), it suffices to show that it is surjective.By [KelMiy21,Cor. 4.21], the associated morphism U 0 → X in MSm k is refined by a composition of minimal ambient morphisms where s is an abstract admissible blow-up, and f is a τ -covering.So we have the solid commutative square As we observed in Recollection 2.1 the morphism φ can be written as a composition φ = g • t −1 for some abstract admissible blowup t and some minimal morphism t, giving the dashed morphisms making a commutative triangle.Since t, f and s are surjective on the total spaces, so is U 0 → X .
Next we treat the case n > 0. For any m < n, consider the canonical morphisms where c is a morphism of schemes and d is a morphism in MSm k .Since we know that U 0 → X is a τ -covering if and only if U 0 → X is an M τ -covering by the base case n = 0, it remains to show that c is a τ -covering if and only if d is an M τ -covering.But by Cor.B.3, we may assume that the underlying scheme of cosk m sk m U • is given by cosk m sk m U • , and hence that d is represented by c.Then the desired assertion follows from the base case n = 0.
Lemma B.5.Let τ ∈ {Zar, ét}, and X a modulus pair over k.Then, for any finite diagram U • : I → X M τ , there exist an abstract admissible blow-up X ′ → X and a finite diagram V Proof.We discuss the étale case, but the same argument works for the Zariski case.As we observed in Recollection B.1, every object of X Mét is of the form where s, t are the images in MSm k of abstract admissible blowups, and V ′ ∈ X ′ ét .So up to replacing U • with an isomorphic diagram, and replacing X with a sufficiently large abstact admissible blowup, we can assume that all U i are in the strict image of X ét → X Mét (not just in the essential image).
Again applying Recollection B.1, for every φ : i → j in I, we can write U φ : U i → U j as By (the proof of) [KelMiy21, Thm.2.13] there exists an abstract admissible blowup X ′ → X in PSm such that when we form the pullbacks in PSm, the morphism ( * ) becomes an isomorphism in PSm.It follows that X ′ × X V • : sd(I) → PSm is actually indexed by I, since all "backwards" edges σ φ of sd(I) are sent to isomorphisms.The horizontal morphisms ( * * ) are abstract admissible blowups, so they assemble to give a natural isomorphism from X ′ × X V • to U • in MSm.So now we have a diagram in PSm /X whose objects are all in X ét .Since the inclusion X ét → PSm /X is fully faithful, our new diagram factors through X ét .
Corollary B.6.Let τ ∈ {Zar, ét}.Let X be a modulus pair over k, and let U • be an n-truncated M τ -hypercovering of X in MSm k for some n ≥ 0. Then there exists an abstract admissible blow-up X ′ → X and an n-truncated τ -hypercovering U ′ • → X ′ such that the induced morphism of simplicial objects U ′ • → X ′ → X is isomorphic to U • → X , where U ′ m := (U ′ m , X ∞ × X U ′ m ) for each m ≤ n.Proof.By Lem.B.5, there exist an abstract admissible blow-up X ′ → X and a simplicial object V • in X ′ τ such that V • → X ′ → X is isomorphic to U • → X , where V • is given by V m := (V m , X ∞ × X V m ).Since U • → X is an n-truncated M τ -hypercovering and since (U • → X ) ∼ = (V • → X ′ ), Lem.B.4 shows that V • → X ′ is an n-truncated τ -hypercovering.Proposition B.7.For any modulus pair X over k and any presheaf of abelian groups F ∈ PSh(X M τ ), we have for all n ∈ Z and τ ∈ {Zar, ét}, where the colimit is over abstract admissible blowups.
Proof.By Verdier's hypercovering theorem, [SGA4, Expose V, Sec.7, Thm.7. the line bundle O(i) is the free sub-k-module of k[t 0 , . . ., t n , t −1 0 , . . ., t −1 n ] generated by monomials t r0 0 . . .t rn n such that r j ≥ 0 for j = k, and r k ≥ i − j =k r j .The intersection n k=0 O(i)(U k ) of these groups is the free abelian group generated by monomials t r0 0 . . .t rn n subject to the condition r 0 , . . ., r n ≥ 0 if i ≤ 0, and subject to the further condition j r j ≥ i if i ≥ 0. Hence, the claim in the statement.
