On the Decomposition of the Hochschild Cohomology Group of a Monomial Algebra Satisfying a Separability Condition

Abstract

In this paper, we consider the finite connected quiver Q having two subquivers Q ⁽¹⁾ and Q ⁽²⁾ with . Suppose that Q ⁽ⁱ⁾ is not a subquiver of Q ^(j) where {i, j} = {1, 2}. For a monomial algebra Λ =kQ/I obtained by the quiver Q, when the associated sequence of paths given by Minsharp _<(I) satisfies a certain separability condition, we propose the method so that we easily construct a minimal projective resolution of Λ as a right Λ ^e -module and calculate the Hochschild cohomology group of Λ.

Key Words:

• Associated sequence of path
• Hochschild cohomology
• Monomial algebra
• Path algebra

2010 Mathematics Subject Classification:

• 16E40
• 16G20

ACKNOWLEDGMENTS

We would like to thank the referee for many valuable comments and suggestions. The referee informed us an example of an infinite-dimensional quiver algebra satisfying the separability condition. We think that it is interesting to study the decomposition of the Hochschild cohomology of such algebras. However, we only consider finite-dimensional algebras in this paper.

Notes

Communicated by D. Zacharia.