Hochschild cohomology and homology of quantum complete intersections. (English) Zbl 1247.16005
From the introduction: This paper gives a general description of the Hochschild cohomology and homology of finite-dimensional quantum complete intersections. Here is an outline:
In Theorems 3.4 and 7.4 we explicitly determine a \(k\)-basis for the Hochschild cohomology and homology, respectively.
Using these results we study the size of the Hochschild cohomology and homology in the following sense: Let \(\mathbb N\) be the set of nonnegative integers (i.e., \(0\in\mathbb N\)). We denote by \[ \gamma (\text{HH}^*(\Lambda))=\inf\left\{t\in\mathbb N\,\Big|\,\limsup\tfrac{\dim_k\text{HH}^n(\Lambda)}{n^{t-1}}<\infty\right\} \] the rate of growth of the Hochschild cohomology (and similarly for the Hochschild homology). In Theorems 4.5 and 8.2, we obtain explicit combinatorial formulas for \(\gamma(\text{HH}^*(\Lambda))\) and \(\gamma(\text{HH}_*(\Lambda))\). In particular it will be shown (Corollary 4.6) that whenever not all commutation parameters are roots of unity we have \(\gamma(\text{HH}^*(\Lambda))\leq c-2\). For \(c=2\) that means that the Hochschild cohomology is finite. This explains why there are essentially only two cases for \(c=2\), while we obtain additional behaviors for larger \(c\).
We will also generalize the result of P. A. Bergh and K. Erdmann [Algebra Number Theory 2, No. 5, 501-522 (2008; Zbl 1205.16011)] 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).
Finally we will study the multiplicative structure of the Hochschild cohomology ring. It will turn out (Theorem 5.5) that it always contains a subring \(\mathcal S\) which is finitely generated over \(k\), and isomorphic to the quotient of the Hochschild cohomology modulo its nilpotent elements. We will give a criterion for when the entire Hochschild cohomology ring is finitely generated over this subring (Theorem 5.9). We will give examples (Examples 6.4 and 6.5) that all the following behaviors occur (for \(c\geq 3\)):
\(\bullet\) \(\mathcal S=k\), but \(\gamma(\text{HH}^*(\Lambda))=c-2\).
\(\bullet\) \(\gamma(\mathcal S)=\gamma(\text{HH}^*(\Lambda))=c-2\), and \(\text{HH}^*(\Lambda)\) is finitely generated over \(\mathcal S\).
\(\bullet\) \(\gamma(\mathcal S)=\gamma(\text{HH}^*(\Lambda))=c-2\), but \(\text{HH}^*(\Lambda)\) is not finitely generated over \(\mathcal S\).
In Theorems 3.4 and 7.4 we explicitly determine a \(k\)-basis for the Hochschild cohomology and homology, respectively.
Using these results we study the size of the Hochschild cohomology and homology in the following sense: Let \(\mathbb N\) be the set of nonnegative integers (i.e., \(0\in\mathbb N\)). We denote by \[ \gamma (\text{HH}^*(\Lambda))=\inf\left\{t\in\mathbb N\,\Big|\,\limsup\tfrac{\dim_k\text{HH}^n(\Lambda)}{n^{t-1}}<\infty\right\} \] the rate of growth of the Hochschild cohomology (and similarly for the Hochschild homology). In Theorems 4.5 and 8.2, we obtain explicit combinatorial formulas for \(\gamma(\text{HH}^*(\Lambda))\) and \(\gamma(\text{HH}_*(\Lambda))\). In particular it will be shown (Corollary 4.6) that whenever not all commutation parameters are roots of unity we have \(\gamma(\text{HH}^*(\Lambda))\leq c-2\). For \(c=2\) that means that the Hochschild cohomology is finite. This explains why there are essentially only two cases for \(c=2\), while we obtain additional behaviors for larger \(c\).
We will also generalize the result of P. A. Bergh and K. Erdmann [Algebra Number Theory 2, No. 5, 501-522 (2008; Zbl 1205.16011)] 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).
Finally we will study the multiplicative structure of the Hochschild cohomology ring. It will turn out (Theorem 5.5) that it always contains a subring \(\mathcal S\) which is finitely generated over \(k\), and isomorphic to the quotient of the Hochschild cohomology modulo its nilpotent elements. We will give a criterion for when the entire Hochschild cohomology ring is finitely generated over this subring (Theorem 5.9). We will give examples (Examples 6.4 and 6.5) that all the following behaviors occur (for \(c\geq 3\)):
\(\bullet\) \(\mathcal S=k\), but \(\gamma(\text{HH}^*(\Lambda))=c-2\).
\(\bullet\) \(\gamma(\mathcal S)=\gamma(\text{HH}^*(\Lambda))=c-2\), and \(\text{HH}^*(\Lambda)\) is finitely generated over \(\mathcal S\).
\(\bullet\) \(\gamma(\mathcal S)=\gamma(\text{HH}^*(\Lambda))=c-2\), but \(\text{HH}^*(\Lambda)\) is not finitely generated over \(\mathcal S\).
MSC:
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16S37 | Quadratic and Koszul algebras |
| 16S80 | Deformations of associative rings |