Next we prove the vanishing assertion.Since R q f * (O(i)) is a coherent sheaf on A n+1 , it suffices to show that its global section vanishes.Consider the short exact sequences 0 → O Bn+1 (i + 1) → O Bn+1 (i) → φ * (O P n (i)) → 0 where φ : P n ֒→ B n+1 is the canonical inclusion of the exceptional divisor.

i
a i t i ∈ I[t].Proposition 4.8.Consider the blowup f : B = Bl A n A d → A of affine space A = Spec(k[t 1 , . . ., t n ]) along a sub-affine space A d ⊂ A n , equip A with the divisor A ∞ = t r1 1 . . .t ri i with r j , i ≥ 1 and equip B with the pullback B ∞ to obtain an abstract admissible blowup B = (B, B ∞ ) → (A, A ∞ ) = A. Then we have M O A ∼ = Rf * M O B where M O A = M O| AZar means restriction to the small Zariski site (and similar for M O B ).
Definition A.3 (cf.[AT19, §1.2]).Suppose that f : Y → X is a abstract admissible blowup between normal crossings modulus pairs, such that Y → X is an actual blowup of noetherian qe regular schemes.A weak factorisation of Y → X is a factorisation of f in MSCH19 t φ as above.This gives a new diagram indexed by the barycentric subdivision sd(I) of the directed graph I. Here, sd(I) is the directed graph which has a span i σ φ ← φ ψ φ → j for every edge i φ → j of I. 21 By construction, this new diagram factors as sd(I) V → PSm → MSm.Now consider the disjoint unions W = ⊔ Ar(I) W φ and U = ⊔ Ar(I) U target(φ) with the canonical morphisms W t → U → X in PSm.
4.1],for any category with finite limits C equipped with a finitary 22 topology τ , and 21 More explicitly, sd(I) has set of vertices the disjoint union V sd(I) = V I ⊔ E I of the vertices and edges of I, and set of edges E sd(I) = E I ⊔ E I two copies of E I .The source morphism E sd(I) → V sd(I) is the identity on both copies of E I .The target is the sum of the source and target morphisms E I ⇒ V I of I.22 Finitary means every covering family {U i → X} admits a finite subfamily which is still a covering family.
PSm k is the category formed by modulus pairs over k, together with ambient k-morphisms.(4) MSm k is the category of modulus pairs over k.It is constructed by formally inverting the class Σ of abstract admissible blowups: 6 i.e., those ambient morphisms The category MSm k admits all 8 categorical fibre products Y = T × S X , [KelMiy21, Thm.1.40].The canonical functor PSm k → MSm k preserves the fibre products in item (5).Remark 2.2.By the universal property of localisation, the category PSh(MSm k ) is canonically identified with the full subcategory of PSh(PSm k ) consisting of those presheaves which send abstract admissible blowups to isomorphisms.Since abstract admissible blowups are categorical monomorphisms in PSm k , this is precisely the category of sheaves for the topology on PSm k generated by abstract admissible blowups.In symbols, PSh(MSm k ) = Shv Σ (PSm k ), cf.[KelMiy21, §A.1].Recollection 2.3 (Modulus topologies).(1)The Zariski topology on PSm k is generated by families (2.1) covering in the classical sense.Zariski coverings on PSm k form a pretopology in the sense of [SGA72, Exposé II].(2) The MZariski topology or MZar-topology on MSm k is generated by images of Zariski coverings under the localisation functor PSm k → MSm k .Zariski coverings do not form a pretopology on MSm k .In general the coverings of the pretopology they generate consists of various iterated compositions of abstract admissible blowups, inverses of abstract admissible 7 Of course if we insist on working in PSm k we also need T • × S • X • to be smooth over k, but in general the pullback along a minimal morphism basically always exists in the larger category of modulus pairs. 8Again, if we insist on working in MSm k then we also need T • × S • X • to be smooth over k, but in general the large category of modulus pairs admits all fibre products.Shv MZar (MSm k ) is canonically identified with the full subcategory of Shv Zar (PSm k ) consisting of those sheaves which send abstract admissible blowups to isomorphisms.In symbols, one could write Shv MZar (MSm k ) = Shv Σ,Zar (PSm k ).
