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 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.