In this work, we study the Hochschild-Mitchell Cohomology of triangular matrix categories. Given a triangular matrix category $\Lambda=\left[ \begin{ smallmatrix} \mathcal{ T} & 0 \\ M & \mathcal{ U} \end{ smallmatrix}\right]$, we investigate the relationship of the Hochschild-Mitchell cohomologies $H^{ i}(\Lambda)$ and $H^{ i}(\mathcal{ U})$ of $\Lambda$ and $\mathcal{ U}$ respectively; and we show that they can be connected by a long exact sequence. This result extend the well-known result of Michelana-Platzeck given in [S. Michelena, M. I. Platzeck. { \it{ Hochschild cohomology of triangular matrix algebras}}. J. Algebra 233, (2000) 502-525].
Categories: math.RT math.CT math.KT