[Paper Review] $K$-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic $p$

Matthew Morrow|arXiv (Cornell University)|Dec 15, 2015

Algebraic Geometry and Number Theory23 references3 citations

TL;DR

This paper establishes pro versions of the Geisser–Levine and Bloch–Kato–Gabber theorems in characteristic $p$, showing that the mod $p^r$ pro $K$-groups of a regular $F_p$-algebra modulo powers of an ideal are isomorphic to the logarithmic Hodge–Witt sheaves on the formal scheme. The key result is an isomorphism $\{K_n(A/I^s)/p^r\}_s \simeq \{W_r\Omega_{A/I^s,\text{log}}^n\}_s$, extending classical $K$-theory isomorphisms to formal schemes via topological cyclic homology and de Rham–Witt theory.

ABSTRACT

We describe the mod $p^r$ pro $K$-groups $\{K_n(A/I^s)/p^r\}_s$ of a regular local $\mathbb F_p$-algebra $A$ modulo powers of a suitable ideal $I$, in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of Geisser-Levine and Bloch-Kato-Gabber. This is achieved by combining the pro Hochschild-Kostant-Rosenberg theorem in topological cyclic homology with the development of the theory of de Rham-Witt complexes and logarithmic Hodge-Witt sheaves on formal schemes in characteristic $p$. Applications include the following: the infinitesimal part of the weak Lefschetz conjecture for Chow groups; a $p$-adic version of Kato-Saito's conjecture that their Zariski and Nisnevich higher dimensional class groups are isomorphic; continuity results in $K$-theory; and criteria, in terms of integral or torsion étale-motivic cycle classes, for algebraic cycles on formal schemes to admit infinitesimal deformations. Moreover, in the case $n=1$, we compare the étale cohomology of $W_rΩ^1_ ext{log}$ and the fppf cohomology of $\mathbfμ_{p^r}$ on a formal scheme, and thus present equivalent conditions for line bundles to deform in terms of their classes in either of these cohomologies.

Motivation & Objective

• To extend classical $K$-theory isomorphisms—previously valid for smooth varieties in characteristic $p$—to the setting of regular formal schemes.
• To establish a pro version of the isomorphism between Milnor $K$-theory, $K$-theory, and logarithmic Hodge–Witt groups in the context of infinitesimal thickenings.
• To develop the theory of de Rham–Witt complexes and logarithmic Hodge–Witt sheaves on formal schemes in characteristic $p$.
• To apply the results to problems in $K$-theory continuity, cycle class deformations, and the weak Lefschetz conjecture.
• To compare étale and fppf cohomologies of $W_r\Omega^1_{\text{log}}$ and $\boldsymbol{\mu}_{p^r}$, providing criteria for line bundle deformations.

Proposed method

• Use the pro Hochschild–Kostant–Rosenberg theorem in topological cyclic homology to relate $K$-theory to differential forms.
• Develop the theory of de Rham–Witt complexes and logarithmic Hodge–Witt sheaves on formal schemes over $\mathbb{F}_p$-algebras.
• Apply the Cartier isomorphism and study the $p$-filtration on the de Rham–Witt complex for regular formal schemes.
• Construct a formal dlog map from Milnor $K$-theory to logarithmic Hodge–Witt sheaves and analyze its pro properties.
• Use coherent duality and the Artin–Rees property to control cohomological vanishing in Zariski and étale topologies.
• Apply the $5$-lemma and homotopy cartesian squares to compare continuous cohomology of $K$-theory and logarithmic Hodge–Witt sheaves.

Experimental results

Research questions

• RQ1 How can the classical isomorphisms between $K$-theory, Milnor $K$-theory, and Hodge–Witt groups be extended to formal schemes in characteristic $p$?
• RQ2 What is the pro $K$-theory of a regular $F_p$-algebra modulo powers of an ideal, and how does it relate to logarithmic Hodge–Witt sheaves?
• RQ3 Under what conditions do algebraic cycles on formal schemes admit infinitesimal deformations, and how can this be detected via étale-motivic cycle classes?
• RQ4 How do the étale cohomology of $W_r\Omega^1_{\text{log}}$ and the fppf cohomology of $\boldsymbol{\mu}_{p^r}$ compare for line bundles on formal schemes?
• RQ5 What continuity properties does $K$-theory satisfy in characteristic $p$, and how do these relate to crystalline cohomology?

Key findings

• The pro abelian group $\{K_n(A/I^s)/p^r\}_s$ is isomorphic to $\{W_r\Omega_{A/I^s,\text{log}}^n\}_s$ for a regular, $F$-finite $\mathbb{F}_p$-algebra $A$ and ideal $I$ such that $A/I$ is gnc and local.
• The kernel of the map $\{K_n^M(A/I^s)/p^r\}_s \to \{K_n(A/I^s)/p^r\}_s$ is controlled and vanishes when $I$ is principal and $A/I$ is regular.
• A short exact sequence is established: $0 \to \{\mathbb{W}_s\Omega_R^{n-1}/p^r\}_s \xrightarrow{\text{dlog} \circ \gamma_n} \{W_r\Omega_{R[t]/t^s,\text{log}}^n\}_s$, linking relative and absolute Hodge–Witt groups.
• The map $H^i_{\text{Zar}}(Y, \{\mathcal{K}_{n,Y_s}\}) \to H^i_{\text{Zar}}(Y, \mathcal{K}_{n,Y})$ has kernel and cokernel killed by a power of $p$ for $i+n < d-1$, implying continuity in $K$-theory.
• The comparison map $H^i_{\text{ét}}(Y, \{W_s\Omega_{Y_s,\text{log}}^n\}_s) \to H^i_{\text{ét}}(Y, \{W_s\Omega_{Y,\text{log}}^n\}_s)$ has kernel and cokernel killed by a power of $p$ when $i+n < d-1$, linking étale and continuous cohomology.
• The weak Lefschetz conjecture for Chow groups is proven in the infinitesimal setting via the isomorphism between pro $K$-theory and logarithmic Hodge–Witt sheaves.