Bounds on cohomological support varieties
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917710868250624 |
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| author | Briggs, Benjamin Grifo, Eloísa Pollitz, Josh |
| author_facet | Briggs, Benjamin Grifo, Eloísa Pollitz, Josh |
| contents | Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$ of finitely generated $R$-modules an algebraic variety $V_R(M)$ that encodes homological properties of $M$. We give lower bounds for the dimension of $V_R(M)$ in terms of classical invariants of $R$. In particular, when $R$ is Cohen-Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$ has finite projective dimension, we also give an upper bound for $ \dim V_R(M)$ in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_15574 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounds on cohomological support varieties Briggs, Benjamin Grifo, Eloísa Pollitz, Josh Commutative Algebra Algebraic Geometry K-Theory and Homology 13D09 (Primary) 13C15, 13D02, 13D07, 13H10, 14M10, 16E45 (Secondary) Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$ of finitely generated $R$-modules an algebraic variety $V_R(M)$ that encodes homological properties of $M$. We give lower bounds for the dimension of $V_R(M)$ in terms of classical invariants of $R$. In particular, when $R$ is Cohen-Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$ has finite projective dimension, we also give an upper bound for $ \dim V_R(M)$ in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. |
| title | Bounds on cohomological support varieties |
| topic | Commutative Algebra Algebraic Geometry K-Theory and Homology 13D09 (Primary) 13C15, 13D02, 13D07, 13H10, 14M10, 16E45 (Secondary) |
| url | https://arxiv.org/abs/2210.15574 |