2019/12/08 by Satoshi Usui, Usui, Satoshi
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1912.03594
openalex publication_date 2019/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study cup product and cap product in Tate-Hochschild theory for a finite dimensional Frobenius algebra. We show that Tate-Hochschild cohomology ring equipped with cup product is isomorphic to singular Hochschild cohomology ring introduced by Wang. An application of cap product occurs in Tate-Hochschild duality; as in Tate (co)homology of a finite group, the cap product with the fundamental class of a finite dimensional Frobenius algebra provides certain duality result between Tate-Hochschild cohomology and homology groups. Moreover, we characterize minimal complete resolutions over a finite dimensional self-injective algebra by means of the notion of minimal complexes introduced by Avramov and Martsinkovsky.