## Syllabus

The syllabus can be found at the syllabus tab.

## Notes

Course notes can be found in the tab to the left.

## Previous semester

This course is a continuation of the Fall 2021 course Derived Categories I.

## Me

If you are interested in a bit about who I am and what I do visit my website.

# 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.

# 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.

# 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 .

# Compact generation and Grothendieck duality

We saw last semester that compact generation of a triangulated category was a very useful property. Here we establish that is compactly generated for a quasi-projective schemes over an affine base. In addition, we provide some useful replacements for deriving common functors.

Lemma. Let be a -graded ring. Then for is a set of compact generators for .

Proof. (Expand to view)

One checks that Thus, if is right orthogonal to all it must be acyclic.

Proposition. If is finitely-generated in degree over , then is compactly generated.

Proof. (Expand to view)

Let . Recall that we [have](2022_01_11.md) a SOD Given by and . Thus, This vanishes for all if and only if is acyclic. As we see that is acyclic also.

Lemma. If is compactly-generated and is an open subscheme, then so is .

Proof. (Expand to view)

Let denote the inclusion. Then the counit is an isomorphism. Thus, if is a set of compact generators for , then is for .

Corollary. Let be quasi-projective over an affine base. Then any compact object is quasi-isomorphic to a bounded complex of locally-free sheaves.

Proof. (Expand to view)

We have seen that form a set of compact generators. As we saw [last semester](https://738.f21.matthewrobertballard.com/notes/2021_10_05/), the compact objects in are summands of iterated cones over maps to objects from the set of compact generators. Such a complex is bounded and has locally-free components.

We saw last time that is the Serre functor on . In fact, this is a consequence of a more general fact known as Grothendieck duality. One formulation of Grothendieck duality is that for a map of schemes possesses a right adjoint to . This is not automatic in full generality. But it is true with mild conditions on and .

For the projection \pi : \mathbb{P}^n_k \to \operatorname{Spec} k} we have

Originally proof of the existence of a right adjoint was arduous. However, [Neeman](https://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174- 9/S0894-0347-96-00174-9.pdf) demonstrated how Brown Representability can be used to give a clean proof.

Thanks to Brown Representability, we know that for to have a right adjoint is equivalent to the natural map being an isomorphism for all sums. This question is local on so we reduce to asking -- when does taking cohomology commute with direct sums?

One useful fact: formation of the Čech complex for an affine cover where all intersections are affine commutes with direct sums.

Proposition. For any map between quasi-projective schemes over an affine base, there is a right adjoint to .

Proof. (Expand to view)

From the discussion above, we reduce to the checking the case where is affine. Here we are computing global sections over a quasi-affine scheme which can be done using a Čech complex.

Remark. Neeman proves the existence of whenver and are quasi-compact and quasi-separated.

## Deriving tensor and sheaf Hom

We have implicitly used both the tensor product and sheaf Hom of -modules.

Recall that is locally-free if each has a neighborhood with Since exactness can be checked locally, preserves exactness of complexes of -modules if is locally-free. Thus, if we can replace any complex with one whose components are locally-free, we can use these adapted objects to derive .

Proposition. For any quasi-projective schemes and any chain complex of quasi-coherent sheaves, there is complex with locally-free components and a quasi-isomorphism .

Proof. (Expand to view)

It suffices to prove this for projective . Any complex of quasi-coherent sheaves on is of the form for some complex of graded modules . There are enough K-projectives in K(\operatarname{GrMod} R). Indeed, the twists of suffice. Thus, we get a -projective and a quasi-isomorphism . Applying gives the result.

Note that despite the notation we cannot find enough K-projective complexes in .

Using this result, we take for some such .

Similarly, we can derive by replacing by a complex of locally-frees or by a K-injective. The choices are naturally quasi-isomorphic so we denote either by

# The projection formula and standard adjunctions

The main functors in play for sheaves of modules are

• for a map
• for
• and

We will also use the notation for both the underived and derived duals over .

We record some back facts about them and we omit the details for the most part.

