We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the selfextension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd FrobeniusPerron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.
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FrobeniusPerron theory for projective schemes
The FrobeniusPerron theory of an endofunctor of a k \Bbbk linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study FrobeniusPerron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the FrobeniusPerron dimension for domestic and tubular weighted projective lines, define FrobeniusPerron generalizations of CalabiYau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain ArtinSchelter regular and finitedimensional algebras.
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 Award ID(s):
 2001015
 NSFPAR ID:
 10414593
 Date Published:
 Journal Name:
 Transactions of the American Mathematical Society
 Volume:
 376
 Issue:
 4
 ISSN:
 00029947
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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