# Grothendieck spectral sequence [duplicate]

Possible Duplicate:
Composing left and right derived functors

Hi,

probably this question is obvious. I apologize for this.

Given functors $F$ and $G$ left exact, with as good properties as you want, on the "correct" category, we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}F\circ G$. I am looking for an analogous for a "mixed version" in the following case: $F$ left exact and $G$ right exact. What appens to $L^pG\circ R^q F$?

Thanks

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## marked as duplicate by Anton GeraschenkoJul 26 '11 at 20:16

Such questions are probably best thought about in terms of composing total derived functors. $L^{p}G(Y)$ is computed by taking a $G$-acyclic resolution of your object $Y$, applying $G$ to each term of that resolution, and then taking cohomology of the resulting complex at $p$. But you could also stop after applying $G$ to each term to get the total derived functor $LG(Y)$, which is a complex. Likewise, you can get the total derived functor $RF(X)$. Now you could try to compose the two total derived functors: given $X$, resolve it with $F$-acyclics, apply $F$ term by term to a complex $Y$, and –  Chris Brav Jul 26 '11 at 11:22
now resolve the complex $Y$ by a complex of $G$-acyclics (meaning replace it with a quasi-isomorphic complex of $G$-acyclics) and apply $G$ term by term. This will give you a composition $LG \circ RF$. Now you could ask how to compute cohomology of the resulting complex $LG \circ RF (X)$. This should be doable in terms of a spectral sequence beginning with $L^{p}G \circ R^{q}F$. –  Chris Brav Jul 26 '11 at 11:25