On the cohomology of type \(A_n\) algebras (Q1600181)

Let \(A_i\), \(1\leq i\leq n\), be \(n\) unitary algebras over a field \(k\) and let \(M_i\) be a non-zero \(A_{i+1}\)-\(A_i\)-bimodule, \(1\leq i\leq n-1\). The associated tensor triangular algebra \(\tau\) is isomorphic to the tensor algebra \(T_A(M)\) where \(A=A_1\times\cdots\times A_n\) and \(M=M_1\oplus\cdots\oplus M_{n-1}\). The paper under review concerns the Hochschild cohomology \(HH^*(\tau)\) of the tensor triangular algebra \(\tau\); the author describes, in particular, a spectral sequence which converges to \(HH^*(\tau)\). This result ``generalizes'' the case \(n=2\) [see \textit{C. Cibils}, Lect. Notes Pure Appl. Math. 210, 35-51 (2000; Zbl 0985.16007)].