Support varieties over skew complete intersections via derived braided Hochschild cohomology

TL;DR

This paper develops a support variety theory for skew complete intersections using derived braided Hochschild cohomology, proving key properties and applications like the Generalized Auslander-Reiten Conjecture.

Contribution

It introduces a novel support theory for skew complete intersections based on derived braided Hochschild cohomology, extending existing frameworks to non-commutative settings.

Findings

Derived braided Hochschild cohomology acts on Ext groups as a noetherian algebra.

Support theory is established when parameters are roots of unity.

Applications include proofs of the Generalized Auslander-Reiten Conjecture and symmetric complexity.

Abstract

In this article we study a theory of support varieties over a skew complete intersection $R$, i.e. a skew polynomial ring modulo an ideal generated by a sequence of regular normal elements. We compute the derived braided Hochschild cohomology of $R$ relative to the skew polynomial ring and show its action on $\mathrm{Ext}_R(M,N)$ is noetherian for finitely generated $R$-modules $M$ and $N$ respecting the braiding of $R$. When the parameters defining the skew polynomial ring are roots of unity we use this action to define a support theory. In this setting applications include a proof of the Generalized Auslander-Reiten Conjecture and that $R$ possesses symmetric complexity.