Bounds on cohomological support varieties

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 in...

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Bibliographic Details
Published in:arXiv.org 2024-07
Main Authors: Briggs, Benjamin, Grifo, Eloísa, Pollitz, Josh
Format: Article
Language:English
Subjects:
Lie groups
Lower bounds
Rings (mathematics)
Upper bounds
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2210.15574