Exceptional collections and Čech cohomology

We have developed most of the background necessary to describe the derived category but we require one more important notion.

Definition. Let be a -linear triangulated category. An object is (-)exceptional if

A sequence of objects is called an exceptional collection if

  • each is exceptional
  • and for all and .

We say an exceptional colletion is strong if for all .

An exceptional collection is full if it generates .

Assume we have an abelian category and a full exceptional collection for .

Since full exceptional collections generate, we can apply the results from last semester to get an equivalence where

If, in addition, the collection is strong then we can take and is equivalent to the category of perfect complexes of modules over an honest -algebra. This algebra is usually non-commutative.

We can now state a fundamental result of Beilinson.

Theorem. For any , the sheaves form a full strong exceptional collection for .

Showing that something is a full (strong) exceptional collection breaks up into two parts:

  • a computation to check that vanishing of the appropriate Hom spaces and
  • a generation check.

We handle the generation statement now and the cohomology computation after introducing Čech cohomology.

Lemma. We have

Proof. (Expand to view)

The Koszul complex yields an exact complex of graded modules Applying gives the exact sequence Twisting by we get an exact sequence Consequently, if we have we can generate both and .


Lemma. We have for any coherent sheaf .

Proof. (Expand to view)

We have for a finitely- generated graded module . From the previous lemma and the fact is exact, it is enough to show we can generate from for .

Inductively, we can construct an exact sequence with Since is regular and graded local, we have the kernel is free. Thus, for , the extension of by is and in the derived category. So we generate .


Lemma. The collection generates .

Proof. (Expand to view)

Any complex with for and is an iterated sequence of cones of the form where for maximal with . Thus, we can generate a bounded complex using its cohomology sheaves -- which we generated in the last lemma.


To finish the proof of Beilinson's result, we need to compute We will do this next.

Čech cohomology

The universal way to "compute" is to

  • replace with an injective resolution,
  • apply , and
  • compute cohomology of the resulting complex of -modules.

Unfortunately, projective resolutions are hard enough to handle and we have some very explicit projective modules. Injective objects are generally more complicated. Having enough control over their explicit structure to compute cohomology of a sequence is a big ask.

However, there is a bit more flexibility in computing derived using adapated objects

If we can construct a resolution with then

Recall that is yields an equivalence between and if . Consequently, if is a quasi-coherent sheaf supported on affine patch, then for general .

For a general and open cover by affines, each is adapted and we have the obvious map Since is a sheaf, this is injective but it is rarely surjective.

To fix it and make a resolution, let and where the map is restricting scaled by .

Lemma. is a resolution.

Proof. (Expand to view)

It suffices to check exactness on stalks where each restriction map inducts an isomorphisms. Inclusion-exclusion says this is exact.


Lemma. If each is affine, then

Proof. (Expand to view)

As discussed above, we have a resolution by sheaves adapted to .


It would be nice to know that given two affines in the intersection is also affine. Such a scheme is called semi-separated.

A separated scheme is one where the diagonal morphism is a closed immersion. This implies semi-separated.

We will sidestep the generalities here because we will use very specific affines: principal opens in . Here the intersection is another principal open so the conditions of the previous lemma are immediate.

In our example of we use the principal opens . Then,

It makes sense to sum over and consider the complex of graded modules all at once.