Example 15.61.1. Let $R$ be a ring. Let $K_\bullet $ be a chain complex of $R$-modules with $K_ n = 0$ for $n \ll 0$. Let $M$ be an $R$-module. Choose a resolution $P_\bullet \to M$ of $M$ by free $R$-modules. We obtain a double chain complex $K_\bullet \otimes _ R P_\bullet $. Applying the material in Homology, Section 12.25 (especially Homology, Lemma 12.25.3) translated into the language of chain complexes we find two spectral sequences converging to $H_*(K_\bullet \otimes _ R^\mathbf {L} M)$. Namely, on the one hand a spectral sequence with $E_2$-page

and differential $d_2$ given by maps $\text{Tor}^ R_ j(H_ i(K_\bullet ), M) \to \text{Tor}^ R_{j - 2}(H_{i + 1}(K_\bullet ), M)$. Another spectral sequence with $E_1$-page

with differential $d_1$ given by maps $\text{Tor}^ R_ j(K_ i, M) \to \text{Tor}^ R_ j(K_{i - 1}, M)$ induced by $K_ i \to K_{i - 1}$.

## Comments (2)

Comment #2096 by Kestutis Cesnavicius on

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