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

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 $p^{r,m}_{r,n}\Omega^{\bullet}_{X_{\centerdot}}$ 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 $p^{r,m}_{r,n}\Omega^{\bullet}_{X_{\centerdot}}$ 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.