Algebraic Geometry – Why IndCoh(X) is Analogous to Set of Distributions on X

Question

$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as a categorificaiton of the space of distributions on $X$, just as the (derived) category of quasi-coherent sheaves is to be thought of as a categorification of functions on $X$.

Why exactly is $\IndCoh(X)$ to be thought of as dual to $\QCoh(X)$? The only relation between these categories that I know of is that $\IndCoh(X)$ is a module over $\QCoh(X)$. Is there some way that an ind-coherent sheaf can act on a quasi-coherent one to yield a categorical analog of a point (for example, an object in $\QCoh(\{ \mathrm{pt} \})$)?

Answer 1

The connection is perhaps a bit more clear if you think about the corresponding $\infty$-categories of compact objects: the claim is that coherent sheaves behave like distributions while perfect sheaves behave like functions. This connection manifests itself in several ways:

• Coherent sheaves pushforward along proper maps, while perfect complexes pullback along arbitrary maps. Similarly, distributions integrate along proper maps while functions pullback along arbitrary maps.
• Skyscrapper sheaves at closed points are coherent but not perfect in general, in the same way that Delta functions are distributions but not functions.
• When $X$ is proper, the $\infty$-category of coherent sheaves on $X$ is dual to the $\infty$-category of perfect complexes on $X$, see theorem 1.1.3 in https://arxiv.org/abs/1312.7164. Concretely, each coherent sheaf $\mathcal{G}$ defines a functional sending a perfect sheaf $\mathcal{F}$ to the global sections of $\mathcal{F} \otimes \mathcal{G}$.
• There is a notion of singular support for coherent sheaves, which is a subset of the shifted cotangent bundle measuring how far the sheaf is from being perfect, see https://arxiv.org/abs/1201.6343. Similarly, distributions have a wave front set, which is a subset of the cotangent bundle measuring how far the distribution is from being a smooth function.
• The Hochschild homology of the $\infty$-category of coherent sheaves on $X$ recovers the space of distributions on the loop space of $X$, while the Hochschild homology of the $\infty$-category of perfect sheaves on $X$ recovers the space of functions on the loop space $X$.