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On the cohomology of reciprocity sheaves

Published online by Cambridge University Press:  30 August 2022

Federico Binda
Affiliation:
Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy; E-mail: federico.binda@unimi.it.
Kay Rülling
Affiliation:
Bergische Universität Wuppertal, Fakultät Mathematik und Naturwissenschaften, Gaußstrasse 20, D-42119 Wuppertal, Germany; E-mail: ruelling@uni-wuppertal.de.
Shuji Saito
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8941, Japan; E-mail: sshuji@msb.biglobe.ne.jp.

Abstract

In this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves.

We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero cycles of degree 1 from the cohomology of reciprocity sheaves.

Type
Algebraic and Complex Geometry
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Introduction

0.1 Overview

It is a well-known fact that a large class of cohomology theories for algebraic varieties can be equipped with an exceptional, covariant functoriality, despite the fact that they are naturally contravariant. The existence of this kind of ‘trace’ or ‘Gysin’ morphism associated to projective (or even proper) maps of smooth schemes is usually manifesting the existence of some Poincaré duality theory for the cohomology one is interested in; if one replaces cohomology with homology, which is naturally covariant, the exceptional functoriality is conversely represented by the existence of a pullback along a certain class of maps. The construction of cohomological Gysin morphisms has occupied vast literature, stemming from Grothendieck’s trace formalism for coherent cohomology [Reference HartshorneHar66].

A classical instance in the homological setting is represented by the Chow groups. If X is a smooth quasi-projective variety over a field k, the Chow groups ${\operatorname {CH}}_*(X)$ are naturally covariant for proper maps and admit contravariant Gysin maps for quasi-projective local complete intersection morphisms [Reference FultonFul98]. Fulton’s construction of the Gysin morphism was later promoted by Voevodsky in the context of his triangulated category of mixed motives $\mathbf { DM}^{\mathrm {eff}}_{{\operatorname {Nis}}}(k)$ over a perfect field k. Associated to a codimension n closed immersion of smooth k-schemes, $i\colon Z\to X$, Voevodsky [Reference VoevodskyVoe00b] constructed a distinguished triangle:

$$\begin{align*}M(X-Z)\to M(X) \xrightarrow{i^*} M(Z)(n)[2n]\xrightarrow{\partial_{X,Z}} M(X-Z)[1], \end{align*}$$

where $i^*$ is the Gysin morphism and $\partial _{X,Z}$ is a residue map. Combining it with a projective bundle formula for motives, also provided by Voevodsky, the classical method of Grothendieck allows one to define exceptional functoriality along an arbitrary projective morphism between smooth k-varieties, factoring it as a closed immersion followed by a projection of a projective bundle. This as well as the naturality properties of Voevodsky’s Gysin maps have been studied in detail by Déglise [Reference DégliseDég08], [Reference DégliseDég12].

In more recent times, Gysin morphisms for generalised cohomology theories have been constructed in the context of $\mathbf {A}^1$-homotopy theory, making use of the full six functor formalism as developed by [Reference AyoubAyo07a], [Reference AyoubAyo07b] and [Reference Cisinski and DégliseCD19] (see [Reference Déglise, Jin and KhanDJK18] for more history and updated developments in that direction).

From the Gysin sequence, the projective bundle formula and the blow-up formula (the latter being also an ingredient in the construction of the first one) in the triangulated category of Voevodsky’s motives, it is possible to get corresponding formulas for every cohomology theory which is representable in $\mathbf {DM}^{\mathrm {eff}}_{{\operatorname {Nis}}}(k)$. This is the case of the sheaf cohomology of any complex of (strictly) $\mathbf {A}^1$-invariant Nisnevich sheaves with transfers.

However, $\mathbf {A}^1$-invariant Nisnevich sheaves do not encompass all of the phenomena that one would like to study. Interesting examples of sheaves which fail to satisfy this property are given by the sheaves of (absolute and relative) differential forms, $\Omega ^i_{-/\mathbb {Z}}, \Omega ^i_{-/k}$, the p-typical de Rham-Witt sheaves of Bloch-Deligne-Illusie, $W_m\Omega ^i$, smooth commutative k-groups schemes with a unipotent part (seen as sheaves with transfer) or the complexes $R\varepsilon _* \mathbb {Z}/p^r(n)$, where $\mathbb {Z}/p^r(n)$ is the étale motivic complex of weight n with $\mathbb {Z}/p^r$ coefficients, $\varepsilon $ is the change of site functor from the étale to the Nisnevich topology and $p>0$ is the characteristic of k. For some of the above examples, instances of an exceptional functoriality have been studied before, with results scattered in the literature. In the case of the sheaves of differential forms, the existence of the pushforward is of course a consequence of general Grothendieck duality (e.g. [Reference HartshorneHar66], [Reference NeemanNee96]). In this paper, we offer a unified approach to treat the cohomology of arbitrary reciprocity sheaves, a notion that includes all of the above examples: this is a particular abelianFootnote 1 subcategory ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$ of the category of Nisnevich sheaves with transfers on the category $\operatorname {\mathbf {Sm}}_k$ of smooth and separated k-schemes. Its objects satisfy, roughly speaking, the property that for any $X\in \operatorname {\mathbf {Sm}}_k$, each section $a\in F(X)$ ‘has bounded ramification’, that is, that the corresponding map $a\colon \mathbb {Z}_{tr}(X)\to F$ factors through a quotient $h_0(\mathcal {X})$ of $\mathbb {Z}_{tr}(X)$, associated to a pair $\mathcal {X} = (\overline {X}, X_{\infty })$, where $\overline {X}$ is a proper scheme over k and $X_{\infty }$ is an effective Cartier divisor on $\overline {X}$, such that $X=\overline {X} - |X_{\infty }|$ (see 1.6 for more details). The category of reciprocity sheaves has been introduced by Kahn-Saito-Yamazaki in [Reference Kahn, Saito and YamazakiKSY22] (see also its precursor [Reference Kahn, Saito and YamazakiKSY16]) and is based on a generalisation of the idea of Rosenlicht and Serre of the modulus of a rational map from a curve to a commutative algebraic group [Reference SerreSer84, Chapter III].

Voevodsky’s category of homotopy invariant Nisnevich sheaves, $\operatorname {\mathbf {HI}}_{{\operatorname {Nis}}}$, is an abelian subcategory of ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$. Heuristically, $\mathbf {A}^1$-invariant sheaves are special reciprocity sheaves with the property that every section $a\in F(X)$ has ‘tame’ ramification at infinity. Slightly more exotic examples of reciprocity sheaves are given by the sheaves $\mathrm {Conn}^1$ (in characteristic zero), whose sections over X are rank $1$-connections, or $\mathrm {Lisse}^1_{\ell }$ (in characteristic $p>0$), whose sections on X are the lisse $\overline {\mathbb {Q}}_{\ell }$-sheaves of rank $1$. Since ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$ is abelian, and it is equipped with a laxFootnote 2 symmetric monoidal structure [Reference Rülling, Sugiyama and YamazakiRSY22], many more interesting examples can be manufactured by taking kernels, quotients and tensor products (see 11.1 for even more examples).

0.2 Cohomology of cube invariant sheaves

In order to formulate our main results, we need a bit of extra notation. In [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a], the authors introduced the category $\operatorname {\mathbf {\underline {M}Cor}}$ of modulus correspondences, whose objects are pairs $\mathcal {X} = (\overline {X}, X_{\infty })$, called modulus pairs, where $\overline {X}$ is a separated scheme of finite type over k equipped with an effective Cartier divisor $X_{\infty }$ (the case $X_{\infty }=\emptyset $ is allowed), such that the interior $\overline {X}-|X_{\infty }| = X$ is smooth. The morphisms are finite correspondences on the interiors satisfying some admissibility and properness conditions (see 1.1). The category $\operatorname {\mathbf {\underline {M}Cor}}$ admits a symmetric monoidal structure, denoted $\otimes $. Let $\operatorname {\mathbf {\underline {M}PST}}$ be the category of additive presheaves of abelian groups on $\operatorname {\mathbf {\underline {M}Cor}}$. Given $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$ and $F\in \operatorname {\mathbf {\underline {M}PST}}$, we write $F_{\mathcal {X}}$ for the presheaf on the small étale site $\overline {X}_{{\operatorname {\acute {e}t}}}$ given by $U\mapsto F(U, U\times _{\overline {X}} X_{\infty })$. We say that F is a Nisnevich sheaf if, for every $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$, the restriction $F_{\mathcal {X}}$ is a Nisnevich sheaf; the full subcategory of Nisnevich sheaves of $\operatorname {\mathbf {\underline {M}PST}}$ is denoted $\operatorname {\mathbf {\underline {M}NST}}$. Thanks to [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a], the inclusion $\operatorname {\mathbf {\underline {M}NST}} \subset \operatorname {\mathbf {\underline {M}PST}}$ has an exact left adjoint (the sheafification functor). Among the objects of $\operatorname {\mathbf {\underline {M}PST}}$, we are interested in a special class, namely, those which satisfy the properties of being cube invariant, semipure and with M-reciprocity (see 1.4). The first two properties are easy to explain. Let ${\overline {\square }}=(\mathbf {P}^1, \infty )\in \operatorname {\mathbf {\underline {M}Cor}}$. Then $F\in \operatorname {\mathbf {\underline {M}PST}}$ is cube invariant, if for any $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$ the natural map:

$$\begin{align*}F(\mathcal{X}) \to F(\mathcal{X}\otimes {\overline{\square}})\end{align*}$$

induced by the projection $\overline {X}\times \mathbf {P}^1\to \overline {X}$ is an isomorphism. We have that F is semipure if the natural map:

$$\begin{align*}F(\mathcal{X}) \to F(X, \emptyset), \quad (X = \overline{X}-|X_{\infty}|)\end{align*}$$

is injective. The last condition of M-reciprocity is slightly more technical, and we refer the reader to the body of the paper. We write $\operatorname {\mathbf {CI}}^{\tau , sp}$ for the category of cube invariant, semipure presheaves with M-reciprocity and $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ for $\operatorname {\mathbf {CI}}^{\tau , sp}\cap \operatorname {\mathbf {\underline {M}NST}}$. It is possible to show (see [Reference Merici and SaitoMS20, Section 1.6], [Reference Kahn, Saito and YamazakiKSY22, Section 2.3.7]) that there is a fully faithful functor:

$$\begin{align*}{\underline{\omega}}^{\operatorname{\mathbf{CI}}}\colon {\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}} \to \operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}\end{align*}$$

admitting an exactFootnote 3 left adjoint, so that one can, in particular, specialise Theorem 0.1 below on cube invariant sheaves to the case of reciprocity sheaves. If $G\in {\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$, we write $\widetilde {G}$ for ${\underline {\omega }}^{\operatorname {\mathbf {CI}}} (G)$, and for $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$, $n\geq 1$, let us write:

$$\begin{align*}\gamma^n F = \operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}(\widetilde{\mathbf{G}_m^{\otimes_{\operatorname{\mathbf{HI}}_{{\operatorname{Nis}}}} n}}, F) \cong \operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}(\widetilde{\mathcal{K}^M_n}, F).\end{align*}$$

This is a form of (negative) twist (see 4.4, called contraction in Voevodsky’s theory [Reference Mazza, Voevodsky and WeibelMVW06, Chapter 23]). The tensor product with subscript $\operatorname {\mathbf {HI}}$ is the tensor product for homotopy invariant Nisnevich sheaves with transfers from [Reference Mazza, Voevodsky and WeibelMVW06, Chapter 8], $\mathcal {K}^M_n$ is the sheaf of improved Milnor K-theory introduced in [Reference KerzKer10] and the isomorphism follows from a result of Voevodsky [Reference Rülling, Sugiyama and YamazakiRSY22, Section 5.5]. See Theorems 11.1 and 11.8 for some computations of the twists. The Bloch formula implies that for any family of supports $\Phi $ and any cycle $\alpha \in CH_{\Phi }^i(X)$ (see 5.1), there is a natural cupping map:

$$\begin{align*}c_{\alpha}\colon (\gamma^i F)_{\mathcal{X}}[-i]\to R{\underline{\Gamma}}_{\Phi}F_{\mathcal{X}} \quad \text{in } D(X_{\operatorname{Nis}}),\end{align*}$$

which is compatible with refined intersection and pullback, see 5.8.

The following theorem summarises parts of our results. Write $\operatorname {\mathbf {\underline {M}Cor}}_{ls}$ for the subcategory of $\operatorname {\mathbf {\underline {M}Cor}}$, whose objects $\mathcal {X} = (X,D)$ satisfy the additional condition that $X\in \operatorname {\mathbf {Sm}}$ and $|D|$ is a simple normal crossing divisor.

Theorem 0.1. Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$, and let $\mathcal {X} = (X, D)\in \operatorname {\mathbf {\underline {M}Cor}}_{ls}$.

