Cibils'spectral sequence for the cohomology of triangular algebras

Abstract

In order to study the Hochschild cohomology of triangular algebras $\mathcal T$, we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with $\mathcal T$, and which converges to $HH^*(\mathcal T)$. We explicit its components, and its differentials which are sums of cup products. In case $n=3$, we study some properties of the differential at level 2. Finally, we apply these results to the paths algebra of a quiver without oriented cycles, and link them with previous results on the incidence algebra of a simplicial complex, and more generally on the morphisms algebra of certain categories.