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We give an explicit description of a diagonal map on the Bardzell resolutionfor any monomial path algebra, and we use this diagonal map to describe the cupproduct on Hochschild cohomology. Then, we prove that the cup product istrivial in positive degrees for triangular monomial path algebras. Our proofuses the graded-commutativity of the cup product on Hochschild cohomology anddoes not rely on explicit computation of the Hochschild cohomology modules.

The Hochschild cohomology ring of monomial path algebras