Descendability of Faithfully Flat Covers of Perfect Stacks

TL;DR

This paper extends classical results on projective dimensions of flat modules to the setting of perfect stacks, showing that certain faithfully flat covers are descendable under specific algebraic conditions.

Contribution

It generalizes Gruson and Jensen's proof to quasicoherent sheaves over perfect stacks, establishing descendability of covers with Noetherian or cardinality constraints.

Findings

Faithfully flat covers over perfect stacks are descendable under specified conditions.

The arguments from classical ring theory apply to the setting of perfect stacks.

Conditions include Noetherian rings of finite Krull dimension or bounded cardinality.

Abstract

In 1981, L. Gruson and C. U. Jensen gave a new proof of the fact that, over a ring which is either Noetherian of Krull dimension $n$ or of cardinality $< \aleph_n$, the projective dimension of any flat module is at most $n$. In this short paper, we observe that their arguments apply to the setting of quasicoherent sheaves over perfect stacks. As a consequence, we show that for any perfect stack $\mathfrak{X}$ with a faithfully flat cover $p : \mathrm{Spec}(R) \to \mathfrak{X}$, where $R$ is a Noetherian $\mathbb{E}_{\infty}$-ring of finite Krull dimension or satisfies the cardinality bound $2^{|\pi_*(R)|} < \aleph_{\omega}$, $p_*(\mathcal{O}_{\mathrm{Spec}(R)})$ is a descendable algebra in $\mathrm{QCoh}({\mathfrak{X}})$.