The Stacks project

Lemma 15.64.4 (Künneth Spectral Sequence). Let $R$ be a ring. Let $K$ and $L$ be objects of $D^ b(R)$. There exists a bigraded bounded spectral sequence $\{ E_ r\} _{r \geq 2}$ with

\[ E_2^{p, q} = \bigoplus \nolimits _{i + j = q} \text{Tor}^ R_{-p}(H^ i(K), H^ j(L)) \]

and $d_ r$ of bidegree $(r, -r + 1)$ converging to $H^{p + q}(K \otimes _ R^\mathbf {L} L)$.

Proof. Let $K^\bullet $ be a complex of $R$-modules representing $K$ and let $L^\bullet $ be a complex of $R$-modules representing $L$. Set $F^ iK^\bullet = \tau _{\leq -i}K^\bullet $ and similarly for $L$, see Homology, Section 12.15. Apply Proposition 15.64.3 noting that $\text{gr}^ iK^\bullet = H^{-i}(K)[i]$ to get a bounded spectral sequence $\{ (E')_ r\} _{r \geq 1}$ with

\[ (E')^{p, q}_1 = \bigoplus \nolimits _{i + j = p} \text{Tor}^ R_{-2p - q}(H^{-i}(K), H^{-j}(L)) \]

converging to the cohomology of $K \otimes _ R^\mathbf {L} L$. By construction this spectral sequence has differentials $d_ r^{p, q} : (E')_ r^{p, q} \to (E')_ r^{p + r, q - r + 1}$ for $r \geq 1$. To get the spectral sequence of the lemma we set for $r \geq 2$

\[ E_ r^{p, q} = (E')_{r - 1}^{-q, p + 2q} \]

We leave it to the reader to show that this works. $\square$

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