Serre depth and local cohomology
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914364152348672 |
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| author | Ficarra, Antonino |
| author_facet | Ficarra, Antonino |
| contents | We introduce a fundamental homological invariant, called Serre depth, which stratifies Serre's conditions in the same way that depth stratifies the Cohen-Macaulay property. We study the Serre depths of modules over arbitrary Noetherian local rings and over standard graded algebras over a field, extending the polynomial ring case due to Muta and Terai. Under mild hypotheses, we show that the $r$-th Serre depth of a finitely generated module $M$ measures the deviation of $M$ from satisfying Serre's condition $(S_r)$. The main results of the paper can be summarized as follows:
(i) We establish the basic properties of Serre depth and prove that it is invariant under completion.
(ii) If the base ring $R$ is a homomorphic image of a Gorenstein ring, we show that a finitely generated $R$-module $M$ is equidimensional and satisfies $(S_r)$ if and only if its $r$-th Serre depth equals its Krull dimension. Analogous statements are obtained for schemes.
(iii) For a homogeneous ideal in a standard graded polynomial ring over a field, we compare its Serre depths with those of its initial ideal.
(iv) We characterize the Serre depths of a monomial ideal in terms of its skeletons and prove that the Serre depths of sufficiently large powers of a monomial ideal stabilize; the proof uses Presburger arithmetic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17240 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Serre depth and local cohomology Ficarra, Antonino Commutative Algebra Algebraic Geometry Combinatorics Logic We introduce a fundamental homological invariant, called Serre depth, which stratifies Serre's conditions in the same way that depth stratifies the Cohen-Macaulay property. We study the Serre depths of modules over arbitrary Noetherian local rings and over standard graded algebras over a field, extending the polynomial ring case due to Muta and Terai. Under mild hypotheses, we show that the $r$-th Serre depth of a finitely generated module $M$ measures the deviation of $M$ from satisfying Serre's condition $(S_r)$. The main results of the paper can be summarized as follows: (i) We establish the basic properties of Serre depth and prove that it is invariant under completion. (ii) If the base ring $R$ is a homomorphic image of a Gorenstein ring, we show that a finitely generated $R$-module $M$ is equidimensional and satisfies $(S_r)$ if and only if its $r$-th Serre depth equals its Krull dimension. Analogous statements are obtained for schemes. (iii) For a homogeneous ideal in a standard graded polynomial ring over a field, we compare its Serre depths with those of its initial ideal. (iv) We characterize the Serre depths of a monomial ideal in terms of its skeletons and prove that the Serre depths of sufficiently large powers of a monomial ideal stabilize; the proof uses Presburger arithmetic. |
| title | Serre depth and local cohomology |
| topic | Commutative Algebra Algebraic Geometry Combinatorics Logic |
| url | https://arxiv.org/abs/2602.17240 |