Philosophical Identity of Derived Correspondences Between Smooth Varieties.

Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γ_f ⊂ X × Y of f, which (as a set) is defined as

Γ_f = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections p_X, p_Y from X × Y to the factors X, Y. Restricted to the subvariety Γ_f, p_X is an isomorphism (since f is a morphism). The fibres of p_Y restricted to Γ_f are just the fibres of f; so for example f is proper iff p_Y | Γ_f is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f_∗ : H(X) → H(Y)

This map can be expressed in terms of the graph Γ_f and the projection maps as

f_∗(α) = p_Y∗ (p_X^∗(α) ∪ [Γ_f]) —– (2)

where [Γ_f] ∈ H (X × Y) is the fundamental class of the subvariety [Γ_f]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that p_X be an isomorphism with some weaker assumption, such as p_X |Γ_f, p_Y | Γ_f finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ_∗ : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf O_Γ on X × Y. Associated to the projection maps p_X, p_Y, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ D^b(X × Y) with support which is proper over both factors. A derived correspondence defines a functor Φ_F by

Φ_F : D^b(X) → D^b(Y)
(−) ↦ Rp_Y∗(Lp_X^∗(−) ⊗^L F)

where (−) could refer to both objects and morphisms in D^b(X). F is sometimes called the kernel of the functor Φ_F.

The functor Φ_F is exact, as it is defined as a composite of exact functors. Since the projection p_X is flat, the derived pullback Lp_X^∗ is the same as ordinary pullback p_X^∗. Given derived correspondences E ∈ D^b(X × Y), F ∈ D^b(Y × Z), we obtain functors Φ_E : D^b(X) → D^b(Y), Φ_F : D^b(Y) → D^b(Z), which can then be composed to get a functor

Φ_F ◦ Φ_E : D^b(X) → D^b(Z)

which is a two-sided identity with respect to composition of kernels.

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