  1. (1) (Projective bundle formula, Theorem 6.3) Let V be a locally free $\mathcal {O}_{X}$-module of rank $n+1$, and let $P=\mathbf { P}(V)\xrightarrow {\pi } X$ be the corresponding projective bundle. Let $\mathcal {P} = (P, \pi ^*D)$. Then there is a natural isomorphism in $D(X_{\operatorname {Nis}})$:

    $$\begin{align*}\sum_{i=0}^n \lambda^i_V\colon \bigoplus_{i=0}^n (\gamma^i F)_{\mathcal{X}}[-i] \to R\pi_* F_{\mathcal{P}},\end{align*}$$
    where $\lambda ^i_V$ is induced by $c_{\xi ^i}$ for the i-fold power $\xi ^i\in {\operatorname {CH}}^i(X)$ of the first Chern class $\xi $ of V.
  2. (2) (Gysin sequence, Theorem 7.16) Let $i\colon Z\hookrightarrow X$ be a smooth closed subscheme of codimension j intersecting D transversally (Definition 2.11), and set $\mathcal {Z}=(Z, D_{|Z})$. Then there is a canonical distinguished triangle in $D(X_{{\operatorname {Nis}}})$:

    (0.1.1)$$ \begin{align}i_*\gamma^j F_{\mathcal{Z}}[-j]\xrightarrow{g_{\mathcal{Z}/\mathcal{X}}} F_{\mathcal{X}}\xrightarrow{\rho^*} R\rho_* F_{(\tilde{X}, D_{|\tilde{X}}+ E)}\xrightarrow{\partial} i_*\gamma^j F_{\mathcal{Z}}[-j+1], \end{align} $$
    where $\rho :\tilde {X}\to X$ is the blow-up of X along Z and $E=\rho ^{-1}(Z)$. The Gysin map $g_{\mathcal {Z}/\mathcal {X}}$ satisfies an excess intersection formula (7.9.1), it is compatible with smooth base change (Proposition 7.9) and the cup product with Chow classes (Proposition 7.8).

We stress the fact that, in constrast to the $\mathbf {A}^1$-invariant setting, our Gysin sequence does not involve the cohomology of the open complement of $Z\subset X$ but rather, the cohomology of a modulus pair constructed by taking the blow-up of X along Z. When $F= \widetilde {G}$ and $G\in \operatorname {\mathbf {HI}}_{{\operatorname {Nis}}}$, one can in fact verify that (0.1.1) gives back the classical Gysin sequence of Déglise and Voevodsky. For non-$\mathbf {A}^1$-invariant sheaves, the existence of the Gysin map is new essentially in all of the above-mentioned examples: for instance, it does not follow from the work of Gros [Reference GrosGro85] for the de Rham-Witt sheaves. Other interesting cases are given by $F = \widetilde {\mathrm {Conn}^1}$ or $\widetilde {\mathrm {Lisse}^1}$ (see Corollary 11.6). We may also apply (0.1.1) for $D=\emptyset $ and F the whole de Rham-Witt complex and obtain in this way a Gysin sequence for the crystalline cohomology $Ru_{X*}\mathcal {O}_{X/W_n}$, where $u_X: (X/W_n)_{\mathrm {crys}}\to X_{{\operatorname {Nis}}}$ is the natural map of sites, which generalises to higher codimension the classical sequence induced by the residue map along a smooth closed divisor(see Corollary 11.10 and the following remark).

The key computation leading to the above results is the vanishing $H^i(Y, F_{(Y,\rho ^*L)})=0$, for $i\ge 1$, where $\rho : Y\to \mathbf {A}^n$ is the blow-up in the origin and $L\subset \mathbf {A}^n$ a hyperplane passing through the origin (see Theorem 2.12). The proof of this theorem occupies almost all of Section $2$ and relies deeply on the theory of modulus sheaves with transfers.

By factoring any projective morphism as a closed embedding followed by a projection from a projective bundle, we can use Theorem 0.1 to construct pushforward maps (in fact, we construct the pushforward with proper support along a quasi-projective morphism; see Definition 8.5 and Proposition 8.6 for the main properties). Note that the pushforward is compatible with composition, smooth base change and cup product with Chow classes (see 9.5 and Theorem 9.7).

For $F = {\underline {\omega }}^{\operatorname {\mathbf {CI}}} W_m\Omega ^i$, the construction gives even a refinement of the pushforward map for cohomology of Hodge-Witt differentials constructed by Gros [Reference GrosGro85] (see Corollary 0.6 below).

0.3 Chow correspondences

When a cohomology theory is equipped with pushforward with proper support and a cup product with cycles, it is possible, with a bit of extra work, to produce an action of Chow correspondences. Let S be a separated k-scheme of finite type, and let $C_S$ be the category whose objects are maps $(f\colon X\to S)$ with the property that the induced map $X\to \operatorname {Spec}(k)$ is smooth and quasi-projective. As for morphisms, we set (if Y is connected):

$$\begin{align*}C_S(X,Y) = {\operatorname{CH}}^{\dim Y}_{\Phi^{\mathrm{prop}}_{X\times_S Y}}(X\times Y),\end{align*}$$

where $\Phi ^{\mathrm {prop}}_{X\times _S Y}$ is the family of supports on $X\times Y$ consisting of closed subsets which are contained in $X\times _S Y$, and that are proper over X. Composition is given by the usual composition of correspondences using the refined intersection product [Reference FultonFul98, Chapter 16]. If $F^{\bullet }$ is a bounded below complex of reciprocity sheaves and $(f\colon X\to S)$ and $(g\colon Y\to S)$ are objects of $C_S$, we can define for $\alpha \in C_S(X,Y)$ a morphism:

$$\begin{align*}\alpha^* \colon Rg_* F^{\bullet}_Y \to Rf_* F^{\bullet}_X \text{ in } D^+(S_{{\operatorname{Nis}}})\end{align*}$$

that is compatible with the composition of correspondences, satisfies a projection formula and gives back the pushforward for reciprocity sheaves when $\alpha = [\Gamma _h^t]$ is the transpose of the graph of a proper S-morphism $h\colon X\to Y$ (see Proposition 9.10).

For homotopy invariant sheaves, the existence of the action of Chow correspondences follows from work of Rost [Reference RostRos96] and Déglise [Reference DégliseDég12] (although, to our knowledge, this has not been spelled out explicitly in the literature).

Previous instances of constructions of an action of Chow correspondences on the cohomology of Hodge and Hodge-Witt differentials can be found in [Reference Chatzistamatiou and RüllingCR11] and [Reference Chatzistamatiou and RüllingCR12]. However, we remark that the approach followed in this paper is conceptually different: in [Reference Chatzistamatiou and RüllingCR11] and [Reference Chatzistamatiou and RüllingCR12], the existence of the whole de Rham and de Rham-Witt complex, with its structure of graded algebra, was used. In contrast, here, the projective pushforward is directly constructed starting from a single reciprocity sheaf F (and its twists). Our statements are also finer, since we get morphisms defined at the level of derived categories, rather than just between the cohomology groups.

0.4 Applications

Let us now discuss how we can apply the formulas established so far to get new interesting invariants.

0.4.1 Obstructions to the existence of zero cycles of degree $1$

In Section 10.1, we explain how to use the proper correspondence action on the cohomology of an arbitrary reciprocity sheaf to construct very general obstructions of Brauer-Manin type to the existence of zero cycles on smooth projective varieties over function fields, recovering the classical obstruction as a special case.

Here is the main result (see Theorem 10.1):

Theorem 0.2. Let $f\colon Y\to X$ be a dominant quasi-projective morphism between connected smooth k-schemes. Assume that there are integral subschemes $V_i\subset Y$, which are proper, surjective and generically finite over X of degree $n_i$, $i=1,\ldots , s$. Set $N=\mathrm {gcd}(n_1,\ldots , n_s)$. Let $F^{\bullet }\in \mathrm {Comp}^+({\operatorname {\mathbf {RSC}}}_{\operatorname {Nis}})$ be a bounded below complex of reciprocity sheaves. Then there exists a morphism $\sigma : Rf_* F_Y^{\bullet }\to F_X^{\bullet }$ in $D(X_{\operatorname {Nis}})$, such that the composition:

$$\begin{align*}F_X^{\bullet}\xrightarrow{f^*} Rf_*F_Y^{\bullet}\xrightarrow{\sigma} F_X^{\bullet} \end{align*}$$

is multiplication with N.

In particular, if f is proper and $f^*\colon H^i(X, F^{\bullet }_X)\to H^i(Y, F^{\bullet }_Y)$ is not split injective, then the generic fibre of f cannot have index $1$, that is, there cannot exist a zero cycle of degree $1$. It is then possible to assemble the morphisms $\sigma $ in order to produce a generalisation of the classical Brauer-Manin obstruction in the case of the function field of a curve (see (10.2.3) and the references there for more details). This is explained in Corollary 10.4.

See also the end of Section 10.1 for a comprehensive list of references to previous works where unramified cohomology groups have been used to study obstructions to the local-global principle for rational points, rather than for zero cycles, over special types of global fields.

0.4.2 Birational invariants

Once we have established an action of Chow correspondences on the cohomology of reciprocity sheaves, this can be used to find birational invariants.

Let us fix again a separated k-scheme of finite type S. We say that $(f\colon X\to S)$ and $(g\colon Y\to S)\in C_S$, with X and Y integral, are properly birational over S if there exists an integral scheme Z (that we call proper birational correspondence) over S and two proper birational S-morphisms $Z\to X$, $Z\to Y$ (note that we don’t assume that f or g is proper). If we let $Z_0\subset X\times Y$ be the image of $Z\to X\times Y$, we can then look at the composition $[Z_0]^*\circ [Z_0^t]^*$ and get, for example, the following result.

Theorem 0.3 (see Theorem 10.10).

Let $F\in {\operatorname {\mathbf {RSC}}}_{\operatorname {Nis}}$, and assume that $F(\xi )=0$, for all points $\xi $ which are finite and separable over a point of X or Y of codimension $\ge 1$. Then any proper birational correspondence between X and Y induces an isomorphism:

$$\begin{align*}Rg_*F_Y\xrightarrow{\simeq} Rf_* F_X.\end{align*}$$

If $Y=S$ in the statement of Theorem 0.3, we get a vanishing $R^i f_* F_X = 0$ for $i\geq 1$ and for any projective birational morphism $f\colon X\to Y$ and F as in the theorem. The prototype example of a sheaf satisfying the condition $F(\xi )=0$ is the sheaf of top differential forms, $\Omega ^{\dim X}_{/k}$. For this, the birational invariance is classical in characteristic zero and follows from Hironaka’s resolution of singularities. In positive characteristic, it was proven in [Reference Chatzistamatiou and RüllingCR11] by using a similar action of Chow correspondences (although the statements in loc.cit. were for the cohomology groups, not for the whole complexes in the derived category; see also [Reference KovácsKov17]). On the other hand, Theorem 0.3 provides a very general class of birational invariants, many of which are new to us: for example, using results of Geisser-Levine [Reference Geisser and LevineGL00], we can consider the cohomology of the étale motivic complexes $R^i\varepsilon _*( \mathbb {Z}/p^n(d))$ (for all i and n, if $\mathrm { char}(k)=p>0$), where $d=\dim X = \dim Y$; see Corollary 11.16 for a more extensive list). Among the other applications, we can use Theorem 0.3 to generalise parts of [Reference PirutkaPir12, Theorem 3.3] (which generalises [Reference Colliot-Thélène and VoisinCTV12, Proposition 3.4]; see Corollary 11.19 for more details).

We remark that the global sections of reciprocity sheaves enjoy a general invariance under proper (stable) birational correspondences, without assuming $F(\xi )=0$ for $\xi $ as above (see Theorem 10.7 and the notations there).

As a byproduct of 0.2, we also get (stably) proper birational invariance (see Definition 10.2) for the n-torsion of the relative Picard scheme, $\operatorname {Pic}_{X/S}[n]$, for all n and any flat, geometrically integral and projective morphism $X\to S$ between smooth connected k-schemes, such that the generic fibre has index $1$. This is classical and known to the experts if S is the spectrum of an algebraically closed field, but it is new for general S (see Corollary 11.24).

0.4.3 Decomposition of the diagonal

In section 10.3, we investigate the implications of the cycle action in case we have a decomposition of the diagonal, a method which was first employed in [Reference Bloch and SrinivasBS83]. For example, we obtain:

Theorem 0.4 (see Theorem 10.13).

Let $f\colon X\to S$ be a smooth projective morphism, where S is the henselisation of a smooth k-scheme in a 1-codimensional point or a regular connected affine scheme of dimension $\leq 1$ and of finite type over a function field K over k. Assume that the diagonal cycle $[\Delta _{X_{\eta }}]$ of the generic fibre $X_{\eta }$ of f has an integral decomposition. Then, for any $F\in {\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$, the pullback along f induces an isomorphism:

$$\begin{align*}H^0(X, F)\cong H^0(S, F).\end{align*}$$

See Remark 10.14 for some conditions under which the diagonal decomposes. Note that in the case $F = R^{i} \varepsilon _* \mathbb {Z}/p^n(j)$, with $(i,j)\neq (0,0)$, and X is defined over an algebraically closed field of characteristic $p>0$ and admits an integral decomposition of the diagonal, we obtain $H^0(X, R^{i} \varepsilon _* \mathbb {Z}/p^n(j))=H^0(\operatorname {Spec} k,R^{i} \varepsilon _* \mathbb {Z}/p^n(j))=0$. (The vanishing follows from [Reference Geisser and LevineGL00].) This immediately implies a positive answer to Problem 1.2 of [Reference Auel and BigazziABBvB19] and reproves Theorem 1 in loc. cit. (see Corollary 11.21; also see the recent work [Reference OtabeOta20], for a different approach).

