Mathematics Asked on December 28, 2020

Let $mathcal{A}$ be an abelian category and denote by $D(mathcal{A})$ its derived category. Let $K_nin D(mathcal{A})$ be an inverse system of complexes in $mathcal{A}$ viewed as elements of the derived category. We have a natural map

$$ lim K_nrightarrow K_n$$

and thus

$$H_m(lim K_n)rightarrow lim H_m(K_n).$$

Since $lim$ is in general not an exact functor, this won’t be an isomorphism. Do we know more about this map, for instance is surjective in general? The only case I can deal with is the short exact sequence of complexes

$$0rightarrow A^bullet rightarrow B^bullet rightarrow C^bulletrightarrow 0$$

because that yields a long exact sequence

$$ldotsrightarrow H_m(lim B^bullet rightarrow C^bulletleftarrow 0)rightarrow H_m(B^bullet)rightarrow H_m(C^bullet)rightarrow ldots$$

and thus we have a surjection

$$H_m(lim B^bullet rightarrow C^bulletleftarrow 0)rightarrow ker left( H_m(B^bullet)rightarrow H_m(C^bullet) right)cong lim left( H_m(B^bullet)rightarrow H_m(C^bullet) leftarrow H_m(0) right) .$$

However, I don’t know how to generalize this.

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