Morita contexts, idempotents, and Hochschild cohomology—with applications to invariant rings

Abstract

Abstract. We investigate how to compare Hochschild cohomology of algebras related by a Morita context. Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent. Along the way, we show that the grade of the stable endomorphism ring as a module over the endomorphism ring controls vanishing of higher groups of selfextensions, and explain the relation to various forms of the Generalized Nakayama Conjecture for Noetherian algebras. As applications of our approach we explore to what extent Hochschild cohomology of an invariant ring coincides with the invariants of the Hochschild cohomology.

Grade, dominant dimension and Gorenstein algebras

“…In order to prove these results, a close relation between dominant dimension and double centraliser properties on the one hand and Auslander-Bridger's concept of grade is being worked out, extending results of Buchweitz [8]. The results give precise connections between these concepts, valid for artin algebras in general.…”

“…Now the proof can be finished as above, using the results of [8] to show that Hom eAe (Ae, eAe) ∼ = eA as right A-modules.…”

“…Then domdim Λ ≥ 2 if and only if the left Λ-module Λe has the double centraliser property which means that End eΛe (Λe) ∼ = Λ (see [26, 7.7]). Buchweitz [8,Proposition 2.9] showed for any idempotent e that Λ ∼ = End eΛe (Λe) if and only if Auslander-Bridger's grade of Λ/ΛeΛ as a right Λ-module is greater than 2.…”

The First Hochschild Cohomology of Square Algebras With it's Stability

“…We denote the generalized matrix algebra by G. Algebraic studying on derivations, generalized derivations and Lie derivations have been studied in (Du, & Wang, 2012), (Li & Wei, 2012), (Li, & Xiao, 2011). For more details and applications see (Buchweitz, 2003). We define generalized matrix algebra…”

Idempotent et cohomologie de Hochschild

“…[4,6]). For the algebra T = [ A M N B ], we also have such a graded morphism ρ * Λ = (α * Λ , β * Λ ) : H * (T , Λ) → H * (A, Y 1 ) × H * (B, X 2 ); in [2], Buchweitz showed that ρ * T = (α * T , β * T ) is an algebra morphism; in [4], Green and Solberg provided a long exact sequence where ρ * T takes place under certain assumptions. We prove the following results.…”