Example 15.64.5. If $R = \mathbf{Z}$ or more generally if $R$ is a Dedekind domain, then $\text{Tor}_ i^ R(M, N) = 0$ for $i \not\in \{ 0, 1\} $ for all $R$-modules $M$ and $N$. Hence the spectral sequence of Lemma 15.64.4 degenerates at the $E_2$ page and we get short exact sequences
\[ 0 \to \bigoplus _{i + j = n} H^ i(K) \otimes _ R H^ j(L) \to H^ n(K \otimes _ R^\mathbf {L} L) \to \bigoplus _{i + j = n + 1} \text{Tor}^ R_1(H^ i(K), H^ j(L)) \to 0 \]
for all $n \in \mathbf{Z}$.
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