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Let $X = \mathbb{A}^1(\mathbb{C}) \cong \mathbb{C}$ and $F$ be a sheaf on $X$. We call $F$ to be a weakly constructible sheaf (in this particular case) if there exists finite set of points say $S$, such that $F|_{X \setminus S}$ is a local system. Note that here we do not assume the finiteness of the local system. Let $T$ be a tree embedded in $X$ such that its set of vertices is the set $S$. Let $i$ denote the inclusion $T \hookrightarrow X$ then there exists the following natural restriction map

$$i^* : \mathrm{H}^i(X, F) \longrightarrow \mathrm{H}^i(T, F|_T)$$

Question : Why is the above map an isomorphism in all degrees $i$?

My attempt : Let $U := X \setminus T$ be the complement of the tree $T$ and let $j$ denote the inclusion $U \hookrightarrow X$. I have tried to write the long exact sequence associated to the standard triangle and have arrived at the conclusion that I need to prove that $\mathrm{H}^i(X, j_!j^*F) = 0$ for all degrees $i$. I can certainly try and relate this to compactly supported cohomology of the sheaf which might be easier to compute but I cannot get any further than that.

Motivation : Once we have the above claimed isomorphism, then using the description of the cohomology of a constructible sheaf on a simplicial set that is $T$, we can in particular prove that the cohomology of any constructible sheaf on an affine line vanishes in degree greater than 1. This is an implication of the Andreotti-Frankel theorem. This can be found for example as Remark 1.4 in Constructible sheaves by M.V.Nori, and as the first Lemma in $\S 7.2$ of Intersection Homology by M. Goresky, R. MacPherson.

Any help with solution or finding a reference is appreciated. Thanks in advance.

random123
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  • Have you tried the case in which S has cardinality 1? Moreover, in the general case one can observe that compact cohomology of X with coefficients in $j_! j^* F$ is the same as compact cohomology of U with coefficients in $j^F$. This is a local system on something which is homotopic to S^1 and should be easy to compute. Then use Poincaré duality to deduce the non compact cohomology of X with coefficients in $j_! j^ F$. I think this could be a good route. – Andrea Marino Sep 18 '19 at 11:57
  • @AndreaMarino If I understand duality correctly, the dual of $\mathrm{H}^i(X, j_!F)$ is $\mathrm{H}^{2-i}c(X, j(\check{(j^F)})$, where by $\check{(j^F)}$, I mean the dual of the local system $j^F$. Or am I making a mistake? – random123 Sep 18 '19 at 12:12
  • I do not know duality for local systems, is it something like (beside indices) $H(X,G)^* = H_c(X,G^)$ right? In this case setting $G=j_!j^F ^$ you get $H(X,j_!j^F^) ^=H_c(X,j_!j^F) =H_c(U,j^F)$ – Andrea Marino Sep 18 '19 at 13:08
  • @Andrea I dont think so. If you can find a reference or a proof of the mentioned statement, I would be happy to see it. – random123 Sep 18 '19 at 13:31
  • I would turn this into a private conversation, do you know how to do it? However, what does duality says for a general sheaf? Can't understand from your comment, neither I can find a reference for duality in non compact case with local systems – Andrea Marino Sep 18 '19 at 13:42
  • @AndreaMarino I am sorry but I don't know how to start a private chat. Let $X$ be a smooth manifold of dimension $n$ and $F$ be a constructible sheaf on it, then $\mathrm{H}^i(X, F) \cong \mathrm{H}^{n-i}(X, D_X(F))$, where $D_X$ is the dualizing functor. You may find more about it here : https://en.wikipedia.org/wiki/Verdier_duality. – random123 Sep 18 '19 at 14:30

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