In the case $S=\operatorname {Spec} k$ and F is $\mathbf {A}^1$-invariant, Theorem 0.4 is classical; Totaro proved that it also holds for $F=\Omega ^i_{-/k}$ (see [Reference TotaroTot16, Lemma 2.2]) and – building on ideas of Voisin and Colliot-Thélène-Pirutka – used this to find many new examples of hypersurfaces that are not stably rational. It is an interesting question, whether the flexibility in the choice of the sheaf F coming from Theorem 0.4 — for example, F can be any quotient of $\Omega ^i_{-/k}$, say $F=\Omega ^N_{-/k}/\operatorname {dlog} K_N^M$ from Corollary 11.16 — can be used to find new examples of nonstably rational varieties.

Results for higher cohomology groups are also obtained if F satisfies certain extra assumptions (see Theorems 10.15 and 10.16 and Corollary 11.22 for examples).

0.4.4 Cohomology of ordinary varieties

Following Bloch-Kato [Reference Bloch and KatoBK86] and Illusie-Raynaud [Reference Illusie and RaynaudIR83], we say that a variety X over a perfect field k of characteristic $p>0$ is ordinary if $H^m(X, B^r_X)=0$ for all m and r, where $B^r_X = \mathrm {Im}(d\colon \Omega ^{r-1}_X\to \Omega ^{r}_X)$. It is equivalent to ask that the Frobenius $F\colon H^q(X, W\Omega ^r_X)\to H^q(X, W\Omega ^r_X)$ is bijective for all q and r. If X is an abelian variety A, this recovers the property that the p-rank of A is the maximum possible, namely, equal to its dimension. For them, we have the following result.

Corollary 0.5 (see Corollary 11.14).

Let $f\colon X\to S$ be a surjective morphism between smooth projective connected k-schemes. Assume that the generic fibre has index prime to p. Then:

$$\begin{align*}X \text{ is ordinary } \Longrightarrow S \text{ is ordinary.}\end{align*}$$

Note that the assumption on the generic fibre is of course guaranteed if $X_{k(S)}$ has a zero cycle of degree prime to p (for example, when $X_{k(S)}$ is an abelian variety). Similar implications hold for the properties ‘X is Hodge-Witt’ or ‘the crystalline cohomology of X is torsion-free’ (see Remark 11.15).

In connection to ordinary varieties, let us also mention the following result (see Corollary 11.12):

Corollary 0.6. Let $f\colon Y\to X$ be a morphism of relative dimension $r\ge 0$ between smooth projective k-schemes. Assume that X is ordinary. Then the Ekedahl-Grothendieck pushfoward (see [Reference GrosGro85, Chapter II, 1]) factors via:

(0.6.1)$$ \begin{align}R\Gamma(Y, W_n\Omega^q_Y)[r]\to R\Gamma(Y, W_n\Omega^q_Y/B_{n,\infty}^q)[r] \xrightarrow{f_*} R\Gamma(X, W_n\Omega^{q-r}_X), \end{align} $$

where $B_{n,\infty }^q= \bigcup _s F^{s-1}d W_{n+s-1}\Omega ^{q-1}$ (see [Reference Illusie and RaynaudIR83, Chapter IV, (4.11.2)]) and $f_*$ is induced by the pushforward from 9.5.

Note that this is an essentially immediate consequence of the fact that the sheaves $B^q_{n, \infty }$ are reciprocity sheaves, our general formalism and the computation of the twists of Theorem 11.8. In fact, even when X is not ordinary, we always obtain a factorisation in top degree:

$$\begin{align*}R\Gamma(Y, W_n\Omega^r_Y)[r]\to R\Gamma(Y, W_n\Omega^r_Y/B_{n,\infty}^r)[r] \xrightarrow{f_*} R\Gamma(X, W_n\mathcal{O}_X) \end{align*}$$

as a byproduct of the proof of Corollary 11.12.

0.4.5 Relationship with logarithmic motives

In [Reference Binda, Park and ØstværBPØ22], Park, Østvær and the first author recently introduced a triangulated category of logarithmic motives over a field k. Similar in spirit to Voevodsky’s construction, the starting point is the category $lSm/k$ of log smooth (fs)-log schemes over k, promoted then to a category of correspondences. The localisation with respect to a new Grothendieck topology, called the dividing-Nisnevich topology, and with respect to the log scheme ${\overline {\square }}$, the log compactification of $\mathbf {A}^1_k$, produces the category denoted by $\mathbf {logDM}^{\mathrm {eff}}_{dNis}(k)$.

A theorem of Saito (see [Reference SaitoSai20b]) shows that there exists a fully faithful exact functor:

$$\begin{align*}\mathcal{L}og\colon {\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}} \to \mathbf{Shv}^{\mathrm{ltr}}_{dNis}(k, \mathbb{Z}),\end{align*}$$

such that $\mathcal {L}og(F)$ is strictly ${\overline {\square }}$-invariant in the sense of [Reference Binda, Park and ØstværBPØ22, Definition 5.2.2], where the target is the category of dividing Nisnevich sheaves with log transfers on $lSm/k$ (see [Reference Binda, Park and ØstværBPØ22, Section 2.4]). This shows that Nisnevich cohomology of reciprocity sheaves is representable in $\mathbf {logDM}^{\mathrm {eff}}_{dNis}(k)$. Formulas like the projective bundle formula, the blow-up formula, the existence of the Gysin sequence and so on in $\mathbf {logDM}^{\mathrm {eff}}_{dNis}(k)$ can then be used to rededuce a posteriori some of the results in the present paper, under some auxiliary assumptions. We warn the reader that in the proof of the main result of [Reference SaitoSai20b], one needs in an essential way the formalism of pushforward maps along projective morphisms that we show in the present work.

Moreover, note that the motivic formulas given in [Reference Binda, Park and ØstværBPØ22] cannot be used to deduce results involving higher modulus, that we do instead systematically in the present paper, and that the projective bundle formula, the blow-up formula and the Gysin triangle (using the identification of the log Thom space) in [Reference Binda, Park and ØstværBPØ22] are only proved under the assumptions of resolution of singularities, which we don’t need. Finally, a general theory of log motives over a base (not just over a field) would be necessary to get the full strength of the sheaf-theoretic version of the results in this work.

Warning. The content of Theorem 0.1 and of other main results in this paper (namely, Corollary 2.19 and Theorem 3.1) are a sheaf theoretic analogue to some of the results on motives with modulus in [Reference Kahn, Miyazaki, Saito and YamazakiKMSY20], more precisely to [Reference Kahn, Miyazaki, Saito and YamazakiKMSY20, Theorem 7.3.2], [Reference Kahn, Miyazaki, Saito and YamazakiKMSY20, Theorem 7.4.3] and [Reference Kahn, Miyazaki, Saito and YamazakiKMSY20, Theorem 7.4.4] (the latter being in fact a theorem of Keiho Matsumoto [Reference MatsumotoMat22], proved only for the inclusion of a smooth divisor Z in X, whereas we consider the case of Z being a smooth closed subscheme of any codimension). We warn the reader that our results cannot be recovered from the existing literature: for this to be the case, it would be necessary to show that the cohomology of ${\overline {\square }}$-invariant sheaves is representable in the category of motives with modulus $\mathbf {\underline {M}DM}^{\mathrm {eff}}(k)$ constructed in [Reference Kahn, Miyazaki, Saito and YamazakiKMSY20]. In view of [Reference Kahn, Miyazaki, Saito and YamazakiKMSY20, Theorem 5.2.4], one would require a positive answer to the following two questions.

Question 0.7.

  1. (1) Is the Nisnevich cohomology of ${\overline {\square }}$-invariant sheaves invariant under blow-up with centre contained in the support of the modulus?

  2. (2) Is a ${\overline {\square }}$-invariant sheaf F equivalent (in the derived category of sheaves) to its derived Suslin complex $RC_*^{{\overline {\square }}}(F)$ defined in [Reference Kahn, Miyazaki, Saito and YamazakiKMSY20, Definition 5.2.3]?

Both questions seem out of reach for general ${\overline {\square }}$-invariant sheaves: note that (1) would amount to answering affirmatively to [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Question 1, p.4], and that a (weaker) version of it is the content of Theorem 2.12, which is one of the crucial technical results of this paper.

Question (2) is equivalent to asking whether the cohomology of a ${\overline {\square }}$-invariant sheaf with transfers is again ${\overline {\square }}$-invariant. For $\mathbb {A}^1$-invariant sheaves with transfers, this is a deep theorem of Voevodsky and boils down to studying a nontrivial interaction between the Nisnevich sheafification functor and the localisation functor $L_{\mathbb {A}^1}(-)$. For semipure sheaves (cf. 1.4 below), this is shown in [Reference SaitoSai20a], but the general case is wide open (the first and third author once claimed the general case in characteristic 0, but a gap was found in its proof). We hope that the main results of this paper are useful in attempts to answer the above open questions.

Moreover, even if both questions are answered positively, in order to get the full statement of Theorem 0.1 from the motivic point of view, it would be necessary to develop the whole theory of motives with modulus over a base, which is not available at the moment.

0.5 Organisation of the paper

We conclude this introduction with a quick presentation of the structure of the paper.

In §1, we discuss some preliminaries and fix the notation. Nothing in this section is new, and it can be found in [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a], [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b], [Reference Merici and SaitoMS20]. In §2, we prove a key ‘descent’ property for ${\overline {\square }}$-invariant sheaves, namely, Proposition 2.5. This is a crucial technical result that allows us to prove the invariance of the cohomology of cube invariant sheaves along a certain class of blow-ups (see Theorem 2.12). Once this is established, we proceed to prove that the cohomology of cube invariant sheaves is also invariant with respect to the product with the modulus pair $(\mathbf {P}^n, \mathbf {P}^{n-1})$, Theorem 2.18. In §3, we prove a smooth blow-up formula; and in §4, we introduce the twist and prove some of its basic properties. In §5, we use Rost’s theory of cycle modules together with a formula for the tensor product of reciprocity sheaves to construct the cup product with Chow classes. In §6, we prove the projective bundle formula, and in §7, we construct the Gysin sequence: for this, we essentially follow the steps of Voevodsky’s construction in [Reference VoevodskyVoe00b], but we also get a finer theory with supports (the local Gysin map). In §8, we assemble the Gysin maps and the morphisms induced by the projective bundle formula to construct general pushforwards. In this section, we make use also of the cancellation theorems of [Reference Merici and SaitoMS20]. In §9, we explain the construction of the action of Chow correspondences on reciprocity sheaves (and complexes of sheaves). Finally, in §10 and §11, we collect the main applications and a list of examples of reciprocity sheaves. The reader who is mostly interested in examples and applications may read the last two sections without having precise knowledge of modulus sheaves with transfers.

In the paper, we use frequently the results from [Reference SaitoSai20a], which plays a fundamental role for us.

1 Preliminaries

1.1 Notations and conventions

In the whole paper, we fix a perfect base field k. We denote by $\operatorname {\mathbf {Sm}}$ the category of smooth separated k-schemes. We write $\mathbf {P}^1= \mathbf {P}^1_k$ etc. and $X\times Y= X\times _k Y$ for k-schemes X, Y. For a function field $K/k$, we denote by $K\{x_1, \ldots , x_n\}$ the henselisation of $K[x_1,\ldots , x_n]_{(x_1,\ldots. x_n)}$. Let R be a regular noetherian k-algebra. By [Reference PopescuPop86, Theorem 1.8] and [Reference Artin, Grothendieck and VerdierAGV72, Exp I, Proposition 8.1.6], we can write $R=\varinjlim _i R_i$, where $(R_i)_i$ is a directed system of smooth k-algebras, and we use the notation $F(R)=\varinjlim _i F(\operatorname {Spec} R_i)$, for any presheaf F on $\operatorname {\mathbf {Sm}}$. If X is a scheme and F is a Nisnevich sheaf on X, we will denote by $H^i(X, F)=H^i(X_{\operatorname {Nis}}, F)$ the ith cohomology group of F on the small Nisnevich site of X, similar with higher direct images. We denote by $X_{(n)}$ (respectively, $X^{(n)}$) the set of n (respectively, co-) dimensional points in X.

1.2 A recollection on modulus sheaves with transfers

We recall some terminology and notations from the theory of modulus sheaves with transfers (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a], [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b], [Reference Kahn, Saito and YamazakiKSY22] and [Reference SaitoSai20a] for details).

