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On the algebraic $K$-theory of smooth schemes over truncated Witt vectors

Xiaowen Hu

Abstract

We study the algebraic $K$-theory of smooth schemes over $W_n(\Bbbk)$, where $\Bbbk$ is a perfect field of characteristic $p>0$. For a $p$-adic smooth scheme $X_{\centerdot}$ over $W_{\centerdot}(k)$, we introduce complexes $p^{r,m}_{r,n}Ω^{\bullet}_{X_{\centerdot}}$ and infinitesimal motivic complexes $\mathbb{Z}_{X_n}(r)$, and for $0 \leq i \leq p-4$, we establish a Chern character isomorphism between the sheaf $\mathcal{K}_{X_n,X_{m},i}$ and the direct sum of certain cohomology sheaves of $p^{r,m}_{r,n}Ω^{\bullet}_{X_{\centerdot}}$ with $1\leq r\leq i$. This leads to a criterion for $K$-theoretic infinitesimal deformations, which is related to Emerton's $p$-adic variational Hodge conjecture. By taking the limit $n \rightarrow \infty$ with $m=1$, we recover a theorem of Bloch, Esnault, and Kerz on continuous relative algebraic $K$-theory. The proof combines Brun's isomorphism relating $K$-theory to derived cyclic homology, computations of relative cyclic homology over $W(\Bbbk)$, and an analysis of multiplicative structures of the mod $p$ relative $K$-theory.

On the algebraic $K$-theory of smooth schemes over truncated Witt vectors

Abstract

We study the algebraic -theory of smooth schemes over , where is a perfect field of characteristic . For a -adic smooth scheme over , we introduce complexes and infinitesimal motivic complexes , and for , we establish a Chern character isomorphism between the sheaf and the direct sum of certain cohomology sheaves of with . This leads to a criterion for -theoretic infinitesimal deformations, which is related to Emerton's -adic variational Hodge conjecture. By taking the limit with , we recover a theorem of Bloch, Esnault, and Kerz on continuous relative algebraic -theory. The proof combines Brun's isomorphism relating -theory to derived cyclic homology, computations of relative cyclic homology over , and an analysis of multiplicative structures of the mod relative -theory.

Paper Structure

This paper contains 47 sections, 92 theorems, 445 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Comparison with the work of Bloch, Esnault, and Kerz
  4. Defining the infinitesimal motivic complexes
  5. The structure of the mod p products
  6. Further problems
  7. Organization of the paper
  8. Notations and conventions
  9. Acknowledgement
  10. Products in Hochschild homology and cyclic homology
  11. The algebraic products
  12. The topological products and a filtered comparison
  13. The Hochschild and cyclic homology of cdga's
  14. Products in Hochschild homology of cdga's
  15. Products in derived Hochschild homology and cyclic homology
  16. ...and 32 more sections

Key Result

Theorem 1.1

Let $\Bbbk$ be a perfect field of characteristic $p$. Let $X_{\centerdot}$ be a $p$-adic smooth scheme separated and of finite type over $W_{\centerdot}(\Bbbk)$. For $0\leq i\leq p-4$ (resp. $0\leq i\leq p-3$) and $n> m\geq 1$, there is a canonical Chern character isomorphism (resp. epimorphism) of Nisnevich sheaves on $X_1$, where $p^{r,m}_{r,n}\Omega^{\bullet}_{X_{\centerdot}}$ is the complex

Figures (1)

  • Figure 1: Multiplication Table

Theorems & Definitions (211)

  • Theorem 1.1: = Theorem \ref{['thm:relative-comparison-local']}
  • Theorem 1.2: = Proposition \ref{['prop:deformationInK-theory']}(i)
  • Remark 1.3
  • Theorem 1.4: = Proposition \ref{['prop:continuity-alg-K-theory']}
  • Definition 2.1
  • Definition 2.2: Lod98
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 201 more