arXiv:0906.1518v1 [math.KT] 8 Jun 2009 TWO CLASSES OF ALGEBRAS WITH INFINITE HOCHSCHILD HOMOLOGY ANDREA SOLOTAR AND MICHELINE VIGU ´ E-POIRRIER Abstract. 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 is not finite. 1. Introduction Let k be a fixed field. All the algebras we consider are associative unital k- algebras. We will denote ⊗ = ⊗ k . It is well known that the homological properties of an algebra are related to the properties of its Hochschild (co)homology groups. For example, if a finite dimen- sional algebra over an algebraically closed field has finite global dimension, then all its higher Hochschild cohomology groups vanish. In [12], D. Happel conjectured that the converse would be true. However, it has been shown in [5] that the con- jecture does not hold for algebras of type A q = k〈x, y〉/(x 2 ,y 2 , xy − qyx), where q ∈ k. In [11], Han proved that the total Hochschild homology of the algebras A q is infinite dimensional. This fact led him to suggest the following conjecture: Conjecture(Han): Let A be a finite dimensional k-algebra. If the total Hochschild homology of A is finite dimensional, then A has finite global dimension. In the same paper, Han provided a proof of this statement for monomial finite dimensional algebras. Avramov and Vigu´ e’s computations in [1] show that Han’s conjecture holds in the commutative case not only for finite dimensional algebras but for essentially finitely generated ones, see also [18]. In [4], Han’s conjecture is proved for graded local algebras, Koszul algebras and graded cellular algebras, provided the characteristic of the ground field is zero. The proof relies on the properties of the graded Cartan matrix and the logarithm and strongly uses the hypothesis on the characteristic of the field. In [3], the authors compute the Hochschild homology groups of quantum com- plete intersections, that is algebras of type A = k〈x, y〉/(x a ,y b , xy − qyx), where q ∈ k ∗ is not a root of unity and a, b ≥ 2 are fixed integers. In particular they prove Han’s conjecture for this class of finite dimensional algebras. Date : June 8, 2009. 2000 Mathematics Subject Classification. Primary 16E40, 16W50. Key words and phrases. global dimension, Hochschild homology theory. This work has been supported by the projects UBACYTX212 and PIP-CONICET 5099. The first author is a research member of CONICET (Argentina) and a Regular Associate of ICTP Associate Scheme. The second author is a research member of University of Paris 13, CNRS, UMR 7539 (LAGA). 1