The Hochschild cohomology ring of monomial algebras.

Hochschild cohomology of an associative algebra HH∗(A) (here A is an associative k-algebra and k is a field) has a rich structure that is useful when studying the representations of the given algebra. It is well known that the Hochschild cohomology can be computed by any resolution of A as an A-Abimodule. There is a cup product “∪” on Hochschild cohomology, which gives HH∗(A) a structure of graded-commmutative k-algebra. Originally the cup product was defined through the bar resolution, which is usually very inefficient for explicit computations. In this article the authors follow the strategy of avoiding the use of bar resolution and using instead a different resolution P of A-A-bimodules (the so-called Bardzell resolution), which is minimal in some sense and is equipped with a diagonal map ∆ : P → P ⊗AP. The Bardzell resolution can be applied for monomial algebras and is described in Section 3. A triangular monomial algebra is a quiver algebra with relations A = kQ/I, such that I can be generated by some finite set of paths and the underlying quiver has no oriented cycles. Using the explicit description of a diagonal map on the Bardzell resolution (which is obtained in Theorem 4.3), the authors show in the main theorem of this article (Theorem 6.5) that for any finite triangular monomial algebra A, the cup product on HH∗(A) is zero in positive degrees