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. $G$.

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