Matthew Morrow, Akhil Mathew, and I have posted our new paper [1] 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 [2] 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.

# References

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

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