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63.8 Filtered derived functors

And then there are the filtered derived functors.

Definition 63.8.1. Let $T: \mathcal{A} \to \mathcal{B}$ be a left exact functor and assume that $\mathcal{A}$ has enough injectives. Define $RT: DF^+(\mathcal{A}) \to D F^+(\mathcal{B})$ to fit in the diagram

\[ \xymatrix{ DF^+(\mathcal{A}) \ar[r]^{RT} & DF^+(\mathcal{B}) \\ K^+(\mathcal{I}) \ar[u] \ar[r]^{T \quad } & K^+(\text{Fil}^ f(\mathcal{B})). \ar[u]} \]

This is well-defined by the previous lemma. Let $G: \mathcal{A} \to \mathcal{B}$ be a right exact functor and assume that $\mathcal{A}$ has enough projectives. Define $LG: DF^+(\mathcal{A}) \to DF^+(\mathcal{B})$ to fit in the diagram

\[ \xymatrix{ DF^-(\mathcal{A}) \ar[r]^{LG} & DF^-(\mathcal{B}) \\ K^-(\mathcal{P}) \ar[u] \ar[r]^{G \quad } & K^-(\text{Fil}^ f(\mathcal{B})). \ar[u]} \]

Again, this is well-defined by the previous lemma. The functors $RT$, resp. $LG$, are called the filtered derived functor of $T$, resp. $G$.

Proposition 63.8.2. In the situation above, we have

\[ \text{gr}^ p \circ RT = RT \circ \text{gr}^ p \]

where the $RT$ on the left is the filtered derived functor while the one on the right is the total derived functor. That is, there is a commuting diagram

\[ \xymatrix{ DF^+(\mathcal{A}) \ar[r]^{RT} \ar[d]_{\text{gr}^ p} & DF^+(\mathcal{B}) \ar[d]^{\text{gr}^ p}\\ D^+(\mathcal{A}) \ar[r]^{RT} & D^+(\mathcal{B}).} \]

Proof. Omitted. $\square$

Given $K^\bullet \in DF^+(\mathcal{B})$, we get a spectral sequence

\[ E_1^{p, q} = H^{p+q}(\text{gr}^ p K^\bullet ) \Rightarrow H^{p+q}(\text{forget filt}(K^\bullet )). \]

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