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Gröbner bases of associative algebras and the Hochschild cohomology

Authors:	Yuji Kobayashi

Institution:	Department of Information Science, Toho University, Funabashi 274-8510, Japan

Abstract:	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 be a finitely generated algebra over a commutative ring with a (possibly infinite) Gröbner base on a free algebra , that is, is the quotient with the ideal of generated by . Given a Gröbner base for an -subbimodule of the free -bimodule $A \cdot X \cdot A = A_K \otimes K \cdot X \otimes_KA$ generated by a set , we have a morphism $\partial$ of -bimodules from the free -bimodule $A \cdot H \cdot A$ generated by to $A \cdot X \cdot A$ sending the generator to the element $h \in H$ . We construct a Gröbner base on $F \cdot H \cdot F$ for the -subbimodule Ker( $\partial$ ) of $A \cdot H \cdot A$ , and with this we have the free -bimodule $A \cdot C \cdot A$ generated by 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 -bimodule itself, we have a free -bimodule resolution of .

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