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 $$\Delta =\{(x,x)\in X\times X\}$$ be the diagonal. Then, $${\int }_X{\delta }_{\Delta }(x,y)f(x)dx=f(y)$$ for the $\delta$-function along $\Delta$.
2. Projection kernels. For a kernel of the form $K(x,y)=e(x)h(y)$, we have $${\int }_{X}K(x,y)f(x)dx=h(y){\int }_{X}e(x)f(x)dx$$ is a type of projection onto $h(y)$.
3. Composition as convolution. For $K(x,y)$ and $Q(y,z)$ the composition of $$f(x)\mapsto {\int }_{X}K(x,y)f(x)dx$$ and $$f(y)\mapsto {\int }_{Y}Q(y,z)f(y)dy$$ is also an integral transformation with kernel $$Q\star K:={\int }_{Y}K(x,y)Q(y,z)dy$$

These three properties still hold integral transforms of derived categories.

Identity Kernels

Given a $k$-algebra $R$, we have a canonical exact sequence of $R{\otimes }_{k}R$ -modules $$0\to I\to R{\otimes }_{k}R\to R\to 0$$ This embeds $\mathrm{Spec}R$ as a closed subscheme of $\mathrm{Spec}R{\times }_{\mathrm{Spec}k}\mathrm{Spec}R$ cut out by the sheaf of ideals $\tilde{I}$.

For a morphism of schemes $X\to Y$, we have a corresponding map
$$\Delta :X\to X{\times }_{Y}X$$ as the diagram $$\begin{array}{ccc}X& \underset{1_X}{\to }& X\\ \downarrow 1_X& & \downarrow \\ X& \to & Y\end{array}$$ commutes.

We set $${\mathcal{O}}_{\Delta }:={\Delta }_{\ast }{\mathcal{O}}_X.$$

Proposition. ${\Phi }_{{\mathcal{O}}_{\Delta }}$ is isomorphic to the identity functor.

Projections Kernels

Given $E\in D(\mathrm{Qcoh}X)$ and $F\in D(\mathrm{Qcoh}Y)$, we set $$E{⊠}_{Z}F:={\pi }_X^{\ast }E\otimes {\pi }_Y^{\ast }F$$ Such an object is called an external product of $E$ and $F$. We will omit $Z$ when the context is clear.

Proposition. Suppose $Z=\mathrm{Spec}k$ and $X$ and $Y$ are Tor-independent over $k$. If $E$ is compact, we have a natural isomorphism $${\Phi }_{E^{\vee }⊠F}(E^{\prime })\cong \mathbf{R}\mathrm{Hom}(E,E^{\prime }){\stackrel{\mathbf{L}}{\otimes }}_{k}F$$

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

Convolution of kernels

Finally, given $Q\in D(\mathrm{Qcoh}X\times Y)$ and $K\in D(\mathrm{Qcoh}Y\times Z)$, we want to construct a object $$K\star Q\in D(\mathrm{Qcoh}X\times Z)$$ such that $${\Phi }_K\circ {\Phi }_K\cong {\Phi }_{K\star Q}$$

Let's label some maps $$\begin{gathered} {\pi }_{X,Y}:X\times Y\times Z\to X\times Y \\ {\pi }_{Y,Z}:X\times Y\times Z\to Y\times Z \\ {\pi }_{X,Z}:X\times Y\times Z\to X\times Z \end{gathered}$$ and set $$K\star Q:={\pi }_{X\times Z\ast }({\pi }_{X\times Y}^{\ast }Q\otimes {\pi }_{Y\times Z}^{\ast }K)$$

Proposition. There is a natural isomorphism $${\Phi }_K\circ {\Phi }_K\cong {\Phi }_{K\star Q}$$

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

• $\star$ is associative and
• ${\mathcal{O}}_{\Delta }$ is a two sided identity.