Frobenius--Witt cotangent complex for derived rings

Abstract

Shimada recently showed that Frobenius--Witt cotangent complex, the animation of Saito's Frobenius--Witt differentials, vanishes on perfectoid rings, thus it serves as a candidate of ``absolute'' cotangent complex. In this article, we give a pullback description of their arithmetic extension, and as a consequence, we give a direct description of Frobenius--Witt cotangent complex, which leads to generalizations to derived rings and animated pre-log rings. This description allows us to compute Frobenius--Witt cotangent complex of derived $δ$-rings as well. We also propose a version of Frobenius--Witt cotangent complex relative to derived $δ$-rings, and establish a vanishing result for prisms, which generalizes vanishing of Frobenius--Witt cotangent complex of perfectoid rings. Finally, independently of previous considerations, we give a regularity criterion via Frobenius--Witt cotangent complex for $p$-local Noetherian (not necessarily local) rings without $F$-finiteness. We also record the flatness of Frobenius--Witt cotangent complex of valuation rings, with essential ideas due to ChatGPT-6 Astra.