Hochschild Cohomology of Quiver Algebras Defined by Two Cycles and a Quantum-Like Relation

We consider quiver algebras A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k, and describe the minimal projective bimodule resolution of A _q . In particular, in the case q = 1, we determine the Hochschild cohomology ring of A ₁ and show that it is a finitely generated k-algebra. Moreover the Hochschild cohomology ring of A ₁ modulo nilpotence is isomorphic to the polynomial ring of two variables.     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

Hochschild Cohomology of <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We compute the Hochschild cohomology of any block of q-Schur algebras. We focus on the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of q-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all q-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derive equivalence.     

On Hochschild Cohomology of a Self-injective Special Biserial Algebra Obtained by a Circular Quiver with Double Arrows

We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra Λ _s obtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of Λ _s by generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of Λ _s is finitely generated as an algebra.     

Hochschild cohomology of ? n -Galois coverings of an algebra

We consider the ? _n -Galois covering ?? _n of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872?C1893]. We calculate the dimensions of all Hochschild cohomology groups of ?? _n and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg??s conjecture.     

On Hochschild cohomology rings of Fibonacci algebras

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.     

Homotopy Theory of A∞ Ring Spectra and Applications to MU-Modules

We give a definition of a derivation of an A ring spectrum and relate this notion to topological Hochschild cohomology. Strict multiplicative structure is introduced into Postnikov towers and generalized Adams towers of A -ring spectra. An obstruction theory for lifting multiplicative maps is constructed. The developed techniques are then applied to show that a broad class of complex-oriented spectra admit structures of MU-algebras where MU is the complex cobordism spectrum. Various computations of topological derivations and topological Hochschild cohomology are made.     

Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type A 1

The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let Λ _t be the Yoneda algebra of a reconstruction algebra of type A ₁ over a field k. In this paper, a minimal projective bimodule resolution of Λ _t is constructed, and the k-dimensions of all Hochschild homology and cohomology groups of Λ _t are calculated explicitly.     

On the Hochschild cohomology of tame Hecke algebras

We explicitly calculate a projective bimodule resolution for a special biserial algebra giving rise to the Hecke algebra H_q(S₄){{\mathcal H}_q(S_4)} when q = −1. We then determine the dimensions of the Hochschild cohomology groups.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

Constructions of stable equivalences of Morita type for finite dimensional algebras II

Suppose k is a field. Let A and B be two finite dimensional k-algebras such that there is a stable equivalence of Morita type between A and B. In this paper, we prove that (1) if A and B are representation-finite then their Auslander algebras are stably equivalent of Morita type; (2) The n-th Hochschild homology groups of A and B are isomorphic for all n≥1. A new proof is also provided for Hochschild cohomology groups of self-injective algebras under a stable equivalence of Morita type.     

On Maximal Diagonalizable Lie Subalgebras of the First Hochschild Cohomology

Let A be a basic connected finite dimensional algebra over an algebraically closed field, with ordinary quiver without oriented cycles. Given a presentation of A by quiver and admissible relations, Assem and de la Peña have constructed an embedding of the space of additive characters of the fundamental group of the presentation into the first Hochschild cohomology group of A. We compare the embeddings given by the different presentations of A. In some situations, we characterise the images of these embeddings in terms of (maximal) diagonalizable subalgebras of the first Hochschild cohomology group (endowed with its Lie algebra structure).     

Singular Hochschild cohomology via the singularity category

We show that the singular Hochschild cohomology (= Tate–Hochschild cohomology) of an algebra A is isomorphic, as a graded algebra, to the Hochschild cohomology of the differential graded enhancement of the singularity category of A. The existence of such an isomorphism is suggested by recent work by Zhengfang Wang.     

Tate–Hochschild cohomology for periodic algebras

This paper is devoted to studying the Tate–Hochschild cohomology for periodic algebras. We will prove that the Tate–Hochschild cohomology ring of a periodic algebra can be written as the localization of the non-negative part of the Tate–Hochschild cohomology ring.

     

Hochschild Cohomology of an Algebra Associated with a Circular Quiver

In this article we show that an algebra A = K Γ/(f(X ^s )) has a periodic projective bimodule resolution of period 2, where KΓ is the path algebra of the circular quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K and X is the sum of all arrows in KΓ. Moreover, by means of this projective bimodule resolution, we compute the Hochschild cohomology group of A, and we give a presentation of the Hochschild cohomology ring HH?(A) by the generators and the relations in the case K is a field.     

Formule d'homotopie entre les complexes de Hochschild et de De Rham

Let k be the field or let M be the space k ⁿ and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hochschild homology H _·(A,A) is isomorphic to the de Rham differential forms over M: this means that the complexes (C _·(A,A),b) and (^·(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy formula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functions over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology.     

The hochschild cohomology ring of a modular group algebra: the commutative case

Let k be a field of characteristic P >0 and let G be a finite abelian group. We determine the structure of the Hochschild cohomology ring of the group algebra k G. Moreover, we prove that for any finite group G the Krull dimension of H H ^*(k G) equals the p-rank of G.     

Hochschild cohomology and “smoothness” in monoidal categories

We introduce and investigate the properties of Hochschild cohomology of algebras in an abelian monoidal category M. We show that the second Hochschild cohomology group of an algebra in M classifies extensions of A up to an equivalence. We characterize algebras of Hochschild dimension 0 (separable algebras), and of Hochschild dimension ≤1 (formally smooth algebras). Several particular cases and applications are included in the last section of the paper.     

Power series rings and projectivity

We show that a formal power series ring A[[X]] over a noetherian ring A is not a projective module unless A is artinian. However, if (A,) is any local ring, then A[[X]] behaves like a projective module in the sense that Ext^p_A(A[[X]], M)=0 for all -adically complete A-modules. The latter result is shown more generally for any flat A-module B instead of A[[X]]. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings. The authors were partly supported by NSERC grant 3-642-114-80 and by the DFG Schwerpunkt ``Global Methods in Complex Geometry'.