1) An additive presheaf on MCor k is called a modulus presheaf with transfers.Let τ ∈ {Zar, Nis, ét}.A modulus presheaf with transfers is called a τ -sheaf with transfers if for any modulus pair X , the presheaf (F | MSm k ) X on the small site X τ is a τ -sheaf, where F | MSm k denotes the restriction via the graph functor MSm k → MCor k .One can prove that Shv MNis (MCor k ) is a Grothendieck abelian category using the usual methods, [KelMiy21, Cor.6.8],i.e., by showing that the forgetful functor admits a left adjoint a tr : PSh(MCor k ) → Shv MNis (MCor k ) there is a unique presheaf on MSm k whose restriction to PSm k agrees with M O on integrally closed modulus pairs.y≥ 0 for which f a and (f a) ny a are in O Y ,y .Since Y is quasi-compact, there is some n which works for all y.So in fact, f a and (f a) n a are in O(Y ).Applying Lemma 4.1 again, it suffices to show that f a and (f a) n a are in O(X).Since X is normal, it suffices to show that they are in the dvrs O X,x for points x ∈ X of codimension one, Lem.4.2.Chose any extension of the valuation of O X,x to L and let O L be the corresponding valuation ring.Since Y → X is proper, the morphism Spec(O L ) → Spec(O X,x ) → X factors as Spec(O L ) → Y , so we find that the images of f a and (f a) n a in L are in fact in O L .That is, they have value ≥ 0. Hence, they are in O X,x .Remark 4.4.Since O = hom(−, A 1 ) is an h-sheaf on the category of normal schemes, [Voe96, Prop.3.2.10], it follows from Prop.3.4 and Prop.4.3 that M O is an h-sheaf on normal modulus pairs, but we will not need this.Notation 4.5.In the following proposition and proof we use M O X = M O| XZar or M O X ét = M O| X ét for the restrictions to the small Zariski, resp.étale sites.
Proof.By definition M O is a Zariski sheaf, so we can assume X is affine and X ∞ has a global generator, sayX ∞ = (f ).Suppose a ∈ O(X • ) is a section whose image in O(Y • ) lies in M O(Y).For all points y ∈ Y , by Lemma 4.1 there is some 14 Cf.[Bar77, pg.128].nProposition 4.6 (Blow-up invariance of M O and its cohomologies).Take X ∈ PSm k normal crossings and suppose that Z ⊆ X is a closed subscheme that has normal crossings with X ∞ (see Def.A.1 for the terminology).Let f : Y → X be the blowup with centre Z, and Y and applying Thm.3.7 again gets us to the desired isomorphism M O X ét ∼ = Rf * M O Y ét .So it now suffices to prove Proposition 4.6 in the special case f 0 By unpacking the definition of M O, we are immediately reduced to showing the equality I[t] = Take an integer N > (d + 1) max{n i }.Then one checks that ( i a i t i ) N ∈ I[t], and hence √ I[t] of ideals of A[t].We first prove I[t] ⊂ √ I[t].Since I[t] is the smallest radical ideal containing I[t], it suffices to show that √ I[t] is a radical ideal.This means by definition that the quotient ring A[t]/ √ I[t] is reduced.But this follows from A[t]/ √ I[t] ∼ = A/ √ I[t].To see the opposite inclusion I[t] ⊃ √ I[t], take any polynomial i a i t i of degree d with a i ∈ √ I. Then for each i, there exists n i > 0 such that a ni i ∈ I.
Note that CE in Example 8.3 is normal, since it is regular in codimension one, and complete intersection.The singularity is contained inside the divisor C ∞ .Of course, BE is also normal, since it's an affine bundle over a smooth curve, and therefore also smooth.