1.1. A modulus pair $\mathcal {X}=(\overline {X}, X_{\infty })$ consists of a separated k-scheme of finite type $\overline {X}$ and an effective (or empty) Cartier divisor $X_{\infty }$, such that $X:= \overline {X}\setminus |X_{\infty }|$ is smooth; it is called proper if $\overline {X}$ is proper over k. Given two modulus pairs $\mathcal {X}=(\overline {X}, X_{\infty })$ and $\mathcal {Y}=(\overline {Y}, Y_{\infty })$, with opens $X:=\overline {X}\setminus |X_{\infty }|$ and $Y:=\overline {Y}\setminus |Y_{\infty }|$, an admissible left proper prime correspondence from $\mathcal {X}$ to $\mathcal {Y}$ is given by an integral closed subscheme $Z\subset X\times Y$ which is finite and surjective over a connected component of X, such that the normalisation of its closure $\overline {Z}^N\to \overline {X}\times \overline {Y}$ is proper over $\overline {X}$ and satisfies:

(1.1.1)$$ \begin{align}X_{\infty|\overline{Z}^N}\ge Y_{\infty|\overline{Z}^N}, \end{align} $$

as Weil divisors on $\overline {Z}^N$, where $X_{\infty |\overline {Z}^N}$ (respectively, $Y_{\infty |\overline {Z}^N}$) denotes the pullback of $X_{\infty }$ (respectively, $Y_{\infty }$) to $\overline {Z}^N$. The free abelian group generated by such correspondences is denoted by $\operatorname {\mathbf {\underline {M}Cor}}(\mathcal {X}, \mathcal {Y})$. By [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Propositions 1.2.3 and 1.2.6], modulus pairs and left proper admissible correspondences define an additive category that we denote by $\operatorname {\mathbf {\underline {M}Cor}}$. We write $\operatorname {\mathbf {MCor}}$ for the full subcategory of $\operatorname {\mathbf {\underline {M}Cor}}$, whose objects are proper modulus pairs. We denote by $\tau $ the inclusion functor $\tau \colon \operatorname {\mathbf {MCor}} \to \operatorname {\mathbf {\underline {M}Cor}}$. The induced category of additive presheaves of abelian groups is denoted by $\operatorname {\mathbf {\underline {M}PST}}$ (respectively, $\operatorname {\mathbf {MPST}}$). We have functors:

$$\begin{align*}\omega\colon \operatorname{\mathbf{MCor}} \to \operatorname{\mathbf{Cor}}, \quad {\underline{\omega}}\colon \operatorname{\mathbf{\underline{M}Cor}} \to \operatorname{\mathbf{Cor}}\end{align*}$$

given by $({\underline {X}}, X_{\infty }) \mapsto {\underline {X}} \setminus |X_{\infty }|$, where $\operatorname {\mathbf {Cor}}$ is the category of finite correspondences introduced by Suslin-Voevodsky (see, e.g. [Reference Mazza, Voevodsky and WeibelMVW06]). Note that there is also a fully faithful functor:

$$\begin{align*}\operatorname{\mathbf{Cor}}\to \operatorname{\mathbf{\underline{M}Cor}}, \quad X\mapsto (X,\emptyset).\end{align*}$$

We will abuse notation by writing:

(1.1.2)$$ \begin{align}X=(X,\emptyset)\in \operatorname{\mathbf{\underline{M}Cor}}, \quad \text{for } X\in \operatorname{\mathbf{Sm}}. \end{align} $$

Write $\tau ^*$ for the restriction functor along $\tau $, and write $\tau _!$ for its left Kan extension. Similarly, write $\omega ^*$ (respectively, ${\underline {\omega }}^*$) for the restriction functor along $\omega $ (respectively, ${\underline {\omega }}$) and $\omega _!$ (respectively, ${\underline {\omega }}_!$) for its left Kan extension. We have the following commutative diagrams at our disposal:

(1.1.3)

Here, ${\operatorname {\mathbf {PST}}}$ is the category of presheaves of abelian groups on $\operatorname {\mathbf {Cor}}$, the functors in the left triangle are left adjoint to the functors in the right triangle, all the functors are exact, the diagrams commute and we have $\tau ^*F(\mathcal {X})=F(\mathcal {X})$, $\underline {\omega }^*F(\mathcal {X})=F(X)$ and:

(1.1.4)$$ \begin{align}\underline{\omega}_!F(X)=F(X,\emptyset)\overset{(\scriptsize{1.1.2})}{=:}F(X) \end{align} $$

for $\mathcal {X} = (\overline {X}, X_{\infty })$ and $X = \overline {X}\setminus |X_{\infty }|$.

We denote by ${\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})$ the presheaf on $\operatorname {\mathbf {\underline {M}Cor}}$ (respectively, $\operatorname {\mathbf {MCor}}$) represented by $\mathcal {X}$ in $\operatorname {\mathbf {\underline {M}Cor}}$ (respectively, in $\operatorname {\mathbf {MCor}}$). We have $\tau _!{\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})={\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})$ and $\underline {\omega }_!{\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})={\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(X)$.

Let $\mathcal {X}=(\overline {X}, X_{\infty })$, $\mathcal {Y}=(\overline {Y}, Y_{\infty })\in \operatorname {\mathbf {\underline {M}Cor}}$. We set:

$$\begin{align*}\mathcal{X}\otimes \mathcal{Y} := (\overline{X}\times \overline{Y}, p^*X_{\infty}+ q^*Y_{\infty}),\end{align*}$$

where p and q are the projections from $\overline {X}\times \overline {Y}$ to $\overline {X}$ and $\overline {Y}$, respectively. In fact, this defines a symmetric monoidal structure on $\operatorname {\mathbf {\underline {M}Cor}}$ (respectively, $\operatorname {\mathbf {MCor}}$) which extends (via Yoneda) uniquely to a right exact monoidal structure $\otimes $ on $\operatorname {\mathbf {\underline {M}PST}}$ (respectively, $\operatorname {\mathbf {MPST}}$). Similarly, there is a monoidal structure on ${\operatorname {\mathbf {PST}}}$. The functors $\underline {\omega }_!$, $\omega _!$, $\tau _!$ are monoidal, since they are all defined as left Kan extensions of the functors $\underline {\omega }, \omega $ and $\tau $, which are clearly monoidal. For $F\in \operatorname {\mathbf {\underline {M}PST}}$, the functor $(-)\otimes F:\operatorname {\mathbf {\underline {M}PST}}\to \operatorname {\mathbf {\underline {M}PST}}$ admits a right adjoint denoted by $\operatorname {\underline {Hom}}_{\operatorname {\mathbf {\underline {M}PST}}}(F,-)$; similar with $F\in \operatorname {\mathbf {MPST}}$ (see, e.g. [Reference Mazza, Voevodsky and WeibelMVW06, Chapter 8]).

1.2. For $F\in \operatorname {\mathbf {\underline {M}PST}}$ and $\mathcal {X}=(\overline {X}, X_{\infty })\in \operatorname {\mathbf {\underline {M}Cor}}$, denote by $F_{\mathcal {X}}$ the presheaf:

(1.2.1)$$ \begin{align}({\operatorname{\acute{e}t}}/\overline{X})^{{\operatorname{op}}}\ni U\mapsto F_{\mathcal{X}}(U):= F(U, X_{\infty|U}), \end{align} $$

where $({\operatorname {\acute {e}t}}/\overline {X})$ denotes the category of all étale maps $U\to \overline {X}$. We say F is a Nisnevich sheaf if $F_{\mathcal {X}}$ is a Nisnevich sheaf, for all $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$. We denote by $\operatorname {\mathbf {\underline {M}NST}}$ the full subcategory of $\operatorname {\mathbf {\underline {M}PST}}$ consisting of Nisnevich sheaves.

We say $F\in \operatorname {\mathbf {MPST}}$ is a Nisnevich sheaf if $\tau _!F$ is and denote the corresponding full subcategory by $\operatorname {\mathbf {MNST}}$. The functors in (1.1.3) restrict to Nisnevich sheaves and have the same adjointness and exactness properties (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b, 4.2.5, 5.1.1, 6.2.1]). Furthermore, there are Nisnevich sheafification functors:

$$\begin{align*}{\underline{a}}_{{\operatorname{Nis}}}: \operatorname{\mathbf{\underline{M}PST}}\to \operatorname{\mathbf{\underline{M}NST}}, \quad a_{\operatorname{Nis}}: \operatorname{\mathbf{MPST}}\to \operatorname{\mathbf{MNST}},\end{align*}$$
$$\begin{align*}a_{{\operatorname{Nis}}}^V: {\operatorname{\mathbf{PST}}}\to \operatorname{\mathbf{NST}},\end{align*}$$

which are left adjoint to the forgetful functors, restrict to the identity on Nisnevich sheaves and satisfy:

(1.2.2)$$ \begin{align} \underline{\omega}_! {\underline{a}}_{\operatorname{Nis}}= a^V_{\operatorname{Nis}}\underline{\omega}_!, \quad \omega_! a_{\operatorname{Nis}}= a^V_{\operatorname{Nis}}\omega_!, \quad \tau_! a_{{\operatorname{Nis}}}= {\underline{a}}_{\operatorname{Nis}} \tau_! ,\end{align} $$

and:

(1.2.3)$$ \begin{align}a_{{\operatorname{Nis}}}\omega^*=\omega^* a_{{\operatorname{Nis}}}^V, \quad {\underline{a}}_{{\operatorname{Nis}}}\underline{\omega}^*=\underline{\omega}^* a^V_{\operatorname{Nis}} \end{align} $$

(see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Theorem 2], [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b, Theorems 4.2.4, 4.2.5 and 6.2.1]; $a_{\operatorname {Nis}}^V$ was constructed by Voevodsky). It follows that $\operatorname {\mathbf {NST}}$, $\operatorname {\mathbf {\underline {M}NST}}$ and $\operatorname {\mathbf {MNST}}$ are Grothendieck abelian categories and that the sheafification functors are exact. For $F\in \operatorname {\mathbf {\underline {M}PST}}$ and $\mathcal {X}=(\overline {X}, X_{\infty })\in \operatorname {\mathbf {\underline {M}Cor}}$, we have:

(1.2.4)$$ \begin{align}{\underline{a}}_{\operatorname{Nis}}(F)(\mathcal{X})= \varinjlim_{f:\overline{Y}\to \overline{X}}F_{(\overline{Y}, f^*X_{\infty}), {\operatorname{Nis}}}(\overline{Y}), \end{align} $$

where the limit is over all proper morphisms $f:\overline {Y}\to \overline {X}$ which restrict to an isomorphism over $X=\overline {X}\setminus |X_{\infty }|$ and $F_{(\overline {Y}, f^*X_{\infty }), {\operatorname {Nis}}}$ denotes the Nisnevich sheafification of the presheaf $F_{(\overline {Y}, f^*X_{\infty })}$ on the site $\overline {Y}_{\operatorname {Nis}}$ (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Theorem 2(1)]. In the following, we will use the notation:

$$\begin{align*}F_{\operatorname{Nis}}:= {\underline{a}}_{\operatorname{Nis}}(F), \quad H_{\operatorname{Nis}}:= a_{\operatorname{Nis}}^V(H), \quad F\in \operatorname{\mathbf{\underline{M}PST}}, \,H\in {\operatorname{\mathbf{PST}}}.\end{align*}$$

Lemma 1.3. A morphism $\varphi :F\to G$ in $\operatorname {\mathbf {\underline {M}NST}}$ is surjective (i.e. has vanishing cokernel) if for all $\mathcal {X}=(\overline {X}, X_{\infty })\in \operatorname {\mathbf {\underline {M}Cor}}$, with $\overline {X}$ normal, and all $x\in \overline {X}$, the morphism:

$$\begin{align*}F(\mathcal{X}_{(x)})\to G(\mathcal{X}_{(x)})\end{align*}$$

is surjective, where $\mathcal {X}_{(x)}=(\overline {X}_{(x)}, X_{\infty |\overline {X}_{(x)}})$ and $\overline {X}_{(x)}= \operatorname {Spec} \mathcal {O}_{\overline {X},x}^h$ is the spectrum of the henselisation of the local ring $\mathcal {O}_{\overline {X},x}$.

Proof. Let C be the cokernel of $\varphi $ in $\operatorname {\mathbf {\underline {M}PST}}$. We want to show ${\underline {a}}_{\operatorname {Nis}}(C)=0$. For $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$, set $C_{\mathcal {X}}= \operatorname {Coker}(\varphi _{\mathcal {X}}: F_{\mathcal {X}}\to G_{\mathcal {X}})$ in the category of presheaves on $({\operatorname {\acute {e}t}}/\overline {X})$; denote by $C_{\mathcal {X},{\operatorname {Nis}}}$ its Nisnevich sheafification. By (1.2.4), it suffices to show $C_{\mathcal {X},{\operatorname {Nis}}}=0$, if $\overline {X}$ is normal. The latter is equivalent to the surjectivity of $\varphi _{\mathcal {X}}$ in the category of Nisnevich sheaves on $\overline {X}$, which is equivalent to the statement.

1.4. Set ${\overline {\square }}:=(\mathbf {P}^1, \infty ) \in \operatorname {\mathbf {MCor}}$. For $F\in \operatorname {\mathbf {\underline {M}PST}}$, we say that:

  1. (1) F is cube invariant if the map $F(\mathcal {X})\to F(\mathcal {X}\otimes {\overline {\square }})$ induced by the pullback along the projection is an isomorphism.

  2. (2) F has M-reciprocity if the counit map $\tau _!\tau ^*F\to F$ is an isomorphism.

  3. (3) F is semipure if the unit map $F\to \underline {\omega }^*\underline {\omega }_! F$ is injective.

