The Stacks project

63.9 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

\[ F^1M = L, \ F^ nM = M \text{ for }n \leq 0\text{, and }F^ nM = 0\text{ for }n \geq 2. \]

By definition, we have

\[ \text{forget filt}(\widetilde M) = M, \quad \text{gr}^0(\widetilde M) = N, \quad \text{gr}^1(\widetilde M) = L \]

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

\[ \text{gr}^ p(RT(\widetilde M)) \stackrel{\text{qis}}{=} \left\{ \begin{matrix} 0 & \text{if} & p \neq 0, 1, \\ RT(N) & \text{if} & p = 0, \\ RT(L) & \text{if} & p = 1. \end{matrix} \right. \]

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

\[ E_1^{p, q} = R^{p+q}T(\text{gr}^ p(\widetilde M)) \Rightarrow R^{p+q}T(\text{forget filt}(\widetilde M)). \]

Unwinding the spectral sequence gives us the long exact sequence

\[ \xymatrix{ 0 \ar[r] & T(L) \ar[r] & T(M) \ar[r] & T(N) \ar@(rd, ul)[rdllllr] \\ & R^1T(L) \ar[r] & R^1T(M) \ar[r] & \ldots } \]

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

\[ \text{Tr}( \pi _ X^\ast | R\Gamma _ c(X_{\bar k}, \mathcal{F})) \in \mathbf{Z}/\ell ^ n \mathbf{Z} \]

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.


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