Lemma. We have an adjunction .

Lemma. We also have an adjunction

Lemma. Tensor products commute with pullbacks

From these, we can derive some less immediate ones.

Proof. (Expand to view)

We have natural isomorphisms

We also have a sheafy version of pull-push adjunction.

Lemma. There is a natural isomorphism

Proof. (Expand to view)

We have From the previous lemma, we have two right adjoints. Thus, they must be isomorphic.

One of the most useful relationships between these functors is one that is not true. Thanks to the map coming from the counit and adjunction we have a natural map However, this is not an isomorphism of quasi-coherent sheaves in general.

For what and is the map an isomorphism? Taking works. Since isomorphism can be detected locally, it suffices that is locally isomorphic to .

Moreover, if for some , then will also be an isomorphism if commutes with coproducts which is true for quasi-projective schemes (and more generally).

Proposition. Let be a map of quasi-projective schemes. Then for any and , the natural map is a quasi-isomorphism.

Proof. (Expand to view)

We can replace with a -injective complex and with complexe of locally-free sheaves. Then, from the discussion, above we know that the induced map on the components is an isomorphism.

The previous result is often called the projection formula.

# Base change

Recall in commutative algebras given a diagram
we can form a pushout (or fiber coproduct) using the tensor product .

Consequently, in the category of affine schemes, we have fiber products (or pullbacks) where, if , , and , we have

For schemes generally, we can use this to build fiber products. Given exists. Given affine , and with , we have an affine chart and these can be glued together appropriately to give the fiber product as a scheme.

A square isomorphic to a fiber product square is commonly called Cartesian.

Given a Cartesian square and a can we compare In general, they will not be quasi-isomorphic. However, under some assumptions, this is the case.

Theorem. Assume that for . Then, there is a natural isomorphism

In particular, if either or is flat over , then we can exchange pushing and pulling.

The previous result is often called (flat) base change. We won't provide more details on this result. Lipman has some very thorough notes on Grothendieck duality which include exposition on Tor-independent base change.

Along with the projection formula, base change will serve as regular tool in our future investigations of derived categories.

# Kernels and integral transforms

We saw last semester that for two rings and the category of -bimodules was an imporant source of functors between and .

Here we globalize this idea. Note that an -bimodule is the same data as a -module. Geometrically, corresponds to the fiber product. Therefore, it is natural to look to objects on the (fiber) product for a source of functors.

For and , we get a functor

We call an integral transform and its kernel. If is an equivalence, we say and are Fourier-Mukai and and are Fourier-Mukai partners.

The language is inspired by the analogy with integral transformations in analysis.

Each integral transform comes with a right adjoint thanks to Grothendieck duality.

Lemma. We have

# Identities, projections, and convolutions

Let's expand out analogy with integral transforms in analysis a bit more. There are three nice properties of such linear transformations:

1. Identity kernels. Let be the diagonal. Then, for the -function along .
2. Projection kernels. For a kernel of the form , we have is a type of projection onto .
3. Composition as convolution. For and the composition of and is also an integral transformation with kernel

These three properties still hold integral transforms of derived categories.

## Identity Kernels

Given a -algebra , we have a canonical exact sequence of -modules This embeds as a closed subscheme of cut out by the sheaf of ideals .

For a morphism of schemes , we have a corresponding map
as the diagram commutes.

We set

Proposition. is isomorphic to the identity functor.

Proof. (Expand to view)

Suppressing the derived decoration, we have Using the projective formula, we get From the universal property of the fiber product, we have so and

## Projections Kernels

Given and , we set Such an object is called an external product of and . We will omit when the context is clear.

Proposition. Suppose and and are Tor-independent over . If is compact, we have a natural isomorphism

Proof. (Expand to view)

We have Using the projection formula gives Since is compact, the natural map is a quasi-isomorphism.

From our assumptions, the Cartesian square is Tor-independent. Thus, And giving the result.

In our analogy here with analysis, the Hom-spaces furnish the analog of a non-degenerate inner product.

## Convolution of kernels

Finally, given and