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Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a functor between abelian categories. Could anyone explain what the cohomological dimension the functor $F$ is?

We may need some additional condition to define cohomological dimension. I am mostly interested in the case when both $\mathcal{C}$ and $\mathcal{D}$ are abelian categories of modules over some rings.

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I guess you want $F$ to be left exact so that you can take its right derived functors. What's wrong with the usual definition as "the least number $d$ such that $R^n F (A) = 0$ for all $n > d$ and all objects $A$"? – Zhen Lin Aug 2 '12 at 10:41
For example, I would like to see the case where $F:\mod(A)\rightarrow \mod(A)$ given by taking $M\mapsto \mathrm{tor}(M)$ where $\mathrm{tor}(M)$ stands for torsion part of $M$. Let me think a bit. – Michel Aug 2 '12 at 16:23

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