# Intuition behind the definition of finite correspondences

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?

• 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 $X$ – Denis Nardin Apr 6 '16 at 14:21

• 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? – Anandam Banerjee Apr 7 '16 at 9:58