The Stacks project

Lemma 13.26.14. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $T : \mathcal{A} \to \mathcal{B}$ be a left exact functor. Assume $\mathcal{A}$ has enough injectives. Let $(K^\bullet , F)$ be an object of $\text{Comp}^{+}(\text{Fil}^ f(\mathcal{A}))$. There exists a spectral sequence $(E_ r, d_ r)_{r\geq 0}$ consisting of bigraded objects $E_ r$ of $\mathcal{B}$ and $d_ r$ of bidegree $(r, - r + 1)$ and with

\[ E_1^{p, q} = R^{p + q}T(\text{gr}^ p(K^\bullet )) \]

Moreover, this spectral sequence is bounded, converges to $R^*T(K^\bullet )$, and induces a finite filtration on each $R^ nT(K^\bullet )$. The construction of this spectral sequence is functorial in the object $K^\bullet $ of $\text{Comp}^{+}(\text{Fil}^ f(\mathcal{A}))$ and the terms $(E_ r, d_ r)$ for $r \geq 1$ do not depend on any choices.

Proof. Choose a filtered quasi-isomorphism $K^\bullet \to I^\bullet $ with $I^\bullet $ a bounded below complex of filtered injective objects, see Lemma 13.26.9. Consider the complex $RT(K^\bullet ) = T_{ext}(I^\bullet )$, see ( Thus we can consider the spectral sequence $(E_ r, d_ r)_{r \geq 0}$ associated to this as a filtered complex in $\mathcal{B}$, see Homology, Section 12.24. By Homology, Lemma 12.24.2 we have $E_1^{p, q} = H^{p + q}(\text{gr}^ p(T(I^\bullet )))$. By Equation ( we have $E_1^{p, q} = H^{p + q}(T(\text{gr}^ p(I^\bullet )))$, and by definition of a filtered injective resolution the map $\text{gr}^ p(K^\bullet ) \to \text{gr}^ p(I^\bullet )$ is an injective resolution. Hence $E_1^{p, q} = R^{p + q}T(\text{gr}^ p(K^\bullet ))$.

On the other hand, each $I^ n$ has a finite filtration and hence each $T(I^ n)$ has a finite filtration. Thus we may apply Homology, Lemma 12.24.11 to conclude that the spectral sequence is bounded, converges to $H^ n(T(I^\bullet )) = R^ nT(K^\bullet )$ moreover inducing finite filtrations on each of the terms.

Suppose that $K^\bullet \to L^\bullet $ is a morphism of $\text{Comp}^{+}(\text{Fil}^ f(\mathcal{A}))$. Choose a filtered quasi-isomorphism $L^\bullet \to J^\bullet $ with $J^\bullet $ a bounded below complex of filtered injective objects, see Lemma 13.26.9. By our results above, for example Lemma 13.26.11, there exists a diagram

\[ \xymatrix{ K^\bullet \ar[r] \ar[d] & L^\bullet \ar[d] \\ I^\bullet \ar[r] & J^\bullet } \]

which commutes up to homotopy. Hence we get a morphism of filtered complexes $T(I^\bullet ) \to T(J^\bullet )$ which gives rise to the morphism of spectral sequences, see Homology, Lemma 12.24.4. The last statement follows from this. $\square$

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 015W. Beware of the difference between the letter 'O' and the digit '0'.