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On the algebraizability of formal deformations in $K$-cohomology

Eoin Mackall

Abstract

We show that algebraizability of the functors $R^1π_*\mathcal{K}^M_{2,X}$ and $R^2π_*\mathcal{K}^M_{2,X}$ is a stable birational invariant for smooth and proper varieties $π:X\rightarrow k$ defined over an algebraic extension $k$ of $\mathbb{Q}$. The same is true for the étale sheafifications of these functors as well. To get these results we introduce a notion of relative $K$-homology for schemes of finite type over a finite dimensional, Noetherian, excellent base scheme over a field. We include this material in an appendix.

On the algebraizability of formal deformations in $K$-cohomology

Abstract

We show that algebraizability of the functors and is a stable birational invariant for smooth and proper varieties defined over an algebraic extension of . The same is true for the étale sheafifications of these functors as well. To get these results we introduce a notion of relative -homology for schemes of finite type over a finite dimensional, Noetherian, excellent base scheme over a field. We include this material in an appendix.
Paper Structure (9 sections, 15 theorems, 61 equations)

This paper contains 9 sections, 15 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Background
  3. In the case when $i=2$
  4. In the case $i=1$
  5. $K$-cohomology and relative $K$-homology
  6. Topological preliminaries
  7. $K$-cohomology
  8. Functorality of $K$-cohomology
  9. Relative $K$-homology

Key Result

Lemma 3.2

Fix an algebraic extension $k/\mathbb{Q}$ and let $\pi:X\rightarrow k$ be a smooth, proper, and geometrically connected scheme. Let $\mathcal{E}$ be a finite rank locally free sheaf on $X$ and $\varphi:\mathbb{P}(\mathcal{E})\rightarrow k$ be the structure map of the associated projective bundle. Th

Theorems & Definitions (54)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Example 3.5
  • ...and 44 more