The Stacks project

64.9 Application of filtered complexes

Let $\mathcal{A}$ be an abelian category with enough injectives, and $0 \to L \to M \to N \to 0$ a short exact sequence in $\mathcal{A}$. Consider $\widetilde M \in \text{Fil}^ f(\mathcal{A})$ to be $M$ along with the filtration defined by

By definition, we have

and $\text{gr}^ n(\widetilde M) = 0$ for all other $n \neq 0, 1$. Let $T: \mathcal{A} \to \mathcal{B}$ be a left exact functor. Assume that $\mathcal{A}$ has enough injectives. Then $RT(\widetilde M) \in DF^+(\mathcal{B})$ is a filtered complex with

and $\text{forget filt}(RT(\widetilde M))\stackrel{\text{qis}}{ = } RT(M)$. The spectral sequence applied to $RT(\widetilde M)$ gives

Unwinding the spectral sequence gives us the long exact sequence

This will be used as follows. Let $X/k$ be a scheme of finite type. Let $\mathcal{F}$ be a flat constructible $\mathbf{Z}/\ell ^ n \mathbf{Z}$-module. Then we want to show that the trace

is additive on short exact sequences. To see this, it will not be enough to work with $R\Gamma _ c(X_{\bar k}, -) \in D^+(\mathbf{Z}/\ell ^ n \mathbf{Z})$, but we will have to use the filtered derived category.