We denote by $\operatorname {\mathbf {\underline {M}PST}}^{\tau }$ the full subcategory of $\operatorname {\mathbf {\underline {M}PST}}$ consisting of the objects with M-reciprocity. Note that for $\mathcal {X}$, a proper modulus pair, we have ${\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})\in \operatorname {\mathbf {\underline {M}PST}}^{\tau }$. We denote by $\operatorname {\mathbf {CI}}^{\tau ,sp}$ the full subcategory of $\operatorname {\mathbf {\underline {M}PST}}$ consisting of the cube invariant semipure objects with M-reciprocity. We set:

$$\begin{align*}\operatorname{\mathbf{\underline{M}NST}}^{\tau}:= \operatorname{\mathbf{\underline{M}PST}}^{\tau}\cap \operatorname{\mathbf{\underline{M}NST}}\quad \text{and}\quad \operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}:=\operatorname{\mathbf{CI}}^{\tau,sp}\cap \operatorname{\mathbf{\underline{M}NST}}.\end{align*}$$

By [Reference SaitoSai20a, Theorem 10.1], the sheafification functor ${\underline {a}}_{\operatorname {Nis}}$ restricts to:

(1.4.1)$$ \begin{align}{\underline{a}}_{\operatorname{Nis}}: \operatorname{\mathbf{CI}}^{\tau,sp}\to \operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}. \end{align} $$

The natural inclusion $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}\hookrightarrow \operatorname {\mathbf {\underline {M}PST}}^{\tau }$ has a left adjoint:

(1.4.2)$$ \begin{align}h^{{\overline{\square}},\mathrm{sp}}_{0,{\operatorname{Nis}}}: \operatorname{\mathbf{\underline{M}PST}}^{\tau}\to \operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}} \end{align} $$

given by:

$$\begin{align*}h^{{\overline{\square}},\mathrm{sp}}_{0,{\operatorname{Nis}}}(F)= {\underline{a}}_{{\operatorname{Nis}}}({\underline{h}}_0^{\overline{\square}}(F)^{\mathrm{sp}}),\end{align*}$$

where for $G\in \operatorname {\mathbf {\underline {M}PST}}$:

  1. (1) ${\underline {h}}_0^{\overline {\square }}(G)\in \operatorname {\mathbf {\underline {M}PST}}$ is the maximal cube invariant quotient of G defined by:

    (1.4.3)$$ \begin{align} {\underline{h}}_0^{\overline{\square}}(G)(\mathcal{X})=\operatorname{Coker}(G(\mathcal{X}\otimes {\overline{\square}})\xrightarrow{i_0^*-i_1^*} G(\mathcal{X})), \end{align} $$
    where $i_{\varepsilon }: \{\varepsilon \}\to {\overline {\square }}$, $\varepsilon \in \{0,1\}$, are induced by the natural closed immersions,
  2. (2) $G^{\mathrm {sp}}=\operatorname {Im}(G\to \underline {\omega }^*\underline {\omega }_! G)$ denotes the semipurification of F.

The left adjointness of (1.4.2) to the natural inclusion follows from [Reference Merici and SaitoMS20, Lemma 1.14(i)] and the adjunction $\tau _!\dashv \tau ^*$. We note that for any $F\in \operatorname {\mathbf {\underline {M}PST}}$, the presheaf $h^{{\overline {\square }},\mathrm {sp}}_{0,{\operatorname {Nis}}}(F)$ is defined and is in fact a cube invariant, semipure Nisnevich sheaf on $\operatorname {\mathbf {\underline {M}Cor}}$.

For $\mathcal {X}$ a proper modulus pair, we set:

(1.4.4)$$ \begin{align}h^{{\overline{\square}},\mathrm{sp}}_{0,{\operatorname{Nis}}}(\mathcal{X}):=h^{{\overline{\square}},\mathrm{ sp}}_{0,{\operatorname{Nis}}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}))\in \operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}. \end{align} $$

Lemma 1.5. Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ and $G,H\in \operatorname {\mathbf {\underline {M}PST}}^{\tau }$. Assume there is a surjection ${\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})\rightarrow \!\!\!\!\!\rightarrow G$, for some $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$. We have:

  1. (1) $\operatorname {\underline {Hom}}_{\operatorname {\mathbf {\underline {M}PST}}}(G, F)\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$;

  2. (2) $\operatorname {Hom}_{\operatorname {\mathbf {\underline {M}PST}}}(H\otimes G, F)= \operatorname {Hom}_{\operatorname {\mathbf {\underline {M}PST}}}(h_{0,{\operatorname {Nis}}}^{{\overline {\square }},\mathrm {sp}}(H), \operatorname {\underline {Hom}}_{\operatorname {\mathbf {\underline {M}PST}}}( G, F))$.

Proof. (1). First assume $G={\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})$, for some $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$. In this case, $\operatorname {\underline {Hom}}(G, F)(\mathcal {Y})=F(\mathcal {X}\otimes \mathcal {Y})$. Clearly, this defines a cube invariant Nisnevich sheaf. It has M-reciprocity by [Reference SaitoSai20a, Lemma 1.27(2)] and has semipurity by [Reference SaitoSai20a, Lemma 1.29(2)]. Hence, $\operatorname {\underline {Hom}}(G,F)\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ in this case. In the general, case consider a resolution:

$$\begin{align*}\bigoplus_j {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{Y}_j)\to \bigoplus_i {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}_i)\to G\to 0.\end{align*}$$

We obtain an exact sequence:

(1.5.1)$$ \begin{align}0\to \operatorname{\underline{Hom}}(G, F)\to \prod_i \operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}_i), F)\to \prod_j \operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{Y}_j), F). \end{align} $$

This directly implies cube invariance and semipurity. The sheaf property holds since ${\underline {i}}_{\operatorname {Nis}}{\underline {a}}_{\operatorname {Nis}}: \operatorname {\mathbf {\underline {M}PST}}\to \operatorname {\mathbf {\underline {M}PST}}$ is left exact, where ${\underline {i}}_{\operatorname {Nis}}$ is the forgetful functor. In general, M-reciprocity won’t hold since $\tau _!\tau ^*$ does not commute with infinite products; however, it clearly holds if the first product in (1.5.1) is finite and by assumption we find such a resolution. (2) follows from (1) and adjunction.

1.6. The full subcategory of ${\operatorname {\mathbf {PST}}}$ given by ${\operatorname {\mathbf {RSC}}}:= \underline {\omega }_!\operatorname {\mathbf {CI}}^{\tau ,\mathrm {sp}}$ is called the category of reciprocity presheaves. The full subcategory of $\operatorname {\mathbf {NST}}$ given by ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}:=\underline {\omega }_!\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ is called the category of reciprocity sheaves. It is direct to see that ${\operatorname {\mathbf {RSC}}}$ is an abelian category, closed under subobjects and quotients in ${\operatorname {\mathbf {PST}}}$. On the other hand, it is a theorem [Reference SaitoSai20a, Theorem 0.1] that ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$ is also abelian. We use the following notation for a proper modulus pair $\mathcal {X}$:

$$\begin{align*}h_0(\mathcal{X}):=\underline{\omega}_!(h_0^{\overline{\square}}(\mathcal{X}))= \underline{\omega}_!(h_0^{{\overline{\square}}, \mathrm{ sp}}(\mathcal{X}))\in {\operatorname{\mathbf{RSC}}},\end{align*}$$

and:

$$\begin{align*}h_{0,{\operatorname{Nis}}}(\mathcal{X}):=\underline{\omega}_!(h_{0,{\operatorname{Nis}}}^{\overline{\square}}(\mathcal{X}))=\underline{\omega}_!(h_{0,{\operatorname{Nis}}}^{{\overline{\square}},\mathrm{ sp}}(\mathcal{X}))\in {\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}.\end{align*}$$

Note that $h_{0,{\operatorname {Nis}}}(\mathcal {X})=h_0(\mathcal {X})_{\operatorname {Nis}}$. By [Reference Merici and SaitoMS20, (1.13)] (see also [Reference Kahn, Saito and YamazakiKSY22, Proposition 2.3.7]), there is an adjunction:

(1.6.1)

where $\underline {\omega }^{\operatorname {\mathbf {CI}}}$ is right adjoint to $\underline {\omega }_!$ and is given by:

$$\begin{align*}\underline{\omega}^{\operatorname{\mathbf{CI}}}(F)=\tau_!\operatorname{Hom}_{\operatorname{\mathbf{MPST}}}(h_0^{{\overline{\square}}}(-), \omega^*F).\end{align*}$$

In the notation of [Reference Kahn, Saito and YamazakiKSY22], we have $\underline {\omega }^{\operatorname {\mathbf {CI}}}=\tau _!\omega ^{\operatorname {\mathbf {CI}}}$.

Recall that Voevodsky’s category of homotopy invariant Nisnevich sheaves, $\operatorname {\mathbf {HI}}_{{\operatorname {Nis}}}$, is an abelian subcategory of ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$, and thanks to [Reference VoevodskyVoe00a, Theorem 5.6], the natural inclusion $\operatorname {\mathbf {HI}}_{{\operatorname {Nis}}}\to \operatorname {\mathbf {NST}}$ has a left adjoint:

(1.6.2)$$ \begin{align} h_{0,{\operatorname{Nis}}}^{\mathbf{A}^1} : \operatorname{\mathbf{NST}} \to \operatorname{\mathbf{HI}}_{\operatorname{Nis}}. \end{align} $$

By [Reference Kahn, Saito and YamazakiKSY22, Proposition 2.3.2], we have:

(1.6.3)$$ \begin{align} h_{0,{\operatorname{Nis}}}^{\mathbf{A}^1}(h_{0,{\operatorname{Nis}}}(\mathcal{X})) =h_{0,{\operatorname{Nis}}}^{\mathbf{A}^1}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\omega\mathcal{X})). \end{align} $$

2 Cohomology of blow-ups and invariance properties

2.1 A lemma on modulus descent

Notation 2.1. For $m,n\ge 1$, we use the following notation:

$$\begin{align*}{\overline{\square}}^{(m,n)}:= (\mathbf{P}^1, m\cdot 0 +n\cdot \infty),\quad {\overline{\square}}^{(n)}:= {\overline{\square}}^{(n,n)}.\end{align*}$$

In particular,

$$\begin{align*}\overline{\square}^{(1)}=(\mathbf{P}^1,0+\infty).\end{align*}$$

Lemma 2.2. Let R be an integral regular k-algebra. For all $m,n\ge 1$, there is an isomorphism:

$$\begin{align*}\theta_{m,n}: h_0({\overline{\square}}^{(m,n)})(R)\xrightarrow{\simeq} ((R[t]/t^m)^{\times}\oplus (R[z]/z^n)^{\times})/R^{\times}\oplus \mathbb{Z},\end{align*}$$

where $R^{\times }$ acts diagonally on the direct sum. If $Z\in \underline {\omega }_!{\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}({\overline {\square }}^{(m,n)})(R)$ is a prime correspondence which we can write as $Z=V(g)$, for an irreducible polynomial $g=a_rt^r+ \ldots + a_1 t+a_0\in R[t]$ with $a_r, a_0\in R^{\times }$, and $r\ge 1$, then:

$$\begin{align*}\theta_{m,n}(Z)=(g(t)/(t-1)^r, g_{\infty}(z)/(1-z)^r, r),\end{align*}$$

where $g_{\infty }(z)= a_0 z^r+\ldots + a_{r-1} z+ a_r$. Furthermore, if $m'\le m$ and $n'\le n$, then we obtain a commutative diagram:

where the vertical map on the left-hand side is induced by ${\overline {\square }}^{(m',n')}\to {\overline {\square }}^{(m,n)}$ in $\operatorname {\mathbf {\underline {M}Cor}}$ and the vertical map on the right is the natural quotient map.

Proof. The map $\theta _{m,n}$ is the composition of the two isomorphisms:

$$ \begin{align*} h_0({\overline{\square}}^{(m,n)})(R)& \xrightarrow{\simeq\, (*)} \operatorname{Pic}(\mathbf{P}^1_R, m\cdot 0+ n\cdot \infty)\\ & \xrightarrow{\simeq \, (**)} ((R[t]/t^m)^{\times}\oplus (R[z]/z^n)^{\times})/R^{\times}\oplus \mathbb{Z}, \end{align*} $$

which are defined as follows. We denote by $F_R:=m\cdot 0_R+ n\cdot \infty _R\subset \mathbf {P}^1_R$ the closed subscheme: (*) is induced by the classical map from Weil to Cartier divisors:

$$\begin{align*}{\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R)\ni D\mapsto (\mathcal{O}(D), {\operatorname{id}}_{\mathcal{O}_{F_R}})\in \operatorname{Pic}(\mathbf{P}^1_R, F_R),\end{align*}$$

where $\mathcal {O}(D)$ is the line bundle on $\mathbf {P}^1_R$ given by $\mathcal {O}(D)(U)=\{f\in R(t)^{\times }\mid \operatorname {div}_U(f)\ge D \}$; it is an isomorphism by [Reference Rülling and YamazakiRY16, Theorem 1.1]. For (**), consider the exact sequence:

$$\begin{align*}H^0(\mathbf{P}^1_R, \mathcal{O}^{\times})\to H^0(F_R, \mathcal{O}^{\times})\to \operatorname{Pic}(\mathbf{P}^1_R, F_R) \to \operatorname{Pic}(\mathbf{P}_R^1)\to \operatorname{Pic}(F_R).\end{align*}$$

The last map decomposes as $\operatorname {Pic}(R)\oplus \operatorname {Pic}(\mathbf {P}^1) \to \operatorname {Pic}(F_R)$ given by:

$$\begin{align*}(M,\mathcal{O}(\{1\})^{\otimes n})\mapsto (M_{|\mathbf{P}^1_R}\otimes (\mathcal{O}(\{1\})^{\otimes n})_{|F_R}=M_{|F_R}.\end{align*}$$

