相关论文: Hochschild homology and trivial extensions
For a truncated quiver algebra over a field of arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finite-dimensional if and only if its global dimension is…
We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof…
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 $\varphi^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$ to that of $C$,…
We obtain a decomposition for the Hochschild cochain complex of a split algebra and we study some properties of the cohomology of each term of this decomposition. Then, we consider the case of trivial extensions, specially of Frobenius…
We consider a class of extensions of associative algebras, which we refer to as ``strongly proj-bounded extensions''. We prove that the finiteness of the left global dimension and the support of the Hochschild homology is preserved by…
This paper presents an infinite family of Koszul self-injective algebras whose Hochschild cohomology ring is finite-dimensional. Moreover, for each $N \geq 5$ we give an example where the Hochschild cohomology ring has dimension $N$. This…
In the paper the class of all solvable extensions of a filiform Leibniz algebra in the infinite-dimensional case is classified. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of residually solvable Leibniz algebra.…
We give two sufficient and necessary conditions for a Hochschild extension of a finite dimensional algebra by its dual bimodule and a Hochschild 2-cocycle to be a symmetric algebra.
We consider associative algebras L over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A - examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split…
We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite…
Let $T$ be a Hochschild extension algebra of a finite dimensional algebra $A$ over a field $K$ by the standard duality $A$-bimodule ${\rm Hom}_K(A,\,K)$. In this paper, we determine the ordinary quiver of $T$ if $A$ is a self-injective…
Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule…
In 1989 Happel conjectured that for a finite-dimensional algebra $A$ over an algebraically closed field $k$, $\gl A< \infty$ if and only if $\hch A < \infty$. Recently Buchweitz-Green-Madsen-Solberg gave a counterexample to Happel's…
It is shown that a bounded quiver algebra having a 2-truncated oriented cycle is of infinite Hochschild homology dimension and global dimension, which generalizes a result of Solotar and Vigu\'{e}-Poirrier to nonlocal ungraded algebras…
We prove without any assumption on the ground field that higher Hochschild homology groups do not vanish for two large classes of algebras whose global dimension in not finite.
An extension of algebras is a homomorphism of algebras preserving identities. We use extensions of algebras to study the finitistic dimension conjecture over Artin algebras. Let $f: B \to A$ be an extension of Artin algebras. We denote by…
Given a finite--dimensional monomial algebra $A$ we consider the trivial extension $TA$ and provide formulae, depending on the characteristic of the field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first Hochschild…
In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to…
We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least…
We generalize a recent result by J.F. Carlson to finite tensor categories having finitely generated cohomology. Specifically, we show that if the Krull dimension of the cohomology ring is sufficiently large, then there exist infinitely many…