Bounds on cohomological support varieties

TL;DR

This paper investigates bounds on the dimensions of cohomological support varieties over local rings, revealing limitations in realizability for certain rings and classifying possible varieties over Golod rings.

Contribution

It provides new lower and upper bounds for the dimension of support varieties, and classifies varieties over Golod rings, advancing understanding of their homological properties.

Findings

Lower bounds for support variety dimensions in terms of ring invariants.

Existence of varieties not realizable as support varieties over certain rings.

Complete classification of support varieties over Golod rings.

Abstract

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.