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