Example 15.62.1 (061Z)—The Stacks project

Example 15.62.1. 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

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

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 June 16, 2016 at 22:08

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 $E_\infty$ -page goes. Also, a nitpicking point: "bounded above" has not been defined for chain complexes (only for cochain complexes).

Comment #2124 by Johan on July 21, 2016 at 12:41

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 15.62.4. The change is here.

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2 comment(s) on Section 15.62: Spectral sequences for Tor

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