Since $F_R\to \operatorname {Spec} R$ has a section, the map $\operatorname {Pic}(R)\to \operatorname {Pic}(F_R)$ is injective. Hence, the above sequence yields an exact sequence:

$$\begin{align*}H^0(\mathbf{P}^1_R, \mathcal{O}^{\times})\to H^0(F_R, \mathcal{O}^{\times})\to \operatorname{Pic}(\mathbf{P}^1_R, F_R) \xrightarrow{d} \mathbb{Z} \to 0,\end{align*}$$

where $d(L,\alpha ):=d(L):=\deg (L_{|\mathbf {P}^1_{{\operatorname {Frac}}(R)}})$; we can choose a splitting of d by $r\mapsto (\mathcal {O}_{\mathbf {P}^1_R}(\{1\})^{\otimes r}, {\operatorname {id}}_{\mathcal {O}_{F_R}})$; the map in the middle sends $u\in H^0(F_R, \mathcal {O}^{\times })$ to $(\mathcal {O}_{\mathbf {P}^1_R}, u\cdot : \mathcal {O}_{F_R}\xrightarrow {\simeq } \mathcal {O}_{F_R})$, where $u\cdot $ is the isomorphism given by multiplication by u. Let $(L,\alpha )$ be a pair with L a line bundle on $\mathbf {P}^1_R$ with $d(L)=r$ and $\alpha : \mathcal {O}_{F_R}\xrightarrow {\simeq } L_{|F_R}$ an isomorphism; we find an isomorphism $\varphi :L\otimes \mathcal {O}(\{1\})^{\otimes -r}\xrightarrow {\simeq } \mathcal {O}_{\mathbf {P}_R^1}$ and define the isomorphism $\alpha '$ as the composition:

$$\begin{align*}\alpha'=(\alpha^{\prime}_{m\cdot 0},\alpha^{\prime}_{n\cdot \infty}): \mathcal{O}_{F_R}\xrightarrow{\simeq\, \alpha}L_{|F_R} = (L\otimes\mathcal{O}(\{1\})^{\otimes -r})_{|F_R} \xrightarrow{\varphi_{|F_R}} (\mathcal{O}_{\mathbf{P}^1})_{|F_R}, \end{align*}$$

where the equality follows from the fact that we have a canonical identification $\mathcal {O}(\{1\})_{|F_R} = \mathcal {O}_{|F_R}$. Hence, $\varphi $ induces an isomorphism $(L\otimes \mathcal {O}(\{1\})^{\otimes -r}, \alpha )\cong (\mathcal {O}_{\mathbf {P}^1_R}, \alpha ')$; the isomorphism (**) is given by:

$$\begin{align*}(L,\alpha)\mapsto (\alpha^{\prime}_{m\cdot0}(1), \alpha^{\prime}_{n\cdot \infty}(1), d(L)).\end{align*}$$

Let $Z=V(g)\in {\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}({\overline {\square }}^{(m,n)})(R)$ be a prime correspondence as in the statement. Write $t=T_0/T_1$, and let $G\in R[T_0, T_1]$ be the homogenisation of g. We have an isomorphism:

$$\begin{align*}\mathcal{O}(Z)\otimes \mathcal{O}(\{1\})^{\otimes -r}= \mathcal{O}_{\mathbf{P}^1_R}\cdot \tfrac{(T_0-T_1)^r}{G} \xrightarrow{\simeq}\mathcal{O}_{\mathbf{P}^1_R},\end{align*}$$

where the second isomorphism is given by multiplication with $G/(T_0-T_1)^r$. Thus, $\theta _{m,n}$ admits the description from the statement, where $z=1/t$. The commutativity of the diagram follows directly from this.

Remark 2.3. Denote by $\mathbb {W}_m$ the ring scheme of big Witt vectors of length m. If A is a ring, we can identify the A-rational points of the underlying group scheme with:

$$\begin{align*}\mathbb{W}_m(A)= (1+tA[t])^{\times}/(1+t^{m+1} A[t])^{\times}.\end{align*}$$

Then the maps $\theta _{m,n}$ from Lemma 2.2, $m,n\ge 1$, induce isomorphisms in $\operatorname {\mathbf {NST}}$:

$$\begin{align*}\theta_{m,n}: h_{0,{\operatorname{Nis}}}({\overline{\square}}^{(m,n)})\xrightarrow{\simeq} \mathbb{W}_{m-1}\oplus \mathbb{W}_{n-1} \oplus \mathbf{G}_m \oplus \mathbb{Z}.\end{align*}$$

Indeed, it follows immediately from Lemma 2.2 that we have such an isomorphism of Nisnevich sheaves. To check the compatibility with transfers, it suffices to check the compatibility with transfers of the limit $\varprojlim _{m,n}\theta ^{m,n}$ (since the transition maps are surjective). Since $\mathbb {W}\oplus \mathbb {W}\oplus \mathbf {G}_m\oplus \mathbb {Z}$ is a $\mathbb {Z}$-torsion-free sheaf on $\operatorname {\mathbf {Sm}}_{{\operatorname {Nis}}}$ for which the pullback along dominant étale maps is injective, the compatibility with transfers follows automatically from [Reference Merici and SaitoMS20, Lemma 1.1].

Lemma 2.4. The unit map:

(2.4.1)$$ \begin{align}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}(\overline{\square}^{(1)})\xrightarrow{\simeq} \underline{\omega}^*\underline{\omega}_! h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}(\overline{\square}^{(1)}) \cong \underline{\omega}^*(\mathbf{G}_m\oplus \mathbb{Z}) \end{align} $$

is an isomorphism in $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$. Furthermore, the natural maps:

(2.4.2)$$ \begin{align}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})\to h_{0,{\operatorname{Nis}}}^{{\overline{\square}},\mathrm{sp}}(\overline{\square}^{(1)}) \end{align} $$

are surjective, for all $m,n\ge 1$, and there exists a splitting in $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$:

$$\begin{align*}s_{m,n} : \underline{\omega}^*(\mathbf{G}_m\oplus \mathbb{Z}) \to h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{ sp}}({\overline{\square}}^{(m,n)})\end{align*}$$

of (2.4.2), such that the following diagram is commutative for integers $m'\geq m$ and $n'\geq n$:

(2.4.3)

Proof. The second isomorphism in (2.4.1) holds by Lemma 2.2 and Remark 2.3; the unit map is injective by semipurity. We show the surjectivity of the composite map:

(2.4.4)$$ \begin{align}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})\to h_{0,{\operatorname{Nis}}}^{{\overline{\square}},\mathrm{sp}}(\overline{\square}^{(1)}) \to \underline{\omega}^*(\mathbf{G}_m\oplus \mathbb{Z}) \end{align} $$

for $m,n\ge 1$. By Lemma 1.3, it suffices to show the surjectivity on $(\operatorname {Spec} R, (f))$, where R is an integral normal local k-algebra and $f\in R\setminus \{0\}$, such that $R_f$ is regular. Denote by:

$$\begin{align*}\psi: {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f)\to R_f^{\times}\oplus\mathbb{Z}\end{align*}$$

the precomposition of (2.4.4) evaluated at $(R,f)$ with the quotient map:

$$\begin{align*}{\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f)\to h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{ sp}}({\overline{\square}}^{(m,n)})(R,f).\end{align*}$$

By Lemma 2.2:

(2.4.5)$$ \begin{align}\psi(V(a_0+ a_1t+\ldots +a_r t^r))= ((-1)^r a_0/a_r, r), \end{align} $$

provided that $Z=V(a_0+ a_1t+\ldots +a_r t^r)$ is an admissible prime correspondence and $a_i\in R_f$. We claim that $\psi $ is surjective. To this end, observe that for $a\in R_f^{\times }$, we find $N\ge 0$ and $b\in R$, such that:

(2.4.6)$$ \begin{align}ab=f^{nN}, \quad \text{and}\quad af^{mN}\in R. \end{align} $$

Set $W:=V(t^{mnN} + (-1)^{mnN} a)\subset \operatorname {Spec} R_f[t, 1/t]$ and $K={\operatorname {Frac}}(R)$. Let $t^{mnN} + (-1)^{mnN} a=\prod _i h_i$ be the decomposition into monic irreducible factors in $K[t,1/t]$, and denote by $W_i\subset \operatorname {Spec} R_f[t, 1/t]$ the closure of $V(h_i)$ (note that $W_i=W_j$ for $i\neq j$ is allowed). The $W_i$ correspond to the components of W which are dominant over $R_f$; since W is finite (the polynomial defining W is monic) and surjective over $R_f$, so are the $W_i$. We claim:

(2.4.7)$$ \begin{align}W_i\in {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f). \end{align} $$

Indeed, let $I_i$ (respectively, $J_i$) be the ideal of the closure of $W_i$ in $\operatorname {Spec} R[t]$ (respectively, $\operatorname {Spec} R[z]$ with $z=1/t$). By (2.4.6):

$$\begin{align*}bt^{nmN} + (-1)^{mnN} f^{nN}\in I_i \quad \text{and} \quad f^{mN} + (-1)^{mnN} f^{mN}a z^{mnN}\in J_i.\end{align*}$$

Hence, $(f/t^m)^{nN}\in R[t]/I_i$ and $(f/z^n)^{mN}\in R[z]/J_i$. It follows that $f/t^m$ (respectively, $f/z^n$) is integral over $R[t]/I_i$ (respectively, $R[z]/J_i$); thus, (2.4.7) holds. Put:

$$\begin{align*}W_a= \sum_i W_i \in{\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f).\end{align*}$$

We claim:

(2.4.8)$$ \begin{align}\psi(W_a)=(a, mnN)\in R_f^{\times}\oplus \mathbb{Z}. \end{align} $$

Indeed, it suffices to show this after restriction to the generic point of R, in which case, it follows directly from the definition of the $W_i$ and (2.4.5). This implies the surjectivity of $\psi $ and that of (2.4.4). Next, we show that (2.4.4) has a splitting. Let $\omega _a^{m,n}\in h_{0,{\operatorname {Nis}}}^{{\overline {\square }}, \mathrm {sp}}({\overline {\square }}^{(m,n)})(R,f)$ be the class of $W_a$ and $\lambda _a^{m,n}=\omega _a^{m,n}-\omega _1^{m,n}$, where $\omega _1^{m,n}$ is defined as $\omega _a^{m,n}$ replacing a by $1$ (and using the same N). By (2.4.8), the image of $\lambda _a^{m,n}$ under the map (2.4.4):

$$\begin{align*}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})(R,f) \to R_f^{\times}\oplus \mathbb{Z}\end{align*}$$

is $(a,0)$.

Claim 2.4.1. $\lambda _a^{m,n}$ is independent of the choice of N, and we have:

(2.4.9)$$ \begin{align}\lambda_{ab}^{m,n} =\lambda_a^{m,n} + \lambda_b^{m,n}\quad\text{ for } a,b \in R_f^{\times}. \end{align} $$

Moreover, for $m'\geq m$ and $n'\geq n$, the image of $\lambda ^{m',n'}_a$ under:

$$\begin{align*}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m',n')}) \to h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})\end{align*}$$

coincides with $\lambda _a^{m,n}$.

By the semipurity of $h_{0,{\operatorname {Nis}}}^{{\overline {\square }}, \mathrm {sp}}({\overline {\square }}^{(m,n)})$ and [Reference SaitoSai20a, Theorem 3.1], we have an injective homomorphism:

(2.4.10)$$ \begin{align}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})(R,f) \hookrightarrow \underline{\omega}_! h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{ sp}}({\overline{\square}}^{(m,n)})(K)=h_0({\overline{\square}}^{(m,n)})(K). \end{align} $$

By Lemma 2.2, the isomorphism:

$$\begin{align*}\theta_{m,n}: h_0({\overline{\square}}^{(m,n)})(K)\xrightarrow{\simeq} ((K[t]/t^m)^{\times} \oplus (K[z]/z^n)^{\times})/K^{\times} \oplus \mathbb{Z}\end{align*}$$

sends $\omega ^{m,n}_a$ to:

$$\begin{align*}\theta_{m,n}(\omega^{m,n}_a) =\left (\frac{(-1)^{mnN}a}{(t-1)^{mnN}}, \frac{1}{(1-z)^{mnN}}, mnN\right).\end{align*}$$

Thus, $\theta _{m,n}(\lambda ^{m,n}_a)= (a,1,0)$, which is independent of N. By the injectivity of (2.4.10), this implies the first two assertions of the claim; similarly, the final assertion of the claim follows from the commutative diagram in Lemma 2.2. Since $\lambda _a^{m,n}$ does not change if we replace f by $uf$ with $u\in R^{\times }$, the map $a\to \lambda _a^{m,n}$ glues to give a global morphism of Nisnevich sheaves which induces the splitting $s_{m,n}$ from the statement. It remains to check that $s_{m,n}$ is compatible with transfers. To this end, it suffices to check that ${\underline {\omega }}_!(s_{m,n})$ is compatible with transfers, and since the transition maps are surjective, it further suffices to show that:

$$\begin{align*}\varprojlim_{m,n} {\underline{\omega}}_!(s_{m,n}): \mathbf{G}_m\oplus \mathbb{Z}\to \varprojlim_{m,n} {\underline{\omega_!}}h^{{\overline{\square}}, \mathrm{sp}}_{0,{\operatorname{Nis}}}({\overline{\square}}^{(m,n)})\end{align*}$$

is compatible with transfers. Since we can identify the target with $\mathbb {W}\oplus \mathbb {W}\oplus \mathbf {G}_m\oplus \mathbb {Z}$ by Remark 2.3, the compatibility holds automatically by [Reference Merici and SaitoMS20, Lemma 1.1].

