Descendability of Faithfully Flat Covers of Perfect Stacks

2025/05/18 by Andy Jiang, Jiang, Andy · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2505.12472

openalex publication_date 2025/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 < ℵ_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 \mathfrakX with a faithfully flat cover p : Spec(R) → \mathfrakX, where R is a Noetherian 𝔼_∞-ring of finite Krull dimension or satisfies the cardinality bound 2^|π_*(R)| < ℵ_ω, p_*(O_Spec(R)) is a descendable algebra in QCoh(\mathfrakX).

Cited by

• Dualizability of derived categories of algebraic stacks
• Report on 𝔼_∞-descendability

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