[Paper Review] Purity of reciprocity sheaves

Shuji Saito|arXiv (Cornell University)|Apr 8, 2017

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper proves a conjecture by Kahn-Saito-Yamazaki on the purity of reciprocity sheaves by extending Voevodsky's homotopy invariance framework to modulus sheaves. It establishes that reciprocity sheaves satisfy ${\overline{\square}}$-invariance and local purity, enabling the construction of a triangulated category of motives with modulus that captures non-homotopy-invariant motivic phenomena.

ABSTRACT

The purpose of this paper is to prove a conjecture on reciprocity sheaves by Kahn-Saito-Yamazaki. This is accomplished by extending Voevodsky's fundamental results on homotopy invariant (pre)sheaves with transfers to its generalizations, reciprocity sheaves and cube-invariant sheaves in the context of theory of modulus (pre)sheave with transfers. The main results of this paper is expected to play a crucial role in deducing the main properties of the triangulated category of motives with modulus, which is a new triangulated category enlarging Voevodsky's triangulated category of motives to encompass non homotopy invariant motivic phenomena.

Motivation & Objective

• To prove the conjecture by Kahn-Saito-Yamazaki on purity for reciprocity sheaves in the context of modulus sheaves.
• To extend Voevodsky's theory of homotopy invariant presheaves with transfers to the broader framework of reciprocity sheaves and cube-invariant sheaves.
• To establish foundational properties of the triangulated category of motives with modulus, which generalizes Voevodsky's category to include non-homotopy-invariant phenomena.
• To show that cohomology presheaves of reciprocity sheaves are ${\overline{\square}}$-invariant and satisfy local injectivity and vanishing theorems.
• To demonstrate that the sheafification process preserves ${\overline{\square}}$-invariance, crucial for the construction of the category of motives with modulus.

Proposed method

• Introduces the category ${\mathbf{RSC}}$ of reciprocity sheaves as a full abelian subcategory of ${\mathbf{PST}}$, generalizing homotopy invariant sheaves.
• Defines $h_0(\mathfrak{X})$ as a quotient of $\mathbb{Z}_{\operatorname{tr}}(X)$ using pairs $\mathfrak{X} = (\overline{X}, X_\infty)$, where $\overline{X}$ is proper and $X_\infty$ is a boundary divisor.
• Establishes ${\overline{\square}}$-invariance of cohomology presheaves via the cokernel formula $h_0(\mathfrak{X})(Y) = \operatorname{Coker}(\mathbf{\underline{M}Cor}(Y \otimes \overline{\square}, \mathfrak{X}) \xrightarrow{i_0^* - i_1^*} \mathbf{Cor}(Y,X))$.
• Applies local injectivity and Gysin map techniques to prove purity theorems for reciprocity sheaves with modulus.
• Uses Nisnevich cohomology and sheafification to show that $H^i(X_{\operatorname{Nis}}, (F_{\operatorname{Nis}})_X) \simeq H^i(X_{\operatorname{Nis}}, (\underline{a}_{\operatorname{Nis}} G)_{(X,\emptyset)})$ for $F \in \mathbf{HI} \cap \mathbf{NST}$.
• Employs technical lemmas on henselian local rings and proper morphisms to control support and finiteness conditions in cohomological arguments.

Experimental results

Research questions

• RQ1 Does the conjecture on purity of reciprocity sheaves hold in the context of modulus sheaves?
• RQ2 Can Voevodsky’s homotopy invariance framework be extended to non-homotopy-invariant sheaves via the theory of reciprocity sheaves?
• RQ3 Are cohomology presheaves of reciprocity sheaves ${\overline{\square}}$-invariant and locally injective?
• RQ4 Does the sheafification of a ${\overline{\square}}$-invariant presheaf preserve ${\overline{\square}}$-invariance?
• RQ5 Can the triangulated category of motives with modulus be constructed using the purity and invariance results for reciprocity sheaves?

Key findings

• The conjecture of Kahn-Saito-Yamazaki on purity of reciprocity sheaves is proven using ${\overline{\square}}$-invariance and local injectivity techniques.
• Cohomology presheaves of reciprocity sheaves are shown to be ${\overline{\square}}$-invariant, generalizing Voevodsky’s homotopy invariance to the modulus setting.
• The sheafification functor preserves ${\overline{\square}}$-invariance, ensuring that the Nisnevich sheafification of a ${\overline{\square}}$-invariant presheaf remains ${\overline{\square}}$-invariant.
• The category $\mathbf{RSC}$ contains $\mathbf{HI}$ and includes important objects like $\Omega^i$, $W_n\Omega^i$, and the presheaf of algebraic groups with unipotent parts.
• The Gysin map and fibration techniques are used to prove that $H^i(X_{\operatorname{Nis}}, (F_{\operatorname{Nis}})_X) \simeq H^i(X_{\operatorname{Nis}}, (\underline{a}_{\operatorname{Nis}} G)_{(X,\emptyset)})$ for $F \in \mathbf{HI} \cap \mathbf{NST}$, supporting the purity result.
• The vanishing theorem for cohomology with support is established, showing that $H^i_c(X, F) = 0$ for $i < 0$ and $F \in \mathbf{RSC}$, which is essential for the triangulated structure of motives with modulus.