Support Varieties and cohomology of Verdier quotients of stable category of complete intersection rings

Let $(A,mathfrak{m})$ be a complete intersection with $k = A/mathfrak{m}$ algebraically closed. Let CMS(A) be the stable category of maximal CM $A$-modules. For a large class of thick subcategories $mathcal{S}$ of CMS(A) we show that there is a theory of support varieties for the Verdier quotient $mathcal{T} = $ CMS(A)$/mathcal{S}$. As an application we show that the analogous version of Auslander-Reiten conjecture, Murthys result, Avramov-Buchweitz result on symmetry of vanishing of cohomology holds for $mathcal{T}$.