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

Abstract

We study the algebraic $K$-theory of smooth schemes over $W_n(\mathbb{k})$, where $\mathbb{k}$ is a perfect field of characteristic $p>0$. For a $p$-adic smooth scheme $X_{\cdot }$ over $W_{\cdot }(k)$, we introduce complexes $p_{r,n}^{r,m}{\Omega }_{X_{\cdot }}^{\bullet }$ and infinitesimal motivic complexes ${\mathbb{Z}}_{X_n}(r)$, and for $0\le i\le p-4$, 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 $p_{r,n}^{r,m}{\Omega }_{X_{\cdot }}^{\bullet }$ with $1\le r\le i$. 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 $n\to \infty$ 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 $W(\mathbb{k})$, and an analysis of multiplicative structures of the mod $p$ relative $K$-theory.

Typos corrected. Section 10.4 is detailed. The abstract and introduction are improved. Comments are welcome!