Hochschild cohomology of algebras arising from categories and from bounded quivers

CLAUDE CIBILS; ANDREA SOLOTAR; MARCELO LANZILOTTA; EDUARDO MARCOS

JOURNAL OF NONCOMMUTATIVE GEOMETRY

EUROPEAN MATHEMATICAL SOC

Lugar: Zürich; Año: 2019 vol. 13 p. 1011 - 1053

1661-6952

The main objective of this paper is to provide a theory for computing theHochschild cohomology of algebras arising from a linear category with finitelymany objects and zero compositions. For this purpose, we consider such acategory using an ad hoc quiver Q, with an algebra associated to each vertexand a bimodule to each arrow. The computation relies on cohomologicalfunctors that we introduce, and on the combinatorics of the quiver. Onepoint extensions are occurrences of this situation, and Happel?s long exactsequence is a particular case of the long exact sequence of cohomology that weobtain via the study of trajectories of the quiver. We introduce cohomologyalong paths, and we compute it under suitable Tor vanishing hypotheses. Thecup product on Hochschild cohomology enables us to describe the connectinghomomorphism of the long exact sequence.Algebras arising from a linear category where the quiver is the round tripone, provide square matrix algebras which have two algebras on the diagonaland two bimodules on the corners. If the bimodules are projective, we showthat five-terms exact sequences arise. If the bimodules are free of rank one,we provide a complete computation of the Hochschild cohomology. On theother hand, if the corner bimodules are projective without producing newcycles, Hochschild cohomology in large enough degrees is that of the productof the algebras on the diagonal.As a by-product, we obtain some families of bound quiver algebras whichare of infinite global dimension, and have Hochschild cohomology zero in largeenough degrees.