Proposition 2.5. Denote by $\psi : \mathbf {A}^1_y\times \mathbf {A}^1_s\to \mathbf {A}^1_x\times \mathbf {A}^1_s$ the morphism induced by the $k[s]$-algebra morphism $k[x,s]\to k[y,s]$, $x\mapsto ys$. We denote by the same symbol, the induced morphism in $\operatorname {\mathbf {\underline {M}Cor}}$:

(2.5.1)$$ \begin{align}\psi: \overline{\square}^{(1)}_y\otimes {\overline{\square}}^{(2)}_s\to \overline{\square}^{(1)}_x\otimes \overline{\square}^{(1)}_s. \end{align} $$

Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ and $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$. Then $\psi ^*$ factors as follows:

where the vertical map is induced by the natural morphism ${\overline {\square }}^{(2)}_s\to \overline {\square }^{(1)}_s$.

Proof. It is direct to check that $\psi $ induces a morphism (2.5.1). To check the factorisation statement, we may replace F by $\operatorname {\underline {Hom}}({\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X}), F)$ to reduce to the case $\mathcal {X}=(\operatorname {Spec} k,\emptyset )$ (see Lemma 1.5(1)). By Yoneda and (1.4.2), we are reduced to show that we have a factorisation as follows:

(2.5.2)

By [Reference Merici and SaitoMS20, Lemma 1.14(iii)] and Lemma 2.4, the map a is surjective. Thus, we have to show $\psi (\operatorname {Ker} a)=0$. By semipurity, it suffices to show that we have a factorisation as in (2.5.2) after applying $\underline {\omega }_!$. By [Reference Rülling, Sugiyama and YamazakiRSY22, Proposition 5.6], we have:

$$\begin{align*}H:=\underline{\omega}_!(h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{ sp}}(\overline{\square}^{(1)}_x\otimes\overline{\square}^{(1)}_s))=h_{0,{\operatorname{Nis}}}(\overline{\square}^{(1)}_x\otimes\overline{\square}^{(1)}_s)=\mathcal{K}^M_2\oplus \mathbf{G}_m\oplus\mathbf{G}_m\oplus \mathbb{Z},\end{align*}$$

where $\mathcal {K}^M_2$ is the (improved) Milnor K-theory sheaf; in particular, H is $\mathbf {A}^1$-invariant. Thus, $\underline {\omega }_!(\psi )$ and $\underline {\omega }_!(a)$ factor via $h^{\mathbf {A}^1}_{0,{\operatorname {Nis}}}(h_{0,{\operatorname {Nis}}}(\overline {\square }^{(1)}_y\otimes {\overline {\square }}^{(2)}_s))$ (cf. (1.6.2)). Thus, we obtain solid arrows in $\operatorname {\mathbf {NST}}$:

(2.5.3)

Since $\bar {a}$ is the composition of the natural isomorphisms (cf. (1.6.3)):

$$\begin{align*}h^{\mathbf{A}^1}_{0,{\operatorname{Nis}}}(h_{0,{\operatorname{Nis}}}(\overline{\square}^{(1)}_y\otimes{\overline{\square}}^{(2)}_s)) \cong h^{\mathbf{A}^1}_{0,{\operatorname{Nis}}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^1_y\setminus\{0\})\otimes_{{\operatorname{\mathbf{PST}}}} {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^1_s\setminus\{0\})) \cong H\end{align*}$$

the dotted arrow exists, which completes the proof.

Remark 2.6. Going through the definitions, one can check that the map $H\to H$ induced by $\bar {\psi }$ in (2.5.3) is on a regular local ring R given by:

$$\begin{align*}(\{a,b\}, c,d, n)\mapsto (\{a,b\}+ \{d,-1\}, cd, d, n),\end{align*}$$

where we use the identification $H(R)= K^M_2(R)\oplus R^{\times }\oplus R^{\times }\oplus \mathbb {Z}$.

2.2 Cohomology of a blow-up centred in the smooth part of the modulus

The goal of this section is to prove Theorem 2.12 below, giving the invariance of the cohomology of cube invariant sheaves along a certain class of blow-ups. This plays a fundamental role in what follows, and it is used in the proof of the $(\mathbf {P}^n, \mathbf {P}^{n-1})$-invariance of the cohomology.

Recall the following definition from [Reference SaitoSai20a, Section 5].

Definition 2.7. Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}$. We define the modulus presheaf $\sigma ^{(n)}(F)$ by:

$$\begin{align*}\sigma^{(n)}(F)(\mathcal{Y}) = \operatorname{Coker} (F(\mathcal{Y})\xrightarrow{\mathrm{pr}^*} F(\mathcal{Y}\otimes (\mathbf{P}^1, n0+\infty))),\end{align*}$$

where $\mathrm {pr}^*$ is the pullback along the projection $\mathrm {pr}\colon \mathcal {Y}\otimes (\mathbf {P}^1, n0+\infty ) \to \mathcal {Y}$. Note that $\mathrm {pr}^*$ is split injective, with left inverse given by the inclusion $i_1\colon \operatorname {Spec} k\hookrightarrow \mathbf {P}^1$ of the 1-section. Hence, we have an isomorphism, natural in $\mathcal {Y}$:

$$\begin{align*}F(\mathcal{Y}\otimes (\mathbf{P}^1, n0+\infty)) \cong \sigma^{(n)}(F)(\mathcal{Y})\oplus F(\mathcal{Y}). \end{align*}$$

Following [Reference SaitoSai20a, Definition 5.6], we write $F^{(n)}_{-1}$ for $\sigma ^{(n)}(F)$ when F is moreover in $\operatorname {\mathbf {\underline {M}NST}}$. Note that we have a natural identification:

$$\begin{align*}F^{(n)}_{-1}=\operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}((\mathbf{P}^1, n\cdot 0+ \infty)/1, F) = F(-\otimes (\mathbf{P}^1, n0+\infty))/F(-),\end{align*}$$

where $(\mathbf {P}^1, n\cdot 0+ \infty )/1= \operatorname {Coker}({\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\operatorname {Spec} k,\emptyset )\xrightarrow {i_1} {\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathbf {P}^1, n\cdot 0+\infty ))$ in $\operatorname {\mathbf {\underline {M}PST}}$. By Lemma 1.5(1), we have $F^{(n)}_{-1}\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ if $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$, so that the association $F\mapsto F_{-1}^{(n)}$ gives an endofunctor of $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$. This construction is the modulus version of Voevodsky’s contraction functor (see [Reference Mazza, Voevodsky and WeibelMVW06, p.191].

Notation 2.8. We denote by $\operatorname {\mathbf {\underline {M}Cor}}_{ls}$ the full subcategory of $\operatorname {\mathbf {\underline {M}Cor}}$ consisting of ‘log smooth’ modulus pairs, that is, objects $\mathcal {X}=(X,D)$, where $X\in \operatorname {\mathbf {Sm}}$ and $|D|$ is a simple normal crossing divisor (in particular, each irreducible component of $|D|$ is a smooth divisor in X). Note that $\otimes $ restricts to a monoidal structure on $\operatorname {\mathbf {\underline {M}Cor}}_{ls}$.

Lemma 2.9. Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ and $\mathcal {X}=(X,D)\in \operatorname {\mathbf {\underline {M}Cor}}_{ls}$. Let $H\hookrightarrow X$ be a smooth divisor, such that $|D|+H$ is SNCD, and denote by $j: U:= X\setminus H\hookrightarrow X$ the inclusion of the complement. Then:

$$\begin{align*}R^i j_* F_{(U, D_{|U})}=0, \quad \text{for all }i\ge 1,\end{align*}$$

where $F_{(U, D_{|U})}$ denotes the Nisnevich sheaf on U defined in (1.2.1).

Proof. This is an immediate consequence of [Reference SaitoSai20a, Corollary 8.6(3)].

Lemma 2.10. Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ and $\mathcal {X}=(X, D)\in \operatorname {\mathbf {\underline {M}Cor}}_{ls}$. Let $E_i\subset \mathbf {A}^1$, $i=1,\ldots , n$, be effective (or empty) divisors, and denote by $\pi : \mathbf {A}^n_X\to X$ the projection. Then:

$$\begin{align*}R^i\pi_* (F_{(\mathbf{A}^1,E_1)\otimes\ldots \otimes(\mathbf{A}^1, E_n)\otimes \mathcal{X}})=0,\quad \text{for all } i \ge 1, \,n\ge 0.\end{align*}$$

Proof. First consider the case $n=1$. Set $E:=E_1$ and ${\overline {\square }}^{(E,r)}:=(\mathbf {P}^1, E+ r\cdot \infty )$, for $r\ge 1$. The natural morphism ${\overline {\square }}^{(E,r)}\to {\overline {\square }}$ induces a map $F_{\mathcal {X}\otimes {\overline {\square }}} \to F_{\mathcal {X}\otimes {\overline {\square }}^{(E,r)}}$. The cohomology sheaves of the cone C of this map are supported in $X\times |E+\infty |$, whence $R^i\overline {\pi }_*C=0$, $i\ge 1$, where $\overline {\pi }:\mathbf {P}^1_X\to X$ is the projection. We obtain surjections:

$$\begin{align*}R^i\overline{\pi}_*F_{\mathcal{X}\otimes{\overline{\square}}}\to R^i\overline{\pi}_*F_{\mathcal{X}\otimes{\overline{\square}}^{(E,r)}}\to 0,\quad \text{for all } i\ge 1.\end{align*}$$

By the cube invariance of cohomology (see [Reference SaitoSai20a, Theorem 9.3]), the left term vanishes. Thus, M-reciprocity (see [Reference SaitoSai20a, Lemma 1.27(1)]) yields:

$$\begin{align*}0=\varinjlim_r R^i\overline{\pi}_*F_{\mathcal{X}\otimes{\overline{\square}}^{(E,r)}}= R^i\overline{\pi}_* j_*F_{(\mathbf{A}^1, E)\otimes\mathcal{X}},\end{align*}$$

where $j:\mathbf {A}^1_X\hookrightarrow \mathbf {P}^1_X$ is the open immersion. Together with Lemma 2.9, we obtain:

$$\begin{align*}R^i\overline{\pi}_* Rj^k_*F_{(\mathbf{A}^1,E)\otimes\mathcal{X}}=0, \quad \text{for all } i\geq 1, k\geq 0.\end{align*}$$

Thus, the vanishing $R^i\pi _*F_{(\mathbf {A}^1,E)\otimes \mathcal {X}}=0$ follows from the Leray spectral sequence.

The general case follows by induction (by factoring $\pi $ as $\mathbf {A}^n_X\xrightarrow {\pi _1} \mathbf {A}^{n-1}_X\xrightarrow {\pi _{n-1}} X$ and observing):

$$\begin{align*}\pi_{1*}(F_{(\mathbf{A}^1, E_1)\otimes \ldots\otimes(\mathbf{A}^1, E_n)\otimes\mathcal{X}})= F_{1, (\mathbf{A}^1, E_2)\otimes\ldots\otimes(\mathbf{A}^1, E_n)\otimes\mathcal{X}},\end{align*}$$

where $F_1:=\operatorname {\underline {Hom}}_{\operatorname {\mathbf {\underline {M}PST}}}({\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathbf {A}^1,E_1), F)$ lies in $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ by Lemma 1.5(1).

2.11. We recall some standard terminology. Let $(X, D)\in \operatorname {\mathbf {\underline {M}Cor}}_{ls}$, $Y\in \operatorname {\mathbf {Sm}}$, and let $f: Y\to X$ be a k-morphism of finite type. We say D is transversal to f, if for any number of irreducible components $D_1, \ldots , D_r$ of the SNCD $|D|$, the morphism f intersects the scheme-theoretic intersection $D_1\cap \ldots \cap D_r$ transversally (i.e. the scheme-theoretic inverse image $f^{-1}(D_1\cap \ldots \cap D_r)$ is smooth over k and of codimension r in Y). Note that f is always transversal to the empty divisor.

If f is a closed immersion, we also say Y and D intersect transversally. Since X is of finite type over a perfect field, this is equivalent to say, that for any point $x\in Y\cap D$, we find a regular sequence of parameters $t_1,\ldots , t_n\in \mathcal {O}_{X,x}$, such that $\mathcal {O}_{Y,x}=\mathcal {O}_{X,x}/(t_1,\ldots , t_s)$ and the irreducible components of $|D|$ containing x are in $\operatorname {Spec} \mathcal {O}_{X,x}$ given by $V(t_{s+1}),\ldots , V(t_r)$, with $1\le s\le r\le n$.

Theorem 2.12. Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ and $\mathcal {X}=(X,D)\in \operatorname {\mathbf {\underline {M}Cor}}_{ls}$. Assume there is a smooth irreducible component $D_0$ of $|D|$ which has multiplicity 1 in D. Let $Z\subset X$ be a smooth closed subscheme which is contained in $D_0$ and intersects $|D-D_0|$ transversally. Let $\rho : Y\to X$ be the blow-up in Z. Then the natural map:

$$\begin{align*}F_{\mathcal{X}}\xrightarrow{\simeq} R\rho_{*} F_{(Y, \rho^*D)}\end{align*}$$

is an isomorphism in the derived category of abelian Nisnevich sheaves on X.

