Descendability of Faithfully Flat Covers of Perfect Stacks

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 $\mathrm{QCoh}({\mathfrak{X}})$.