15.62 Spectral sequences for Tor
In this section we collect various spectral sequences that come up when considering the Tor functors.
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}.
Example 15.62.2. Let R \to S be a ring map. Let M be an R-module and let N be an S-module. Then there is a spectral sequence
\text{Tor}^ S_ n(\text{Tor}^ R_ m(M, S), N) \Rightarrow \text{Tor}^ R_{n + m}(M, N).
To construct it choose a R-free resolution P_\bullet of M. Then we have
M \otimes _ R^{\mathbf{L}} N = P^\bullet \otimes _ R N = (P^\bullet \otimes _ R S) \otimes _ S N
and then apply the first spectral sequence of Example 15.62.1.
Example 15.62.3. Consider a commutative diagram
\xymatrix{ B \ar[r] & B' = B \otimes _ A A' \\ A \ar[r] \ar[u] & A' \ar[u] }
and B-modules M, N. Set M' = M \otimes _ A A' = M \otimes _ B B' and N' = N \otimes _ A A' = N \otimes _ B B'. Assume that A \to B is flat and that M and N are A-flat. Then there is a spectral sequence
\text{Tor}^ A_ i(\text{Tor}_ j^ B(M, N), A') \Rightarrow \text{Tor}^{B'}_{i + j}(M', N')
The reason is as follows. Choose free resolution F_\bullet \to M as a B-module. As B and M are A-flat we see that F_\bullet \otimes _ A A' is a free B'-resolution of M'. Hence we see that the groups \text{Tor}^{B'}_ n(M', N') are computed by the complex
(F_\bullet \otimes _ A A') \otimes _{B'} N' = (F_\bullet \otimes _ B N) \otimes _ A A' = (F_\bullet \otimes _ B N) \otimes ^{\mathbf{L}}_ A A'
the last equality because F_\bullet \otimes _ B N is a complex of flat A-modules as N is flat over A. Hence we obtain the spectral sequence by applying the spectral sequence of Example 15.62.1.
Example 15.62.4. Let K^\bullet , L^\bullet be objects of D^{-}(R). Then there is a spectral sequence with
E_2^{p, q} = H^ p(K^\bullet \otimes _ R^{\mathbf{L}} H^ q(L^\bullet )) \Rightarrow H^{p + q}(K^\bullet \otimes _ R^{\mathbf{L}} L^\bullet )
and another spectral sequence with
E_2^{p, q} = H^ p(H^ q(K^\bullet ) \otimes _ R^{\mathbf{L}} L^\bullet ) \Rightarrow H^{p + q}(K^\bullet \otimes _ R^{\mathbf{L}} L^\bullet )
Both spectral sequences have d_2^{p, q} : E_2^{p, q} \to E_2^{p + 2, q - 1}. After replacing K^\bullet and L^\bullet by bounded above complexes of projectives, these spectral sequences are simply the two spectral sequences for computing the cohomology of \text{Tot}(K^\bullet \otimes L^\bullet ) discussed in Homology, Section 12.25.
Comments (2)
Comment #5790 by Brad Dirks on
Comment #5805 by Johan on