On the homology of a tensor product of local rings

For quotient rings $A$ and $B$ of a regular ring $R$ the connection between the Hopf algebras $\operatorname{Tor}^A(k,k)$, $\operatorname{Tor}^B(k,k)$ and $\operatorname{Tor}^{A\underset R\otimes R}(k,k)$ is investigated. In general, this connection is expressed by a spectral sequence. Criteria are obtained for
$$ \operatorname{Tor}^{A\underset R\otimes R}(k,k)\cong\operatorname{Tor}^A(k,k)\underset{\operatorname{Tor}^R(k,k)}\otimes\operatorname{Tor}^B(k,k). $$

Bibliography: 18 items.