TateHochschild cohomology

General Introduction
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Search results
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Online References
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1109.4019 TateHochschild homology and cohomology of Frobenius algebras from arXiv Front: math.KT by Petter Andreas Bergh, David A. Jorgensen We study TateHochschild homology and cohomology for a twosided Noetherian Gorenstein algebra. These (co)homology groups are defined for all degrees, nonnegative as well as negative, and they agree with the usual Hochschild (co)homology groups for all degrees larger than the injective dimension of the algebra. We prove certain duality theorems relating the TateHochschild (co)homology groups in positive degree to those in negative degree, in the case where the algebra is Frobenius. We explicitly compute all TateHochschild (co)homology groups for certain classes of Frobenius algebras, namely, certain quantum complete intersections.
arXiv:1209.4888 Tate and TateHochschild Cohomology for finite dimensional Hopf Algebras from arXiv Front: math.KT by Van C. Nguyen Let A be any finite dimensional Hopf algebra over a field k. We generalize the notion of Tate cohomology for A, which is defined in both positive and negative degrees, and compare it with the TateHochschild cohomology of A that was presented by Bergh and Jorgensen. We introduce cup products that make the Tate and TateHochschild cohomology of A become graded rings. We establish the relationship between these rings, which turns out to be similar to that in the ordinary nonTate cohomology case. As an example, we explicitly compute the TateHochschild cohomology for a finite dimensional (cyclic) group algebra. In another example, we compute both the Tate and TateHochschild cohomology for a Taft algebra, in particular, the Sweedler algebra H_4.
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