  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.
