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6 added 12 characters in body

I think here is how one should understand the last paragraph of the wiki.

Consider $f:X\to pt$. We have (all functors are derived and my $Hom$ are sheaf $Hom$) $$Hom_{pt}(f_!F,F)=f_*Hom_X(F,f^!F)$$

$f_!F=\Gamma_c(X,F)$, so the l.h.s. computes the dual of the cohomology with compact support of the constant sheaf $F$, i.e. the dual of the cohomology with compact support of $X$.

$Hom_X(F,f^!F)= \Gamma(f^!F)$ and thus the r.h.s. is $f_*(\Gamma(f^!F))=\Gamma(X,f^!F)$, the homology of $X$.

EDIT : to answer precisely the question I include a summary of the comments.

1. there is a typo in the wiki: $[F,X[k]]$ should be understood as $[F,F[k]]$.

2. for any sheaf of $F$-modules $S$ (concentrated in degree 0), $[F[−k],S]=H^k(X,S)$.

3. Contrary to what is claimed in the wiki, there is no duality between $H^k(X,F)=[F[−k],F]$ and $H_k(X,F)=[F[−k],D_X]$ (where $D_X:=f^!F$ for $f:X\to pt$). The duality is, either between $H^k_c(X,F)$ and $H_k(X,F)$, or between $H^k(X,F)$ and $H_k^{BM}(X,F)$. And as far as I understand $H^*(X,S)$ is dual to $H_c^*(X,S^\vee)$.

5 deleted 1 characters in body; added 27 characters in body

I think here is how one should understand the last paragraph of the wiki.

Consider $f:X\to pt$. We have (all functors are derived and my $Hom$ are sheaf $Hom$) $$Hom_{pt}(f_!F,F)=f_*Hom_X(F,f^!F)$$

$f_!F=\Gamma_c(X,F)$, so the l.h.s. computes the dual of the cohomology with compact support of the constant sheaf $F$, i.e. the cohomology with compact support of $X$.

$Hom_X(F,f^!F)= \Gamma(f^!F)$ and thus the r.h.s. is $f_*(\Gamma(f^!F))=\Gamma(X,f^!F)$, the homology of $X$.

EDIT : to answer precisely the question I include a summary of the comments.

1. there is a typo in the wiki: $$[F,X[k]] [F,X[k]] should be understood as [F,F[k]]. 2. for any sheaf of F-modules S, S (concentrated in degree 0), [F[−k],S]=H^k(X,S). 3. Contrary to what is claimed in the wiki, there is no duality between H^k(X,F)=[F[−k],F] and H_k(X,F)=[F[−k],D_X] (where D_X:=f^!F for f:X\to pt). The duality is, either between H^k_c(X,F) and H_k(X,F), or between H^k(X,F) and H_k^{BM}(X,F). And as far as I understand H^*(X,S) is dual to H_c^*(X,S^\vee). 4 added 559 characters in body I think here is how one should understand the last paragraph of the wiki. Consider f:X\to pt. We have (all functors are derived and my Hom are sheaf Hom)$$ Hom_{pt}(f_!F,F)=f_*Hom_X(F,f^!F) $$f_!F=\Gamma_c(X,F), so the l.h.s. computes the dual of the cohomology with compact support of the constant sheaf F, i.e. the cohomology with compact support of X. Hom_X(F,f^!F)= \Gamma(f^!F) and thus the r.h.s. is f_*(\Gamma(f^!F))=\Gamma(X,f^!F), the homology of X. EDIT : to answer precisely the question I include a summary of the comments. 1. there is a typo in the wiki:$$[F,X[k]]$should be understood as$[F,F[k]]$. 2. for any sheaf of$F$-modules$S$,$[F[−k],S]=H^k(X,S)$. 3. Contrary to what is claimed in the wiki, there is no duality between$H^k(X,F)=[F[−k],F]$and$H_k(X,F)=[F[−k],D_X]$(where$D_X:=f^!F$for$f:X\to pt$). The duality is, either between$H^k_c(X,F)$and$H_k(X,F)$, or between$H^k(X,F)$and$H_k^{BM}(X,F)$. And as far as I understand$H^*(X,S)$is dual to$H_c^*(X,S^\vee)$. 3$*$replaced by$pt\$ to avoid ambiguity