The proof is given in 2.16. The key point is to understand the case of the blow-up of $\mathbf {A}^2$ in the origin with $D_0$ a line, which is established in the next Lemma. Here, after some preliminary steps, we are reduced to prove the vanishing of the cohomology of the pushforward of F along the projection from the blow-up to the exceptional divisor. This is where the modulus descent, in that, Proposition 2.5, is crucially used.

Lemma 2.13. Let $F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ and $\mathcal {X}=(X,D)\in \operatorname {\mathbf {\underline {M}Cor}}_{ls}$. Let $\rho : Y\to \mathbf {A}^2$ be the blow-up in the origin $0\in \mathbf {A}^2$, and let L be a line containing $0$. Then:

$$\begin{align*}R^i\rho_{X*} F_{(Y,\rho^*L)\otimes \mathcal{X}}=0,\quad \text{for all }i\ge 1,\end{align*}$$

where $\rho _X:=\rho \times {\operatorname {id}}_X: Y\times X\to \mathbf {A}^2\times X$ is the base change of $\rho $.

Proof. We can assume X is henselian local and:

$$\begin{align*}L=V(x)\subset \mathbf{A}^2=\operatorname{Spec} k[x,y].\end{align*}$$

Set:

$$\begin{align*}\mathcal{F}:=F_{(Y, \rho^*L)\otimes \mathcal{X}};\end{align*}$$

it is a Nisnevich sheaf on $Y\times X$. For $i\ge 1$, the higher direct images $R^i\rho _{X*}\mathcal {F}$ are supported in $0\times X$, whence:

$$\begin{align*}H^j(\mathbf{A}^2_X, R^i\rho_{X*}\mathcal{F})=0, \quad \text{for all }i,j\ge 1,\end{align*}$$

and:

$$\begin{align*}R^i\rho_{X*}\mathcal{F}=0\Longleftrightarrow H^0(\mathbf{A}^2_X, R^i\rho_{X*}\mathcal{F})=0.\end{align*}$$

Furthermore, $\rho _{X*} \mathcal {F}= F_{(\mathbf {A}^2, L)\otimes \mathcal {X}}$, since $(Y, \rho ^*L)\otimes \mathcal {X}\cong (\mathbf {A}^2, L)\otimes \mathcal {X}$ in $\operatorname {\mathbf {\underline {M}Cor}}$ (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Section 1.7]). Hence, by Lemma 2.10:

$$\begin{align*}H^i(\mathbf{A}^2_X, \rho_{X*}\mathcal{F})=H^i(\mathbf{A}^2_X, F_{(\mathbf{A}^2, L)\otimes \mathcal{X}})=0.\end{align*}$$

Thus, the Leray spectral sequence yields:

(2.13.1)$$ \begin{align}H^0(\mathbf{A}_X^2, R^i\rho_{X*}\mathcal{F})= H^i(Y\times X, \mathcal{F}), \quad i\ge 0,\end{align} $$

and we have to show that this group vanishes for $i\ge 1$. Write:

$$\begin{align*}Y=\operatorname{Proj} k[x,y][S,T]/(xT-yS)\subset \mathbf{A}^2\times \mathbf{P}^1,\end{align*}$$

and denote by:

$$\begin{align*}\pi: Y\times X\hookrightarrow \mathbf{A}^2\times \mathbf{P}^1_X \to \mathbf{P}^1_X=\operatorname{Proj} \mathcal{O}_X[S,T]\end{align*}$$

the morphism induced by projection. In order to show that (2.13.1) vanishes, we can project along $\pi $ and use the Leray spectral sequence:

$$\begin{align*}H^{i-j}(\mathbf{P}^1_X, R\pi^j_*\mathcal{F})\Rightarrow H^i(Y\times X, \mathcal{F})\end{align*}$$

to reduce the problem to showing that:

(2.13.2)$$ \begin{align}H^i(\mathbf{P}^1_X, R^j\pi_*\mathcal{F})=0, \quad i\ge 1, j\geq 0. \end{align} $$

The terms $R^j\pi _* \mathcal {F}$ for $j\geq 1$ are easy to handle using Lemma 2.10. Indeed, set $s=S/T$, and write:

$$\begin{align*}\mathbf{P}^1\setminus \{\infty\}= \mathbf{A}^1_s:= \operatorname{Spec} k[s],\quad \mathbf{P}^1\setminus\{0\}= \operatorname{Spec} k[\tfrac{1}{s}].\end{align*}$$

Set $U:=\mathbf {A}^1_s \times X$ and $V:=(\mathbf {P}^1\setminus \{0\})\times X$ and:

$$\begin{align*}\mathcal{U}:=(\mathbf{A}^1_s,0)\otimes\mathcal{X}, \quad \mathcal{V}:=(\mathbf{P}^1\setminus\{0\}) \otimes \mathcal{X}.\end{align*}$$

We have:

$$\begin{align*}\pi^{-1}(U)= \mathbf{A}^1_y\times U, \quad \pi^{-1}(V)=\mathbf{A}^1_x\times V,\end{align*}$$

and the restriction of $\pi $ to these open subsets is given by projection. Furthermore by construction,

(2.13.3)$$ \begin{align} \mathcal{F}_{|\pi^{-1}(U)}=F_{(\mathbf{A}^1_y,0)\otimes \mathcal{U}},\quad \mathcal{F}_{|\pi^{-1}(V)}= F_{(\mathbf{A}^1_x, 0)\otimes \mathcal{V}}. \end{align} $$

Thus, Lemma 2.10 (in the case $n=1$) yields:

$$\begin{align*}R^j\pi_*\mathcal{F}=0, \quad j\ge 1.\end{align*}$$

It remains to show:

(2.13.4)$$ \begin{align}H^i(\mathbf{P}^1_X, \pi_*\mathcal{F})=0, \quad i\ge 1. \end{align} $$

Set:

(2.13.5)$$ \begin{align}F_1:=\operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^1_x, 0), F ).\end{align} $$

Note that $F_1\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ by Lemma 1.5(1). Let $j: V\hookrightarrow \mathbf {P}^1_X$ be the open immersion. Its base change along $\pi $ induces a morphism:

(2.13.6)$$ \begin{align}\iota: (\mathbf{A}^1_x,0)\otimes \mathcal{V}\to (Y,\rho^*L)\otimes \mathcal{X} \quad \text{in } \operatorname{\mathbf{\underline{M}Cor}}. \end{align} $$

This yields an exact sequence of Nisnevich sheaves on $\mathbf {P}^1_X$:

$$\begin{align*}0\to \pi_*\mathcal{F}\xrightarrow{\pi_*(\iota^*)} j_*F_{1,\mathcal{V}}\to \Gamma\to 0,\end{align*}$$

defining $\Gamma $; here, the first map is injective by the semipurity of F. Since $\Gamma $ is supported on $0\times X$, we obtain for $i\ge 2$:

$$ \begin{align*} H^i(\mathbf{P}^1_X, \pi_*\mathcal{F}) & = H^i(\mathbf{P}^1_X, j_*F_{1,\mathcal{V}})\\ &= H^i(V, F_{1,\mathcal{V}}), & & \text{by Lemma}\ {2.9},\\ &=0, &&\text{by Lemma}\ {2.10}. \end{align*} $$

It remains to prove the vanishing (2.13.4) for $i=1$. This will occupy the rest of the proof. Let:

$$\begin{align*}a\colon Y\times X\to \mathbf{A}^1_x\times \mathbf{P}^1\times X\end{align*}$$

be induced by the base change of the closed immersion $Y\hookrightarrow \mathbf {A}^2\times \mathbf {P}^1$ followed by the base change of the projection $\mathbf {A}^2\to \mathbf {A}^1_x$. The map a induces a morphism:

(2.13.7)$$ \begin{align}\alpha: (Y,\rho^*L)\otimes \mathcal{X}\to (\mathbf{A}^1_x, 0)\otimes \mathbf{P}^1_{\mathcal{X}}\quad \text{in }\operatorname{\mathbf{\underline{M}Cor}}, \end{align} $$

where $\mathbf {P}^1_{\mathcal {X}}:=\mathbf {P}^1\otimes \mathcal {X}$ and which precomposed with $\iota $ from (2.13.6) yields the morphism:

(2.13.8)$$ \begin{align}\alpha\iota: (\mathbf{A}^1_x,0)\otimes \mathcal{V}\to (\mathbf{A}^1_x, 0)\otimes \mathbf{P}^1_{\mathcal{X}} \end{align} $$

induced by the open immersion $\mathbf {A}^1_x\times V\hookrightarrow \mathbf {A}^1_x\times \mathbf {P}^1_X$. This gives a factorisation:

where the diagonal morphism is injective by [Reference SaitoSai20a, Theorem 3.1(2)] and the semipurity of $F_1$. This implies that the morphism labeled $\pi _*(\alpha ^*)$ is injective too. Similarly, the embedding $\mathcal {V} \to (\mathbf {P}^1, 0)\otimes \mathcal {X}$ induces another injective morphism $F_{1, (\mathbf {P}^1, 0)\otimes \mathcal {X}} \to j_*F_{1,\mathcal {V}}$. In total, we obtain the following commutative diagram:

(2.13.9)

with exact rows, defining the cokernels ${\underline {\Sigma }}$, ${\underline {\Lambda }}$ and ${\underline {\Lambda }}(0)$, as well as the map $\varphi $. Applying $R\Gamma (\mathbf {P}^1_X,-)$ yields:

with exact rows and in which the $\partial _i$ are the connecting homomorphisms and where:

$$\begin{align*}\Sigma:= H^0(\mathbf{P}^1_X, {\underline{\Sigma}}), \quad \Lambda:= H^0(\mathbf{P}^1_X, {\underline{\Lambda}}), \quad \Lambda(0):=H^0(\mathbf{P}^1_X, {\underline{\Lambda}}(0)).\end{align*}$$

The group $H^1(\mathbf {P}^1, F_{1, (\mathbf {P}^1, 0)\otimes \mathcal {X}})$ vanishes by the cube invariance of cohomology (see [Reference SaitoSai20a, Theorem 9.3]), thus, $\partial _{2|\Lambda (0)}$ is surjective, the vanishing (2.13.4) for $i=1$ will follow, if we can show:

(2.13.10)$$ \begin{align}\Lambda(0)\subset \varphi(\Sigma). \end{align} $$

Note that ${\underline {\Sigma }}$, ${\underline {\Lambda }}$ and ${\underline {\Lambda }}(0)$ have support in $0\times X\subset U$, so we can compute the global sections on U instead of $\mathbf {P}^1$ to show (2.13.10). Now, since $H^1(U, F_{1,\mathcal {U}})=0$, by Lemma 2.10, unravelling the definitions, we obtain from (2.13.3) and (2.13.9) with $G:=F(-\otimes \mathcal {X})$ and $\mathbf {A}^1_s=\mathbf {P}^1\setminus \{\infty \}$ the following descriptions:

$$\begin{align*}\Sigma= \frac{G((\mathbf{A}^1_y,0)\otimes (\mathbf{A}^1_s,0))} {\alpha^* G((\mathbf{A}^1_x,0)\otimes \mathbf{A}^1_s)}, \end{align*}$$
$$\begin{align*}\Lambda= \frac{G((\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s\setminus\{0\},\emptyset))} { G((\mathbf{A}^1_x,0)\otimes \mathbf{A}^1_s)}, \end{align*}$$
(2.13.11)$$ \begin{align}\Lambda(0)= \frac{G((\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s,0))} { G((\mathbf{A}^1_x,0)\otimes \mathbf{A}^1_s)}. \end{align} $$

By [Reference SaitoSai20a, Lemma 5.9], we have isomorphisms (see Notation 2.1):

(2.13.12)$$ \begin{align}\frac{G(\overline{\square}^{(1)}_x\otimes (\mathbf{A}^1_s,0))}{G((\mathbf{P}^1_x,\infty)\otimes(\mathbf{A}^1_s,0))} \xrightarrow{\simeq} \frac{G((\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s,0))}{G(\mathbf{A}^1_x\otimes(\mathbf{A}^1_s,0))},\end{align} $$
(2.13.13)$$ \begin{align}\frac{G(\overline{\square}^{(1)}_x\otimes \overline{\square}^{(1)}_s)}{G(\overline{\square}^{(1)}_x\otimes(\mathbf{P}^1_s,\infty))} \xrightarrow{\simeq} \frac{G(\overline{\square}^{(1)}_x\otimes (\mathbf{A}^1_s,0))}{G(\overline{\square}^{(1)}_x\otimes \mathbf{A}^1_s)}.\end{align} $$

Write j for the open immersion $(\mathbf {A}^1_x,0) \hookrightarrow \overline {\square }^{(1)}_x$. The base change of $j^*$ induces a commutative diagram:

(2.13.14)

The horizontal composite morphism is zero by (2.13.11), hence, the kernel of the diagonal arrow contains $G(\overline {\square }^{(1)}_x\otimes \mathbf {A}^1_s)$. Next, note that from (2.13.12), we get the surjective morphism:

(2.13.15)$$ \begin{align} G(\overline{\square}^{(1)}_x\otimes (\mathbf{A}^1_s,0)) \oplus G(\mathbf{A}^1_x \otimes (\mathbf{A}^1_s, 0)) \to G( (\mathbf{A}^1_x,0) \otimes (\mathbf{A}^1_s, 0))\to 0. \end{align} $$

Combining (2.13.15), (2.13.13) and (2.13.14), we get a surjection:

(2.13.16)