Gröbner bases of associative algebras and the Hochschild cohomology

We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let $A$ be a finitely generated algebra over a commutative ring $K$ with a (possibly infinite) Gröbner base $G$ on a free algebra $F$, that is, $A$ is the quotient $F/I(G)$ with the ideal $I(G)$ of $F$ generated by $G$. Given a Gröbner base $H$ for an $A$-subbimodule $L$ of the free $A$-bimodule $A \cdot X \cdot A = A_K \otimes K \cdot X \otimes_KA$ generated by a set $X$, we have a morphism $\partial$ of $A$-bimodules from the free $A$-bimodule $A \cdot H \cdot A$ generated by $H$ to $A \cdot X \cdot A$ sending the generator $[h]$to the element $h \in H$. We construct a Gröbner base $C$ on $F \cdot H \cdot F$ for the $A$-subbimodule Ker($\partial$) of $A \cdot H \cdot A$, and with this $C$ we have the free $A$-bimodule $A \cdot C \cdot A$ generated by $C$ and an exact sequence $A \cdot C \cdot A \rightarrow A \cdot H \cdot A \rightarrow A \cdot X \cdot A$. Applying this construction inductively to the $A$-bimodule $A$ itself, we have a free $A$-bimodule resolution of $A$.