From the introduction: For which algebras does symmetry in the vanishing of cohomology hold? That is, given an algebra \(\Lambda\), does the implication \[ \text{Ext}^i_\Lambda(M,N)=0\text{ for }i\gg 0\Rightarrow\text{Ext}^i_\Lambda(N,M)=0\text{ for }i\gg 0 \] hold for finitely generated modules \(M\) and \(N\)? As shown by L. L. Avramov and R.-O. Buchweitz [in Invent. Math. 142, No. 2, 285-318 (2000; Zbl 0999.13008)], this implication holds for finitely generated modules over commutative local complete intersections.
We show in this paper that Ext-symmetry holds for all graded modules over quantum complete intersections, provided all the defining commutators are roots of unity. We also show that, if such an algebra is symmetric, that is, if it is isomorphic as a bimodule to its own dual, then symmetry holds for all modules.