AI 中文总结
本文对Frobenius--Witt余切复形的算术扩张给出拉回描述，刻画了该复形并将其推广到导出环等，还建立了棱柱的消失结果，推广了完美畴环的相关消失性。
AI 中文摘要
Shimada近期证明，Saito的Frobenius--Witt微分的动画（animation）——Frobenius--Witt余切复形在完美畴（perfectoid）环上消失，因此它可作为“绝对”余切复形的候选。本文中，我们给出其算术扩张的拉回描述，由此直接刻画Frobenius--Witt余切复形，进而推广到导出环和预动画预对数环（animated pre-log rings）。该描述还使我们能计算导出δ-环的Frobenius--Witt余切复形；我们还提出相对于导出δ-环的Frobenius--Witt余切复形版本，并建立棱柱（prisms）的消失结果，该结果推广了完美畴环的Frobenius--Witt余切复形的消失性。
英文摘要
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.
Comments21 pages, added an appendix on flatness of Frobenius--Witt cotangent complex of valuation rings, after discussion with ChatGPT-6 Astra
