Purity of reciprocity sheaves

Saito, Shuji

doi:10.1016/j.aim.2020.107067

Cited by 23 publications

(94 citation statements)

References 9 publications

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“…Hence our assumption implies that a K comes from G 1,Nis P 1 K ,∞ and the desired assertion follows from the cube invariance of G 1,Nis (see [45,Theorem 10.1]) and Remark 4.11:…”

Section: The Following Claim Clearly Implies Axiom (C4)mentioning

confidence: 83%

“…3.4. We recall some more definitions and results from [24,25,45] related to Nisnevich sheaves. For F ∈ MPST and X = X,X ∞ ∈ MCor we denote by F X the presheaf on Xé t defined by…”

Section: It Is Shown Inmentioning

confidence: 99%

“…Lemma 4.10 says that τ ! CI(F ) ⊂ τ CI sp , in the notation of[45].https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1474748021000074 Downloaded from https://www.cambridge.org/core.…”

mentioning

confidence: 99%

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Reciprocity Sheaves and Their Ramification Filtrations

Rülling¹,

Saito²

2021

J. Inst. Math. Jussieu

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We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on $F(L)$ , where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank $1$ connections (in characteristic $0$ ), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

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“…Hence our assumption implies that a K comes from G 1,Nis P 1 K ,∞ and the desired assertion follows from the cube invariance of G 1,Nis (see [45,Theorem 10.1]) and Remark 4.11:…”

Section: The Following Claim Clearly Implies Axiom (C4)mentioning

confidence: 83%

“…3.4. We recall some more definitions and results from [24,25,45] related to Nisnevich sheaves. For F ∈ MPST and X = X,X ∞ ∈ MCor we denote by F X the presheaf on Xé t defined by…”

Section: It Is Shown Inmentioning

confidence: 99%

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Reciprocity Sheaves and Their Ramification Filtrations

Rülling¹,

Saito²

2021

J. Inst. Math. Jussieu

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“…We briefly recall some terminology and notations from the theory of modulus sheaves with transfers; see [14], [15], [18] and [28] for details. A modulus pair X = (X,X ∞ ) consists of a separated k-scheme of finite type X and an effective (or empty) Cartier divisor X ∞ such that X := X \ |X ∞ | is smooth; it is called proper if X is proper over k. Given two modulus pairs X = (X,X ∞ ) and Y = (Y ,Y ∞ ), with opens X := X \ |X ∞ | and Y := Y \ |Y ∞ |, an admissible left proper prime correspondence from X to Y is given by an integral closed subscheme Z ⊂ X × Y that is finite and surjective over a connected component of X, such that the normalisation of its closure Connectivity and purity for logarithmic motives…”

Section: Notations and Recollections On Reciprocity Sheavesmentioning

confidence: 99%

Connectivity and Purity for Logarithmic Motives

Binda¹,

Merici²

2021

J. Inst. Math. Jussieu

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The goal of this article is to extend the work of Voevodsky and Morel on the homotopy t-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel’s connectivity theorem and show a purity statement for $({\mathbf {P}}^1, \infty )$ -local complexes of sheaves with log transfers. The homotopy t-structure on ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ is proved to be compatible with Voevodsky’s t-structure; that is, we show that the comparison functor $R^{{\overline {\square }}}\omega ^*\colon {\operatorname {\mathbf {DM}^{eff}}}(k)\to {\operatorname {\mathbf {logDM}^{eff}}}(k)$ is t-exact. The heart of the homotopy t-structure on ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn-Saito-Yamazaki and Rülling.

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“…Its affirmative answer would simplify the right hand side of Theorem 2 (2) under two additional conditions (i) and (ii) below. (These conditions turn out to be essential in [Sai20].) Question 1.…”

Section: H Imentioning

confidence: 99%