Descendability of Faithfully Flat Covers of Perfect Stacks
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910951615234048 |
|---|---|
| author | Jiang, Andy |
| author_facet | Jiang, Andy |
| contents | 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 $2^{|π_*(R)|} < \aleph_ω$, $p_*(\mathcal{O}_{\mathrm{Spec}(R)})$ is a descendable algebra in $\mathrm{QCoh}({\mathfrak{X}})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Descendability of Faithfully Flat Covers of Perfect Stacks Jiang, Andy Algebraic Geometry 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 $2^{|π_*(R)|} < \aleph_ω$, $p_*(\mathcal{O}_{\mathrm{Spec}(R)})$ is a descendable algebra in $\mathrm{QCoh}({\mathfrak{X}})$. |
| title | Descendability of Faithfully Flat Covers of Perfect Stacks |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2505.12472 |