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 $2^{|{\pi }_{\ast }(R)|}<{\aleph }_{\omega }$, $p_{\ast }({\mathcal{O}}_{\mathrm{Spec}(R)})$ is a descendable algebra in $\mathrm{QCoh}(\mathfrak{X})$.

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