  For an extension of associative algebras $B\subset A$ 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.
