The Stacks project

Proof. In case (1) the filtration on $T^\bullet $ is finite and the lemma follows immediately from Homology, Lemma 12.24.11. In case (2) choose $a < b$ such that $F^ iK^\bullet $ and $F^ jL^\bullet $ are acyclic for $i, j > b$ and $F^ iK^\bullet \to K^\bullet $ and $F^ jL^\bullet \to L^\bullet $ are quasi-isomorphisms for $i, j < a$. We claim that in this case the complex $F^ nT^\bullet $ is acyclic for $n > 2b$ and that $F^ nT^\bullet \to T^\bullet $ is a quasi-isomorphism for $n < 2a - 1$. Since the claim shows that Homology, Lemma 12.24.13 applies we see that our lemma is true.

Proof of the claim. Given $n$ and integers $i_1 \leq i_2$ consider

viewed as a subcomplex of $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$. Observe that $F^ nT^\bullet = \mathop{\mathrm{colim}}\nolimits S^{n, i_1, i_2}$ is a filtered colimit. Thus to show the claim it suffices to show that $S^{n, i_1, i_2}$ is acyclic for $n > 2b$ and that $S^{n, i_1, i_2} \to \text{Tot}(K^\bullet \otimes L^\bullet )$ is a quasi-isomorphism for $n < 2a - 1$ and a cofinal set of choices of pairs $i_1, i_2$.

If $i_1 = i_2 = i$ and $n > 2b$, then

is acyclic: either $i > b$ and this complex is acyclic by our assumption that $F^{n - i}L^\bullet $ is K-flat, or $i \leq b$ and then $n - i \geq n - b > b$ and the same holds. Using our flatness assumptions the reader shows that there is a short exact sequence

of complexes. Thus we see by induction on $i_2 - i_1$ that all of the complexes $S^{n, i_1, i_2}$ for $n > 2b$ are acyclic.

Assume $n < 2a - 1$. Observe that

is a quasi-isomorphism because both $a - 1 < a$ and $n - a + 1 < a$ and all our complexes are K-flat hence both sides compute the same object of $D(R)$. Observe that for $i_2 \geq a - 1$ and $n < 2a - 1$ the map

is a quasi-isomorphism because left and right hand side are quasi-isomorphic to $\text{Tot}(F^{i_2 + 1}K^\bullet \otimes L^\bullet )$. Thus using the short exact sequence of complexes in the previous paragraph we find that $S^{n, a - 1, a - 1} \to S^{n, a - 1, t}$ is a quasi-isomorphism for all $t \geq a - 1$. On the other hand, there are similarly short exact sequences

of complexes. Similarly to the above, we note that for $i_1 \leq a - 1$ the map

is a quasi-isomorphism because left and right hand side are quasi-isomorphic to $\text{Tot}(K^\bullet \otimes F^{n - i_1 + 1}L^\bullet )$. Thus we conclude that $S^{n, s, t} \to S^{n, a - 1, t}$ is a quasi-isomorphism for all $s \leq a - 1$. Combined we find that $S^{n, s, t}$ for $s \leq a - 1$ and $t \geq a - 1$ maps quasi-isomorphically to $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$. This finishes the proof of the lemma. $\square$

Proof. Let us say a filtered complex $L^\bullet $ is basic if each $L^ n$, $F^ iL^ n$, $\text{gr}^ iL^ n$ is a free $R$-module and if all differentials are zero.

There exists a basic filtered complex $P_0^\bullet $ and a map $f_0 : P_0^\bullet \to K^\bullet $ of filtered complexes such that $f_0$ and $F^ if_0$, $i \in \mathbf{Z}$ are surjective on cohomology. To see this set

with zero differentials and the map $f_0$ defined by $f_0(\xi _ z) = z$. As for the filtration, we set

We leave it to the reader to check that this gives $f_0 : P_0^\bullet \to K^\bullet $ as claimed.

By induction on $m \geq 0$ we are going to construct embeddings

of filtered complexes and maps $f_ m : P_ m^\bullet \to K^\bullet $ with the following properties

1. the filtered complex $P_{m + 1}^\bullet / P_ m^\bullet $ is basic,

2. the kernel of $H^ n(f_ m)$ and $H^ n(F^ if_ m)$ maps to zero in $H^ n(P_{m + 1}^\bullet )$ and $H^ n(F^ iP_{m + 1}^\bullet )$,

3. the map $f_{m + 1} : P_{m + 1}^\bullet \to K^\bullet $ extends the map $f_ m$.

To to this, set

Note that $\Omega _{n, m}$ surjects onto the kernel of $H^ n(f_ m)$ and that $\Omega _{n, i, m}$ surjects onto the kernel of $H^ n(F^ if_ n)$. For each $z \in \Omega _{n, m}$, resp. $z \in \Omega _{n, i, m}$ we choose a $y_ z \in K^{n - 1}$, resp. $y_ z \in F^ iK^{n - 1}$ such that $d_ K(y_ z) = f_ m(z)$. Then we set

We set $d_{P_ m}(\eta _ z) = z$ and we set $f_{m + 1}(\eta _ z) = y_ z$. Finally, we set

We leave it to the reader to check that this gives $P_ m^\bullet \subset P_{m + 1}^\bullet $ and $f_{m + 1} : P_{m + 1}^\bullet \to K^\bullet $ as claimed.

At this point we simply take

as a filtered complex with map $f : P^\bullet \to K^\bullet $ given by $\bigcup f_ m$.

Part (1) of the statement of the lemma holds because $P^ n$ as a filtered module is isomorphic to the direct sum of $P_0^ n$ and $P_{m + 1}^ n/P_ m^ n$ for $m \geq 0$. Small detail omitted.

Part (3) of the statement. Observe that $H^ n(P^\bullet ) = \mathop{\mathrm{colim}}\nolimits H^ n(P_ m^\bullet )$. The map $f_0$ is surjective on cohomology; whence the same holds for each $f_ m$. For each $m$ by construction the embedding $P_ m^\bullet \subset P_{m + 1}^\bullet $ kills the kernel of $H^ n(f_ m)$. Combining these facts the reader easily concludes that $H^ n(f)$ is an isomorphism. Similarly for $H^ n(F^ if_ n)$. Then also $H^ n(\text{gr}^ if)$ must be an isomorphism because of the short exact sequence $0 \to F^{i + 1} \to F^ i \to \text{gr}^ i \to 0$ (of functors on the category of filtered complexes, say). Small detail omitted.

Part (2) of the statement. To see that $P^\bullet $ is K-flat, by Lemma 15.59.8, it suffices to show that $P_ m^\bullet $ is K-flat. By Lemma 15.59.6 and induction it suffices to note that a complex with zero differentials and free terms is K-flat. The same argument works to show that $F^ iP^\bullet $ is K-flat for all $i \in \mathbf{Z}$. Finally, we see that $\text{gr}^ iP^\bullet $ is K-flat by another application of Lemma 15.59.6. $\square$

Proof. Choose $P^\bullet \to K^\bullet $ and $Q^\bullet \to L^\bullet $ as in Lemma 15.64.2. Then we use the spectral sequence for the filtered complex

described in the text of this section and in Lemma 15.64.1. $\square$

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

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 $r \geq 2$

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