Computing sheaf cohomology on

Last time we saw that computing could be achieved by computing the cohomology of the (augmented) Čech complex

Let's look at the case to get a foothold. This reduces to Since is regular, there is no . We have which as a -module has a filtration whose associated graded pieces are In particular, is a free -module. It is also graded with .

The general complex is isomorphic to the tensor product of complexes over . Since the homologies of the complexes are flat over , we have From our computation above we see that

Theorem. We have

Proof. (Expand to view)

We have an isomorphism of functors which gives the isomorphims of derived functors

Now we use the computation above. Recall that where . Thus, we have and we get the vanishing for . For , from the discussion above,
forms a -basis for . Rewriting this as we get a bijection between and .


Fixing the basis vector gives an isomorphism From our computations, we see that composition is given by multiplication up to isomorphism.

Corollary. Composition is a perfect pairing. In other words, this induces an isomorphism

For a scheme , we can ask if the functor is quasi-isomorphic to for some -linear auto-equivalence . In the case is simply a field, then this is just a Serre functor. In this statement, it is an enhanced version of a Serre functor.

Proposition. For any objects there are natural quasi-isomorphisms where .

Proof. (Expand to view)

Some care is required for a general which we won't take. From composition we have Using the natural map lands us in \mathbf{R}\operatorame{Hom}(\mathcal O, \mathcal O(-n-1)[n]) which is quasi-isomorphic to .

This gives a natural map which we see is a quasi-isomorphism for a collection of generators . Therefore, it is for all objects of .