Homology and cohomology of quantum complete intersections

Abstract

Dedicated to Luchezar Avramov on the occasion of his sixtieth birthdayWe construct a minimal projective bimodule resolution for every finite-dimensional quantum complete intersection of codimension two. Then we use this resolution to compute both the Hochschild cohomology and homology for such an algebra. In particular, we show that the cohomology vanishes in high degrees, while the homology is always nonzero.

Hochschild cohomology and homology of quantum complete intersections

“…We will also generalize the result of [Bergh and Erdmann 2008] in another way: It will be shown that whenever the commutation parameters are sufficiently generic the Hochschild cohomology of the quantum complete intersection is finite (see Example 6.2).…”

“…Buchweitz, Green, Madsen, and Solberg [Buchweitz et al 2005] gave a finitedimensional quantum complete intersection as the first example of an algebra of infinite global dimension which has finite Hochschild cohomology. This result was generalized in [Bergh and Erdmann 2008], which showed that a finite-dimensional quantum complete intersection of codimension 2 (c = 2 in the description above) has an infinite Hochschild cohomology if and only if the commutation parameter is a root of unity.…”

Tate-Hochschild homology and cohomology of Frobenius algebras

“…For example, suppose that A is either a finite dimensional commutative complete intersection of the form Proof. The dimensions of HH n (A, A) for n ≥ 1 follow from [BeE,Theorem 3.1], and so by Theorem 3.7, we only need to calculate the dimension of HH 0 (A, A). By Lemma 4.3, this homology group is the homology of the complex…”

“…In fact, just as for homology, when n ≥ 1, then dim HH n (A, A) is given by a polynomial of degree c − 1 (where c is the number of defining generators for A). On the other hand, if A is as in Theorem 4.7, then it follows from [BeE,Theorem 3.2] that HH n (A, A) = 0 when n ≥ 3. Consequently, there is no cohomological counterpart to Theorem 4.6: there is no universal lower bound for the dimensions of the Tate-Hochschild cohomology groups of quantum complete intersections.…”

“…We have shown that when n ≥ 1, then dim Ker δ n = ab n+2 2 − 1 for n even ab Proof. For n ≥ 1, the dimensions follow from [BeE,Theorem 3.2]. Moreover, by and HH…”