Seminar

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For a finite dimensional Hopf algebra A over a field k the cohomological support for the singularity category Sing A can be defined via the action of  the cohomology algebra $H^*(A,k)$ with little reference to the tensor structure. Yet, for various finite tensor categories the cohomological support turns out to respect that structure via the “tensor product property”: $supp(M \otimes N) = supp M \cap supp N$. When the property holds, it often appears to be intimately connected with some kind of alternative description of the cohomological support, “a rank variety”. I’ll describe such an alternative construction, {\it the hypersurface support}, which goes back to the work of Eisenbud, Avramov, Buchweitz and Iyengar in commutative algebra, for the Hopf algebras which are ``non-commutative complete intersections”. One application of this construction is to the open question of “whether tensor product property holds for small quantum groups”, another to calculations of the Balmer spectrum. Joint work with Cris Negron.

Links to notes:

- Part 1: https://jamboard.google.com/d/10l-t2fEa5zP2feJJW7UclTyaO6f0aMJBjNyu8vf-DMI/edit?usp=sharing

- Part 2: https://jamboard.google.com/d/1O9mDdqfgAnSeaA2B54mHArRo4k8GTXJ8z72UEAARGls/edit?usp=sharing

Fellowship Of The Ring: Support Theories For Non-Commutative Complete Intersections