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 , we want to construct a object such that

Let's label some maps and set

Proposition. There is a natural isomorphism

We suppress the proof which is much like the previous few. We should note though that

  • is associative and
  • is a two sided identity.