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    Tate Cohomology of Finite Dimensional Hopf Algebras

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    NGUYEN-DISSERTATION-2014.pdf (527.2Kb)
    Date
    2014-06-19
    Author
    Nguyen, Van Cat
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    Abstract
    Let A be a finite dimensional Hopf algebra over a field k. In this dissertation, we study the Tate cohomology Ĥ* (A, k) and Tate-Hochschild cohomology (HH) ̂* (A, A) of A, and their properties. We introduce cup products that make them become graded-commutative rings and establish the relationship between these rings. In particular, we show Ĥ* (A, k) is an algebra direct summand of (HH) ̂* (A, A) as a module over Ĥ* (A, k). When A is a finite group algebra RG over a commutative ring R, we show that the Tate-Hochschild cohomology ring (HH) ̂* (RG, RG) of RG is isomorphic to a direct sum of the Tate cohomology rings of the centralizers of conjugacy class representatives of G. Moreover, our main result provides an explicit formula for the cup product in (HH) ̂* (RG, RG) with respect to this decomposition. When A is symmetric, we show that there are finitely generated A-modules whose Tate cohomology is not finitely generated over the Tate cohomology ring Ĥ* (A, k) of A. It turns out that if a module in a connected component of the stable Auslander-Reiten quiver associated to A has finitely generated Tate cohomology, then so does every module in that component.
    URI
    http://hdl.handle.net/1969.1/153306
    Subject
    Tate cohomology
    stable cohomology
    Hopf algebras
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    • Electronic Theses, Dissertations, and Records of Study (2002– )
    Citation
    Nguyen, Van Cat (2014). Tate Cohomology of Finite Dimensional Hopf Algebras. Doctoral dissertation, Texas A & M University. Available electronically from http : / /hdl .handle .net /1969 .1 /153306.

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