# 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:

**Identity kernels**. Let $Δ={(x,x)∈X×X}$ be the diagonal. Then, $∫_{X}δ_{Δ}(x,y)f(x)dx=f(y)$ for the $δ$-function along $Δ$.**Projection kernels**. For a kernel of the form $K(x,y)=e(x)h(y)$, we have $∫_{X}K(x,y)f(x)dx=h(y)∫_{X}e(x)f(x)dx$ is a type of projection onto $h(y)$.**Composition as convolution**. For $K(x,y)$ and $Q(y,z)$ the composition of $f(x)↦∫_{X}K(x,y)f(x)dx$ and $f(y)↦∫_{Y}Q(y,z)f(y)dy$ is also an integral transformation with kernel $Q⋆K:=∫_{Y}K(x,y)Q(y,z)dy$

These three properties still hold integral transforms of derived categories.

## Identity Kernels

Given a $k$-algebra $R$, we have a canonical exact sequence of $R⊗_{k}R$ -modules $0→I→R⊗_{k}R→R→0$ This embeds $SpecR$ as a closed subscheme of $SpecR×_{Speck}SpecR$ cut out by the sheaf of ideals $I$.

For a morphism of schemes $X→Y$, we have a corresponding map

$Δ:X→X×_{Y}X$
as the diagram
$X↓⏐ 1_{X}X 1_{X} X↓⏐ Y $
commutes.

We set $O_{Δ}:=Δ_{∗}O_{X}.$

**Proposition**. $Φ_{O_{Δ}}$ is isomorphic to the identity functor.

##
**Proof**. (Expand to view)

Suppressing the derived decoration, we have $Φ_{O_{Δ}}(E):=π_{X∗}(π_{X}E⊗Δ_{∗}O_{X})$ Using the projective formula, we get $π_{X∗}(π_{X}E⊗Δ_{∗}O_{X})≅π_{X∗}Δ_{∗}(O_{X}⊗Δ_{∗}π_{X}E)$ From the universal property of the fiber product, we have $π_{X}∘Δ=1_{X}$ so $π_{X∗}Δ_{∗}≅Id$ and $Δ_{∗}π_{∗}≅Id.$

▢## Projections Kernels

Given $E∈D(QcohX)$ and $F∈D(QcohY)$,
we set
$E⊠_{Z}F:=π_{X}E⊗π_{Y}F$
Such an object is called an *external product* of $E$ and $F$. We will omit $Z$
when the context is clear.

**Proposition**. Suppose $Z=Speck$ and $X$ and $Y$ are
Tor-independent over $k$. If $E$ is compact, we have a natural isomorphism
$Φ_{E_{∨}⊠F}(E_{′})≅RHom(E,E_{′})⊗L _{k}F$

##
**Proof**. (Expand to view)

We have $Φ_{E_{∨}⊠F}(E_{′}):=π_{Y∗}(π_{X}E_{′}⊗π_{X}E_{∨}⊗π_{Y}F)$ Using the projection formula gives $π_{Y∗}π_{X}(E_{∨}⊗E_{′})⊗F$ Since $E$ is compact, the natural map $E_{∨}⊗E_{′}→Hom(E,E_{′})$ is a quasi-isomorphism.

From our assumptions, the Cartesian square $X×_{k}Y↓⏐ p_{Y}X p_{X} Y↓⏐ Speck $ is Tor-independent. Thus, $π_{Y∗}π_{X}Hom(E,E_{′})⊗F≅p_{X}p_{Y∗}Hom(E,E_{′})⊗F$ And $p_{Y∗}Hom(E,E_{′})≅Hom(E,E_{′})$ 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 $Q∈D(QcohX×Y)$ and $K∈D(QcohY×Z)$, we want to construct a object $K⋆Q∈D(QcohX×Z)$ such that $Φ_{K}∘Φ_{K}≅Φ_{K⋆Q}$

Let's label some maps $π_{X,Y}:X×Y×Z→X×Yπ_{Y,Z}:X×Y×Z→Y×Zπ_{X,Z}:X×Y×Z→X×Z$ and set $K⋆Q:=π_{X×Z∗}(π_{X×Y}Q⊗π_{Y×Z}K)$

**Proposition**. There is a natural isomorphism
$Φ_{K}∘Φ_{K}≅Φ_{K⋆Q}$

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

- $⋆$ is associative and
- $O_{Δ}$ is a two sided identity.