Gröbner bases of associative algebras and the Hochschild cohomology

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 A A be a finitely generated algebra over a commutative ring K K with a (possibly infinite) Gröbner base G G on a free algebra F F , that is, A A is the quotient F / I ( G ) F/I(G) with the ideal I ( G ) I(G) of F F generated by G G . Given a Gröbner base H H for an A A -subbimodule L L of the free A A -bimodule A ⋅ X ⋅ A = A K ⊗ K ⋅ X ⊗ K A A \cdot X \cdot A = A_K \otimes K \cdot X \otimes _KA generated by a set X X , we have a morphism ∂ \partial of A A -bimodules from the free A A -bimodule A ⋅ H ⋅ A A \cdot H \cdot A generated by H H to A ⋅ X ⋅ A A \cdot X \cdot A sending the generator [ h ] [h] to the element h ∈ H h \in H . We construct a Gröbner base C C on F ⋅ H ⋅ F F \cdot H \cdot F for the A A -subbimodule Ker( ∂ \partial ) of A ⋅ H ⋅ A A \cdot H \cdot A , and with this C C we have the free A A -bimodule A ⋅ C ⋅ A A \cdot C \cdot A generated by C C and an exact sequence A ⋅ C ⋅ A → A ⋅ H ⋅ A → A ⋅ X ⋅ A A \cdot C \cdot A \rightarrow A \cdot H \cdot A \rightarrow A \cdot X \cdot A . Applying this construction inductively to the A A -bimodule A A itself, we have a free A A -bimodule resolution of A A .