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Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over a field $k$. An elementary finite correspondence from $X$ to $Y$ is defined to be an irreducible closed subset $W$ of $X\times Y$ whose associated integral subscheme $\widetilde{W}$ is finite and surjective over $X$. The group $Cor_k(X,Y)$ of finite correspondences is the free abelian group on the elementary finite correspondences. My question is:

Why does one puts the conditions of being finite and surjective on the morphism $\widetilde{W}\to X$?

Given elementary finite correspondences $V$ and $W$ from $X$ to $Y$ and $Y$ to $Z$ respectively, it is shown in Mazza-Voevodsky-Weibel Lemma 1.7 that $V\times Z$ and $X\times W$ intersect properly and also $\widetilde{V}\times_Y\widetilde{W}$ is finite and surjective over $X$, so that composition of correspondences can be defined.

Is this statement true in the reverse direction? Is this the only reason fro the choice of the definition of finite correspondences?

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    $\begingroup$ The intuition is that you should think of a finite correspondence as a "multi-valued function", so that to every point of $X$ you can associate several (but finitely-many) points of $Y$. The graph of such a thing is exactly a cycle finite and surjective over $X$ $\endgroup$
    – Denis Nardin
    Commented Apr 6, 2016 at 14:21
  • $\begingroup$ @DenisNardin They can be thought as partial functions too when $X$ is not connected right? $\endgroup$
    – Hypatia du Bois-Marie
    Commented Mar 15, 2023 at 18:12

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Traditionally correspondences were defined simply as cycles on the product, but then you need a moving lemma just to define composition. This limits you to working on smooth varieties. The beauty of finite correspondences is that composition can be defined much more directly, essentially as a composition of multivalued functions (cf Denis Nardin's comment). So by limiting the kind of correspondences you allow, you increase the range applicability.

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    $\begingroup$ I would rather say that you cannot define Voevodsky motives if you move cycles since you want to treat correspondences instead of their rational equivalence classes. The possibility to treat singular schemes is just a bonus here. $\endgroup$
    – Mikhail Bondarko
    Commented Apr 7, 2016 at 7:20
  • $\begingroup$ The question I really want to ask is what property should an irreducible closed subset $W$ of $X\times Y$ satisfy such that if $V$ in $Y\times Z$ also satisfies it, then $W\times Z$ and $X\times V$ intersect properly and composition can be defined without a moving lemma. Is $W\to X$ being finite surjective the only such property? $\endgroup$
    – Anandam Banerjee
    Commented Apr 7, 2016 at 9:58

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