A new proof of the cohomological criterion for noetherian regular local rings

TL;DR

This paper presents a new proof of the cohomological criterion characterizing noetherian regular local rings, utilizing the change-of-ring spectral sequence in the derived category.

Contribution

It introduces a novel proof method for the cohomological criterion using spectral sequences, enhancing understanding of regular local rings.

Findings

Equivalence of regularity with vanishing of Tor for large i

Application of change-of-ring spectral sequence in the proof

Clarification of the cohomological criterion for regular local rings

Abstract

Let $(A,\mathfrak{m}, k=A/\mathfrak{m})$ be a noetherian local ring. Then it is equivalent $n = \dim A = \dim_k \mathfrak{m}/\mathfrak{m}^2$ and $\mathrm{Tor}^A_i(k,k) = 0$ for all $i \gg 0$. The article gives a proof with the change-of-ring spectral sequence in the derived category form.