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.



Cohomology theory "from" Grothendieck's six operations?
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.


