Finite generation for Hochschild cohomology of Gorenstein monomial algebras

Abstract

We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall-Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann-Holloway-Snashall-Solberg-Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.

Tate–Hochschild cohomology for periodic algebras

“…Our main result requires only the eventual periodicity of a minimal projective resolution of a given Gorenstein algebra. Hence it turns out that [10,Corollary 6.4] and [18,Corollary 3.4] can be obtained from our main result, because monomial Gorenstein algebras and periodic algebras are both eventually periodic Gorenstein algebras. Finally, using tensor algebras, we will provide one of the constructions of eventually periodic Gorenstein algebras.…”

“…0-Gorenstein algebras). On the other hand, it follows from the proof of [10,Corollary 6.4] that monomial Gorenstein algebras are eventually periodic algebras. It also follows from the formula gl.dim Λ = proj.dim Λ e Λ (see [14,Section 1.5]) that algebras of finite global dimension are eventually periodic algebras.…”

“…It was proved in [20] that the Tate-Hochschild cohomology carries a structure of a graded commutative algebra. There are studies on the ring structure of the Tate-Hochschild cohomology, such as [10,17,18,19]. Recently, Dotsenko, Gélinas and Tamaroff proved in [10,Corollary 6.4] that, for a monomial Gorenstein algebra Λ, the Tate-Hochschild cohomology ring HH…”

“…Applying Theorem 3.5 to monomial Gorenstein algebras and to periodic algebras, one obtains[10, Corollary 6.4] and[18, Corollary 3.4], respectively. For an eventually periodic Gorenstein algebra Λ, one can obtain all of the dim k HH * (Λ) by using Theorem 3.5 and the Hochschild cohomology HH• (Λ) := i≥0 Ext i Λ e (Λ, Λ) of Λ (see Example 4.7).…”

Characterization of eventually periodic modules in the singularity categories

“…On the other hand, finite dimensional eventually periodic algebras can be found in several papers. For example, Dotsenko, Gélinas and Tamaroff [14,Corollary 6.4] showed that monomial Gorenstein algebras Λ having infinite projective dimension over Λ e are eventually periodic; and the author [33, Sections 3. 1 and 4] provided examples of finite dimensional eventually periodic algebras (that are not periodic), and he also confirmed that there is an eventually periodic algebra that is not Gorenstein.…”