# Famous Fourier-Mukai Results II (Orlov’s Result and the Beilinson Resolution)

This a continuation of this post, and this post follows the paper of Orlov. I’m going to give a rough outline to the following result

(Orlov’s Result) Any functor which is full, faithful and exact is represented by an object on the product.

The proof is long and complicated so I’ll only attempt to give a flavor of the ideas used in the proof. One central part of the proof is the Beilinson resolution of the diagonal of projective space:

(0)

where is the structure sheaf on and .

### Constructing the resolution

Start with what is sometimes called the Euler sequence on $\mathbb{P}^N$ (here is the tangent bundle):

locally:

(1)

Note for and from the les in cohomology. Now from the Kunneth formula in algebraic geometry:

I’ll make use of the global section corresponding to the identity. Now . Locally the identity corresponds to a map

so at this is

and in view of (1) its locally the inclusion followed by the projection:

the only point in this local description is that iff , i.e. , in other words vanishes exactly on the diagonal.

If is a vector then there is a contraction map via

.

Now contraction with gives a map and these are exactly the maps that appear in the Beilinson resolution.

### A rough outline

Now to Orlov’s result. From the data I need to produce an object .

The first step is to use that is projective to get an embedding and consider the functor

Now that has entered the picture we can utilize the Beilinson resolution (0), and using obtain a complex

(2)

where . I’m brushing a lot under the rug, it takes a bit of work to actually come up with this complex.

### Convolution

Now I need a powerful tool which I don’t have a firm grasp of and I can’t really explain here.

Let be a bounded complex. A left Postnikov system of is a diagram:

where the stared triangles are distinguished and the triangles with circles are commutative. An object is a left convolution of if there is a left Postnikov system such that . Denote by the class of all convolutions of .

Prop. If for and then has a convolution . Further, if and then all convolutions are canonically isomorphic.

Remark: I don’t have a great way of motivating this convolution business but its not a very geometric tool and in this post I’m trying to focus on the geometry that goes into this proof.

Continuing, the idea is now to the use the proposition with the complex (2) to obtain an object . Next one can show that basically by showing that both are convolutions of the same complex. With this result one can ultimately show . I’m not including the details because I want to focus on the rough idea and I want to avoid making an overly long post.

### What’s Left

The functor has been represented by an object on the product. Using general Fourier-Mukai properties see e.g. this post, it remains to find an object such that .

The object is produced in much the same way as is produced. Using and ample line bundle on obtain a resolution for the diagonal on and using obtain a complex on much like (2), then use the proposition to get a convolution . The details are a little different and more complicated because is understandably more explicit than a general smooth projective variety .

The object does not represent . Instead, using cohomological properties, one decomposes and and then a another argument is needed go show .

I’m tempted to say that this was much less then even a rough outline of the proof. But I really only wanted to discuss the Beilinson resolution and even though I was brief with Orlov’s result I think its clear that the Beilinson resolution is one of the key ideas behind it.

## About this entry

You’re currently reading “Famous Fourier-Mukai Results II (Orlov’s Result and the Beilinson Resolution),” an entry on Math Meandering

- Published:
- June 20, 2010 / 10:58 pm

- Category:
- alg. geo.

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