# Base change

Recall in commutative algebras given a diagram

$B↓⏐ S R$
we can form a pushout (or fiber coproduct)
$B↓⏐ S R↓⏐ S⊗_{B}R $
using the tensor product $S⊗_{B}R$.

Consequently, in the category of affine schemes, we have fiber products (or pullbacks) $X×_{Z}Y↓⏐ X Y↓⏐ Z $ where, if $X=SpecR$, $Y=SpecS$, and $Z=SpecB$, we have $X×_{Z}Y=Spec(S⊗_{B}R).$

For schemes generally, we can use this to build fiber products. Given $X f Y↓⏐ gZ $ $X×_{Z}Y$ exists. Given affine $U⊂X,V⊂Y$, and $W⊂Z$ with $f(U),g(V)⊂W$, we have an affine chart $U×_{W}V$ and these can be glued together appropriately to give the fiber product as a scheme.

A square
$W↓⏐ X Y↓⏐ Z $
isomorphic to a fiber product square is commonly called *Cartesian*.

Given a Cartesian square $W↓⏐ g~ X f~ f Y↓⏐ gZ $ and a $E∈D(QcohX)$ can we compare $Lg_{∗}Rf_{∗}EandRf~ _{∗}Lg~ _{∗}E$ In general, they will not be quasi-isomorphic. However, under some assumptions, this is the case.

**Theorem**. Assume that
$Tor_{i}(O_{X},O_{Y})=0$
for $i=0$. Then, there is a natural isomorphism
$Lg_{∗}Rf_{∗}E≅Rf~ _{∗}Lg~ _{∗}E$

In particular, if either $O_{X}$ or $O_{Y}$ is flat over $O_{Z}$, 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 $S$ and $R$ the category of $(S,R)$-bimodules was an imporant source of functors between $D(ModS)$ and $D(ModR)$.

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

For $X←π_{X}X×_{Z}Y→π_{Y}Y$ and $Q∈D(QcohX×_{Z}Y)$, we get a functor $Φ_{Q}:D(QcohX)E →D(QcohY)↦Rπ_{Y∗}(Lπ_{X}E⊗L Q) $

We call $Φ_{Q}$ an *integral transform* and $Q$ its *kernel*. If $Φ_{Q}$ is an equivalence, we
say $Φ_{Q}$ and $Q$ are *Fourier-Mukai* and $X$ and $Y$ are *Fourier-Mukai* partners.

The language is inspired by the analogy with integral transformations in analysis. $f(x)↦∫_{X}Q(x,y)f(x)dx.$

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

**Lemma**. We have
$Φ_{Q}⊣Ψ_{Q}:=Rπ_{X∗}RHom_{X×Y}(Q,π_{Y}(−))$