Bounds on cohomological support varieties

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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 VR (M) that encodes homological properties of M. We give lower bounds for the dimension of VR (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 VR (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, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.

Original languageEnglish
JournalTransactions of the American Mathematical Society Series B
Pages (from-to)703-726
Publication statusPublished - 2024

Bibliographical note

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© 2024 by the author(s).

    Research areas

  • Cohomological support variety, dg algebras, Golod rings, homotopy Lie algebra, levels, Lusternik–Schnirelmann category, thick subcategories

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