08 Fakultät Mathematik und Physik

Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9

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Institute: search.filters.UBSInstitut.Institut für Algebra und ZahlentheorieAuthor: search.filters.author.Cruz, Tiago

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    A characterisation of Morita algebras in terms of covers
    (2021) Cruz, Tiago
    A pair (A, P) is called a cover of EndA(P)op if the Schur functor HomA(P,-) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P. By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P)op is self-injective whenever (A, P) is a cover of EndA(P)op for a faithful projective-injective module P.
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    Higher Morita-Tachikawa correspondence
    (2024) Cruz, Tiago
    Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo‐symmetric algebras. Explicitly, the Morita-Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over finite‐dimensional algebras are exactly the finite‐dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi‐generators and quasi‐cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita-Tachikawa correspondence that take into account relative dominant dimension with respect to a self‐orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring.