Matthew Morrow, Akhil Mathew, and I have posted our new paper  on the algebraic $\K$-theory of smooth algebras over perfectoid rings. The main results all say that the $p$-adic $\K$-theory of smooth algebras over perfectoid rings which contain a perfectoid valuation ring behaves like the $\K$-theory of smooth commutative rings.

Here are the main theorems, in the mixed characteristic case.

Theorem. Let $\Oscr$ be a perfectoid valuation ring of mixed characteristic and let $A$ be a perfectoid $\Oscr$-algebra. If $R$ is a smooth $A$-algebra of fiber dimension $\leq d$, then

• $\K(A;\bZ_p)\rightarrow\K(A[\tfrac{1}{p}];\bZ_p)$ is $0$-truncated,
• $\K(R;\bZ_p)\rightarrow\K(R[\tfrac{1}{p}];\bZ_p)$ is $d$-truncated,
• $\K(R;\bZ_p)\we\mathrm{KH}(R;\bZ_p)$,
• and if $A=\Oscr$, then in fact $\K(R;\bZ_p)\we\K(R[\frac{1}{p}];\bZ_p)$.

Note that recent results of Clausen–Bhatt–Mathew and Land–Mathew–Meier–Tamme imply that the $\K(1)$-local $\K$-theory of $R$ and $R[\tfrac{1}{p}]$ agree for any commutative ring $R$. However, to get the concrete bounds as in the theorem requires further hypotheses.

The strategy of the proofs is to first use $\mathrm{arc}_p$-descent to reduce the case of general perfectoid $\Oscr$-algebras to perfectoid valuation rings. Then, a combination of commutative algebra results and classical arguments in higher algebraic K-theory are used to show that $p$-torsion modules do not contribute to $K$-theory.

One of the ideas in the paper is to tilt’ the stable $\infty$-category of perfect $p$-torsion complexes to reduce to a characteristic $p$ situation, where we argue directly.

A similar result was proved by Nizioł in  in the case where $A=\Oscr$ is the ring of integers in the $p$-completed algebraic closure of $\overline{\bQ}_p$ and was used by her to give a new proof of the crystalline comparison theorem in $p$-adic Hodge theory.

One question in commutative algebra came up in our work which we did not answer.

Question. Suppose that $R$ is a commutative ring and that $t\in R$ is a non-zero divisor such that $R/t$ is weakly regular stable coherent. Does every finitely presented projective $R[\tfrac{1}{t}]$-module $M$ extend to a finitely presented and $t$-torsion free $R$-module?

At some point, we make an argument with a Bass-style delooping argument which would be more straightforward if the answer to the question is yes’. If you know, please email me and I’ll include an answer here.

 Antieau, Mathew, and Morrow, The K-theory of perfectoid rings, arXiv:2203.06472.

 Nizioł, Crystalline conjecture via K-theory, Ann. sci. Ec. Norm. Sup. (4), 31 (1998), 659-681.