How, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos?
I would like some references.
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Sign up to join this communityHow, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos?
I would like some references.
Assuming you have the usual 6 operations we have the following recipes for co/homologies of $f \colon X \to Y$ with coefficients $\mathcal{F}$ over $X$, $\mathcal{G}$ over $Y$ and $\mathcal{O}$ denoting the monoidal unit.
Cohomology:........................................... $\mathrm{Hom}_X(\mathcal{O}_X,\mathcal{F})$
Homology:.............................................. $\mathrm{Hom}_X(\mathcal{O}_X,f^!\mathcal{G})$
Cohomology with compact support:....... $\mathrm{Hom}_Y(\mathcal{O}_Y,f_!\mathcal{F})$
Borel-Moore homology:......................... $\mathrm{Hom}_Y(\mathcal{O}_Y,f_!f^!\mathcal{G})$
The defining relations between the 6 functors yield the usual dualities and comparison maps between the four theories. Notice that when f is proper $f_! = f_*$ and the latter pair becomes the first.