The Stacks project

Example 15.60.1. 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 12.22 (especially Homology, Lemma 12.22.6) 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

\[ (E_2)_{i, j} = \text{Tor}^ R_ j(H_ i(K_\bullet ), M) \Rightarrow H_{i + j}(K_\bullet \otimes ^{\mathbf{L}}_ R M) \]

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

\[ (E_1)_{i, j} = \text{Tor}^ R_ j(K_ i, M) \Rightarrow H_{i + j}(K_\bullet \otimes ^{\mathbf{L}}_ R M) \]

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

More details on this example would be great. For instance, it would be nice to mention how the differentials go and which way the filtration on the -page goes. Also, a nitpicking point: "bounded above" has not been defined for chain complexes (only for cochain complexes).

Comment #2124 by on

OK, I tried to improve the exposition a little bit. But to be more precise (for example to deal with signs of differentials) we would have to make a duplicate of the section on double complexes in the setting of chain complexes. Instead you can look at Example 15.60.4. The change is here.


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