TY - JOUR

T1 - Bounds on cohomological support varieties

AU - Briggs, Benjamin

AU - Grifo, Eloísa

AU - Pollitz, Josh

N1 - Publisher Copyright: © 2024 by the author(s).

PY - 2024

Y1 - 2024

N2 - 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.

AB - 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.

KW - Cohomological support variety

KW - dg algebras

KW - Golod rings

KW - homotopy Lie algebra

KW - levels

KW - Lusternik–Schnirelmann category

KW - thick subcategories

U2 - 10.1090/btran/182

DO - 10.1090/btran/182

M3 - Journal article

AN - SCOPUS:85188842096

VL - 11

SP - 703

EP - 726

JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

SN - 2330-0000

ER -