## A glimpse of the future

We step into the future of the class and attempt to explain the terminology of kernels and integral transformations.

Let $p: X \to S$ be a scheme over a base scheme $B$. Here you should think $X$ and $B$ are “locally look like rings” but are have global geometry.

The replacement for modules over rings will be the category of quasi-coherent sheaves $\operatorname{Qcoh} X$ on $X$. Like modules over a ring, we can tensor two such over the structure sheaf $\mathcal O_X$ of $X$.

Given a map $f: X \to Y$ of schemes over $B$, we get a few operations on quasi-coherent sheaves.

The most familiar one, from discussion so far, is pullback $$f^\ast : \operatorname{Qcoh} Y \to \operatorname{Qcoh} X$$ Locally $f$ corresponds to a map of rings $$\phi : R \to S$$ and $f^\ast$ is $S \otimes_R -$.

The other one is pushforward $$f_\ast : \operatorname{Qcoh} X \to \operatorname{Qcoh} Y$$ If $X$ and $Y$ were just rings (ie affine schemes), then $f_\ast$ would be the restriction of scalars along the ring homomorphism $\phi$.

In general, $f_\ast$ is more global. Indeed, it can viewed as akin to integration along the fibers of the map $f: X \to Y$.

We still have the adjunction $$f^\ast \vdash f_\ast$$

The globalization of tensoring with a bimodule comes from take a quasi-coherent sheaf $\mathcal K$ on the fiber product $X \times_B Y$. The fiber product has two maps $$p : X \times_B Y \to X$$ and $$q : X \times_B Y \to Y$$

From this we get a functor \begin{aligned} \Phi_{\mathcal K} : D(\operatorname{Qcoh} X) & \to D(\operatorname{Qcoh} Y) \\ \mathcal E & \mapsto \mathbf{R}q_\ast \left( \mathcal K \overset{\mathbf{L}}{\otimes} \mathbf{L}p^\ast \mathcal E \right) \end{aligned} called an integral transform with kernel $\mathcal K$.

For affine schemes, this is, up to equivalence, the functor $$E \mapsto M \otimes_R E$$

The language is motivated by the analogy with integral transformations of functions: $$e(x) \mapsto \int_X k(x,y)e(x) \ dx$$

Before we dive in the details sufficient to start working with such things, we want to introduce some common concepts used for such $D(X)$ and for which we have enough tools to understand.