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For an extension of associative algebras B⊂A over a field and an A‐bimodule X, we obtain a Jacobi–Zariski long nearly exact sequence relating the Hochschild homologies of A and B, and the relative Hochschild homology, all of them with coefficients in X. This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness. [ABSTRACT FROM AUTHOR]

Assume that HT ht is a bounded HT ht -bimodule, with index of nilpotency HT ht and projective dimension HT ht . The Theorem and its proof read as follows: 6.5 Theorem Let HT ht be an extension of HT ht -algebras and let HT ht be an HT ht -bimodule. [Extracted from the article]