# Quasi-coherent sheaves on projective varieties

Let be a -graded commutative ring.

We constructed a functor where and a functor in the other direction

Let's think a little more about their relationsip. For any , we get a map Applying , we get a map and hence an element of . This gives a natural transformation

Similarly, for any -module , there is a natural transformation which we describe on the principal opens for with . We have Each element of can be written as where and . Restricting to we get an element .

On , we have an isomorphism . Thus, we get an element giving an element of .

Proposition. The natural transformations and induce an adjunction .

Proof. (Expand to view)

Using a [theorem](https://738.f21.matthewrobertballard.com/notes/2021-08-31/) from last semester, we have to check that and are both identity maps.

For the , we check the map is the identity for sections over each principal open . Given , we have restricting back to gives .

For , if we take , then corresponds to global section of which is on each . Applying , we see we get back.

Under some mild assumptions, we see that hits all quasi-coherent sheaves on .

Lemma. If is finitely-generated in degree as a graded -algebra and is quasi-coherent, then is an isomorphism.

Proof. (Expand to view)

From the above, we see that if and only if . Thus, away from .

Let denote our generators in degree . We have a affine cover and is a hypersurface in each . Then is a module and . We know that in so there is some with on . Taking the max of the , we see that there is some with on all of . Thus, in x already and is injective.

A similar argument shows that for any there is some so that lifts to . This shows that is surjective.

Recall that for a graded -module , we have its torsion submodule and we say a is torsion if The module is torsion-free if

There is a short exact sequence Since there is a natural map

Lemma. Assume is finitely-generated over in degree by and that each is not a zero-divisor. Then M is injective

Proof. (Expand to view)

The kernel of consists of such that in for all homogeneous .

Let denote the generators of over in degree . Let be an element of degree . Then for each and for some . Thus, for large enough we have and so .

We can describe as where is the internal Hom for graded modules.

There are also the exact sequences which gives an exact sequence similar to the sequence which leads us to compare and

Since are all isomorphisms, we see that any induces a well-defined and hence an element of .

Let's denote this map by

Lemma. Assume that is finitely-generated over by elements of degree . Then is an isomorphism for all .

Proof. (Expand to view)

An element of is a collection of homogeneous elements for each homogeneous so that - there is such that has degree and - for any in .

Let denote a set of homogeneous generators for over . We claim that the function which assigns extends to a well-defined module map for some . This gives the inverse to .

Let's check that this construction yields a well-defined homomorphism. Up to replacing with and with , we can assume that in . If we have a relation of the form then in we have Thus, is annihilated by some power of each . Passing to a sufficient high power of forces this to vanish.

As a corollary, we get short exact sequences.

Corollary. Assume that is finitely-generated by elements of degree over . Then, for each injective graded -module we have a short exact sequence

Proof. (Expand to view)

Since is injective, we get short exact sequences taking direct sums and passing to the colimit gives the result.

Note the map is not always onto.

Example. Take with . Then,

As consequence, we get a semi-orthogonal decomposition relating graded -modules and quasi-coherent sheaves on .

Proposition. Assume that is finitely-generated by elements of degree over . Then there are semi-orthogonal decompositions where .

Here denotes the abelian subcategory of torsion graded -modules.

Proof. (Expand to view)

For each complex of injectives , we have a short exact sequence of complexes and thus a triangle Each component of is torsion.

Thanks to adjunction we have so we have semi-orthogonality.

It remains to check that is fully-faithful and the derived functors of and preserve cohomological boundedness.

Using adjunction, we have And since is an isomorphism.

Finally, from the triangle we see that for if and only if for . We will see [next]({% link notes/2022_01_13.md %}) that for where is the number of generators of over .

Remark.

• One can remove the condition that the finite set of generators of are of degree if we work with the quotient stack with the quasi-affine instead of .
• One would like to replace and with their finite versions and . This can be done but one needs to truncate the grading.