The Stacks project

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 #8706 by Yassin Mousa on

There is a tricky typo in Example 0662. The indices for the second spectral seqeunce are interchanged.

There are also:

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

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View Example 15.62.4 as pdf