The Stacks project

Lemma 13.26.8. Let $\mathcal{A}$ be an abelian category with enough injectives. Let $0 \to A \to B \to C \to 0$ be a short exact sequence in $\text{Fil}^ f(\mathcal{A})$. Given filtered quasi-isomorphisms $A[0] \to I^\bullet $ and $C[0] \to J^\bullet $ where $I^\bullet , J^\bullet $ are complexes of filtered injective objects with $I^ n = J^ n = 0$ for $n < 0$, then there exists a commutative diagram

\[ \xymatrix{ 0 \ar[r] & A[0] \ar[r] \ar[d] & B[0] \ar[r] \ar[d] & C[0] \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I^\bullet \ar[r] & M^\bullet \ar[r] & J^\bullet \ar[r] & 0 } \]

where the lower row is a termwise split sequence of complexes.

Proof. As $A[0] \to I^\bullet $ and $C[0] \to J^\bullet $ are filtered quasi-isomorphisms we conclude that $a : A \to I^0$, $c : C \to J^0$ and all the morphisms $d_ I^ n$, $d_ J^ n$ are strict, see Homology, Lemma 13.13.4. We are going to step by step construct the south-east and the south arrows in the following commutative diagram

\[ \xymatrix{ B \ar[r]_\beta \ar[rd]^ b & C \ar[r]_ c \ar[rd]^{\overline{b}} & J^0 \ar[d]^{\delta ^0} \ar[r] & J^1 \ar[d]^{\delta ^1} \ar[r] & \ldots \\ A \ar[u]^\alpha \ar[r]^ a & I^0 \ar[r] & I^1 \ar[r] & I^2 \ar[r] & \ldots } \]

As $A \to B$ is a strict monomorphism, we can find a morphism $b : B \to I^0$ such that $b \circ \alpha = a$, see Lemma 13.26.4. As $A$ is the kernel of the strict morphism $I^0 \to I^1$ and $\beta = \mathop{\mathrm{Coker}}(\alpha )$ we obtain a unique morphism $\overline{b} : C \to I^1$ fitting into the diagram. As $c$ is a strict monomorphism and $I^1$ is filtered injective we can find $\delta ^0 : J^0 \to I^1$, see Lemma 13.26.4. Because $B \to C$ is a strict epimorphism and because $B \to I^0 \to I^1 \to I^2$ is zero, we see that $C \to I^1 \to I^2$ is zero. Hence $d_ I^1 \circ \delta ^0$ is zero on $C \cong \mathop{\mathrm{Im}}(c)$. Hence $d_ I^1 \circ \delta ^0$ factors through a unique morphism

\[ \mathop{\mathrm{Coker}}(c) = \mathop{\mathrm{Coim}}(d_ J^0) = \mathop{\mathrm{Im}}(d_ J^0) \longrightarrow I^2. \]

As $I^2$ is filtered injective and $\mathop{\mathrm{Im}}(d_ J^0) \to J^1$ is a strict monomorphism we can extend the displayed morphism to a morphism $\delta ^1 : J^1 \to I^2$, see Lemma 13.26.4. And so on. We set $M^\bullet = I^\bullet \oplus J^\bullet $ with differential

\[ d_ M^ n = \left( \begin{matrix} d_ I^ n & (-1)^{n + 1}\delta ^ n \\ 0 & d_ J^ n \end{matrix} \right) \]

Finally, the map $B[0] \to M^\bullet $ is given by $b \oplus c \circ \beta : M \to I^0 \oplus J^0$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05TV. Beware of the difference between the letter 'O' and the digit '0'.