Support varieties and cohomology of Verdier quotients of the stable category of complete intersection rings

Let (𝐴,𝔪) be a complete, complete intersection with 𝑘=𝐴/𝔪 algebraically closed. Let CM¯⁢(𝐴) be the stable category of maximal Cohen-Macaulay A-modules. For a large class of thick subcategories 𝒮 of CM¯⁢(𝐴) we show that there is a theory of support varieties for the Verdier quotient 𝒯=CM¯⁢(𝐴)/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 𝒯