# Are finite correspondances flat?

In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible closed subset $W$ of $X\times Y$ whose associated integral subscheme is finite and surjective over $X$."

As far as I know $W$ would be flat over $X$ if it was Cohen-Macaulay so...

1.- ¿Is $W$ flat over $X$?

If not,

2.- why isn't this a common sense assumption? Could anyone give an example of why nonflat elementary correspondaces should be allowed?

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Well. it seems easier to do the theory without the flatness restriction.:) The question is whether the (new) Suslin complexes you obtain by imposing the flatness assumption will be quasi-isomorphic to the 'original' ones. If they will not be so, you will obtain another category of motives, that will probably be worse than Voevodsky's. – Mikhail Bondarko Jan 13 '11 at 22:46
Dear Quetzalcube, I am not very knowledgeable about Voevodsky's work, but one thing to say is that flatness is a somewhat delicate condition, whereas finiteness is much more robust. Also, the notion of correspondence is supposed to rigorously capture the idea of a (finitely-but-)multi-valued map, and this is what the definition of elementary correspondence gives: one thinks of $W$ as the graph of such a map. – Emerton Jan 14 '11 at 12:59
If you want a reference for Emerton's claim that elementary correspondence captures rigorously the idea of a multi-valued map, have a look at Theorem 6.8 in Singular homology of abstract algebraic varieties. – name Mar 5 '13 at 6:29

I believe the answer to your first question is no. Here's an example sketch: let $X$ be $A^2$, and let $W$ be two copies of $A^2$ glued at the origin (realized as the union of two transverse linear subspaces of $A^4$, say), mapping to $X$ by the "fold" map (projection to a third linear subspace, say). Actually that's not an example, because $W$ isn't irreducible. But it should become irreducible, without affecting formal-local behavior at the origin (and therefore without affecting the non-flatness), if we just perturb the equations defining $W$ in $A^4$ a bit by adding high order terms (like how one goes from the union of two lines in A^2 given by xy = 0 to the nodal cubic x^3 + xy + y^3 =0.)