Induced Natural transformation on right derived functors

Question

Let $\mathcal A$ and $\mathcal B$ be two abelian categories.

Let $F_1, F_2$ be left-exact functors from $\mathcal A$ to $\mathcal B$ such that there is a natural transformation $\eta$ from $F_1$ to $F_2$.

Is there an induced natural transformation on the right derived functors $R^nF_1$ and $R^nF_2$?

Thank you for any reference.

Answer 1

Picking injective resolutions of objects in $\mathcal{A}$ will give morphisms $R^nF_1(A)\to R^nF_2(A)$, to see the naturality you can use compatible resolutions of other objects given by the Comparison Theorem of Weibel's book on Homological Algebra (Theorem 2.3.7).

Answer 2

More abstractly, the total right derived functor $RF$ is the left Kan extension of $F$'s natural extension to chain complexes along the localization functor from chain complexes in $\mathcal A$ to the derived category of $\mathcal A.$ Thus we get an induced natural transformation $RF_1\to RF_2$ from the functoriality of the left Kan extension along a fixed functor, and the natural transformations on $R^n$ arise from this.

Answer 3

The right derived functor $R^nF_1$ of $F_1$ is the universal $\delta$-functor of $F_1$. This means that if you have any other delta functor $T^n$ and a natural transformation $\alpha: R^0F_1 \Rightarrow T^0$, then there is a unique morphism of $\delta$-functors $\alpha^n: R^nF_1 \Rightarrow T^n$ with $\alpha^0=\alpha$.

So your transformation $\eta: F_1\Rightarrow F_2$ yields a transformation $R^0F_1 \cong F_1 \Rightarrow F_2 \cong R^0F_2$, which in turn yields transformations $\eta^n: R^nF_1 \Rightarrow R^nF_1$. Even better, the $\eta^n$ are compatible with the connecting morphisms.

Here is how $\eta^{n+1}$ is defined if you already have $\eta^n$. Given $A$ take a short exact sequence $0\to A \to I\to Q\to 0$ with $I$ injective. Look at the following diagram

Here you use that $R^{n+1}F_1I = 0$. $\eta_A^{n+1}$ is defined by the universal property of cokernels. You can immediately see that $\eta^{n+1}$ is uniquely determined by the requirement that $\eta^*$ be a morphism of $\delta$-functors. It takes some effort to show that $\eta^{n+1}$ is natural and that the definition does not depend on the particular choice of short exact sequence. Of course all of this only works if the category has enough injectives.

References are: Grothendiecks Tohoku paper; there is a pdf on http://therisingsea.org/ which is called derived functors; Also "A course in homological algebra" by Hilton and Stammbach contains these kind of details.