# 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?

Thanks in advance

• 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

## 1 Answer

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.)

For your second question, I wish I knew an answer! That is, I wish I knew where exactly in the theory these non-flat correspondences arise. Hopefully someone can tell us. Meantime all I can offer up is a sort of moral argument that I've learned to accept: in Voevodsky's theory we try to get at motives from the ground up in several steps. In the first step (the beginning) we have varieties, which are of course rife with geometry. In the second step we try to forget this geometry and just remember the topology. This is accomplished through the h-topology, which captures topological descent (for instance, it's probably true that a map of complex varieties is an h-cover if and only if it is a topological epimorphism on complex points). Then in the third step we go from topology to homotopy theory, which involves making A^1 contractible as well as the introduction of a new topology (Nisnevich or cdh) suitable for homotopical, rather than topological, descent. And then of course one may go further to capture stable or linear phenomena, and the final step down to abelian phenomena (the Grothendieck story) is not yet achieved...

But anyway the key step for us is the second one. These finite correspondences from X to Y are intending to capture maps from X to the free abelian group on Y, the idea being that, in classical topology, the topological free abelian group on a topological space Y is a topological space that, up to homotopy, knows exactly the integral homology of Y. Thus this finite correspondence stuff is really happening in the "topological" section of this story, after algebraic geometry but before homotopy theory proper. Thus the important point is to require our construction to satisfy h-descent; and there's the crux, because by "platification par eclatement", even if we only started with flat guys, after h-sheafification we would have arbitary ones.