Hochschild cohomology of relation extension algebras

IBRAHIM ASSEM; M. ANDREA GATICA; RALF SCHIFFLER; RACHEL THAILLEFER

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2016 vol. 220 p. 2471 - 2499

0022-4049

Let B be the split extension of a finite dimensional algebra C by a C-C-bimodule E. We define a morphism of associative graded algebras ϕ∗: HH∗(B) → HH∗(C) from the Hochschild cohomology of B to that of C, extending similar constructions for the first cohomology groups made and studied by Assem,Bustamante, Igusa, Redondo and Schiffler. In the case of a trivial extension B = C ⋉E, we give necessary and sufficient conditions for each ϕ^n to be surjective. We prove the urjectivity of ϕ^1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of ϕ^1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.