[Paper Review] Moving lemmas in mixed characteristic and applications

Ivan Panin|arXiv (Cornell University)|Feb 2, 2022

Advanced Topics in Algebra4 citations

TL;DR

This paper establishes new geometric moving lemmas in mixed characteristic settings, enabling significant cohomological applications. It proves the Gersten conjecture for $K_2$, verifies the Grothendieck–Serre conjecture for $SL_{1,D}$, and derives Suslin’s exact sequence, yielding finiteness and isotropy results for quadratic forms and a generalized Cousin complex for cohomology theories in the sense of Panin–Smirnov.

ABSTRACT

The present paper contains new geometric theorems in mixed characteristic case. We derive a bunch of cohomological consequences using these geometric theorems. Among them an isotropy result for quadratic spaces, a purity result for quadratic spaces, a result on the Grothendieck--Serre conjecture for the special linear group of an Azumaya algebra. The Gersten conjecture for the functor K2 is proved. Bloch-Ogus type result is obtained as well. Suslin's exact sequence is derived and its application to a finiteness result is given. A version of the Roitman theorem is proved.

Motivation & Objective

• To develop geometric moving lemmas in mixed characteristic for schemes over a discrete valuation ring (d.v.r.)
• To derive cohomological consequences from these lemmas, particularly for $K$-theory and quadratic forms
• To prove the Gersten conjecture for the $K_2$ functor and the Grothendieck–Serre conjecture for $SL_{1,D}$, where $D$ is an Azumaya algebra
• To establish a generalized Cousin complex for any cohomology theory in the Panin–Smirnov sense
• To derive Suslin’s exact sequence and apply it to finiteness results in étale cohomology

Proposed method

• Utilizes a geometric presentation theorem (Theorem 3.1) to construct affine neighborhoods with finite surjective morphisms to projective space over $A$
• Employs a diagram involving a smooth, irreducible $Ω$-scheme $Ω$ over an open $U'$ with a finite surjective morphism to $Ω$
• Applies the canonical sheaf $ω_{Ω/V} \cong \mathcal{O}_{Ω}$ to ensure duality properties
• Uses the proper base change theorem and properties of $l$-adic cohomology to relate cohomology groups on $X$ and its special fiber $X_v$
• Applies the Weil conjectures and Galois cohomology to prove finiteness of $H^3_{Ét}(X, \mathbb{Q}_l/\mathbb{Z}_l(2))$
• Relies on the structure of $\mathbb{Z}_l$-modules in étale cohomology and the finitely generated property of $H^i_{Ét}(\bar{X}_v, \mathbb{Z}_l(2))$

Experimental results

Research questions

• RQ1 Can moving lemmas be established in mixed characteristic for smooth schemes over a d.v.r.?
• RQ2 Does the Gersten conjecture for $K_2$ hold in this mixed characteristic setting?
• RQ3 Can the Grothendieck–Serre conjecture be proven for $SL_{1,D}$ with $D$ an Azumaya algebra over a d.v.r.?
• RQ4 What are the implications of the Cousin complex for cohomology theories in the Panin–Smirnov framework?
• RQ5 Is Suslin’s exact sequence valid in this context, and what finiteness results follow?

Key findings

• The Gersten conjecture for the $K_2$ functor is proved for smooth schemes over a d.v.r. in mixed characteristic
• The Grothendieck–Serre conjecture is verified for $SL_{1,D}$, where $D$ is an Azumaya algebra over a d.v.r.
• Suslin’s exact sequence is derived, and it is applied to show that $H^3_{Ét}(X, \mathbb{Q}_l/\mathbb{Z}_l(2))$ is finite
• The map $\beta: H^3_{Ét}(X, \mathbb{Q}_l/\mathbb{Z}_l(2)) \to {}_{\{l\}}H^2_{Zar}(X, \underline{K}_2)$ is an isomorphism
• The map $\rho: H^1_{Zar}(X, \underline{K}_2) \otimes \mathbb{Q}_l/\mathbb{Z}_l \to H^1_{Zar}(X_v, \underline{K}_2) \otimes \mathbb{Q}_l/\mathbb{Z}_l$ is shown to be an isomorphism
• A generalized Cousin complex is established for any cohomology theory in the sense of Panin–Smirnov, extending known results to mixed characteristic