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.

  • 4
    $\begingroup$ The most general reference I know of is this: arxiv.org/abs/0912.2110, but I hope someone will show up and give a more beginner-friendly one. $\endgroup$ – Denis Nardin May 6 '16 at 2:44
  • 1
    $\begingroup$ I don't think I can parse your question, but cohomology is (derived) pushing forward to a point- which is one of the 6 operations. $\endgroup$ – Dylan Wilson May 6 '16 at 13:27
  • 1
    $\begingroup$ @DylanWilson I am interpreting the question by asking to reduce the usual properties of a cohomology theory (e.g. duality, projection formula, cycle class map...) to the formalism of six operations rather than just defining the cohomology theory. $\endgroup$ – Denis Nardin May 6 '16 at 15:01

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.

  • $\begingroup$ I am confused by the role of O here. Shouldn't the first two rows be just f_*F and f_*f^! G ? While the last become f_!F and f_! f^! G? $\endgroup$ – Matthias Volkov May 6 '16 at 15:41
  • $\begingroup$ @Mattias Volkov If $Y$ is affine, then your definitions agree with mine (after taking the associated sheaf). In the general case you give a sheafified version of co/homology while my definitions provide abelian groups, in fact modules over $\Gamma(X, \mathcal{O}_X)$ for the first two and over $\Gamma(Y, \mathcal{O}_Y)$ for the remaining ones. $\endgroup$ – Leo Alonso May 6 '16 at 15:50
  • $\begingroup$ I understand that, but I don't see Y playing any role in your definition. For example, when f is the identity, don't your definitions of "cohomology with compact support" and "cohomology" with coefficients in F coincide? That just seems wrong to me. That's why I thought you should replace the definition with my comment above, which gives the correct answer when Y is a point. [also, in your comment I think you want Y a point (or maybe contractible) because in general affine varieties have non-trivial cohomology.] $\endgroup$ – Matthias Volkov May 6 '16 at 17:35
  • 3
    $\begingroup$ The philosophy behind these definitions is that of relative objects. They express invariants "along the fibers". For an absolute cohomology, use a contractible base or the absolute base. The 6 operations formalism is geared towards a relative version of co/homology in the spirit of Grothendieck. $\endgroup$ – Leo Alonso May 6 '16 at 22:14
  • $\begingroup$ It's missing a $f_{*}$ in the first one if you want something relative. $\endgroup$ – user40276 Sep 28 '16 at 6:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.