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 dimVR(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.Comment: 23 pages. Comments welcom