Let f : X —> S be a morphism of schemes. A *flattening stratification for f* is a disjoint union decomposition of S into locally closed subschemes S_i such that for a morphism of schemes T —> S with T connected we have that T times_S X —> T is flat if and only if T —> S factors through S_i for some i.

There is also the notion of a flattening stratification for F where F is a quasi-coherent sheaf on X. In the case that X and S are affine this leads to the notion of a flattening stratification of Spec(A) for a module M over a ring B relative to a ring map A —> B.

Flattening stratifications do not always exist, but here are some examples where it does:

- If A = B and M is a finitely presented A-module, then the flattening stratification corresponds to the stratification of A given by the fitting ideals of M.
- If (A, m) is a complete local Noetherian ring, A—> B arbitrary, and M is m-adically complete, then the closed stratum of the flattening stratification for M in Spec(A) exists. (Intentionally vague statement; haven’t worked it out precisely.)

What you should keep in mind is that the flattening stratification does exist whenever the module is finite or formal locally in general.

Here is an example where the flattening stratification does not exist. Namely, take the ring map C[x, y] —> C[s, 1/(s + 1)] given by x |—> s – s^3 and y |—> 1 – s^2. Let f : X —> S be the associated morphism of affine schemes. Note that the image of f is contained in the curve D : x^2 – y^2 + y^3 = 0. Note that D has an ordinary double point at (0, 0). The problem is the stratum which contains the point (0, 0) of S. Namely, working infinitesimally around (0, 0) this is going to give you one of the two branches of the curve D at (0, 0), namely the one with slope 1. But globally, there is no locally closed sub scheme which gives you just that one branch!

The example above is not so bad yet, because there is a stratification of S by monomorpisms which does the job. Here is a simpler, somehow worse example. Namely, let S = Spec(C[x, y]) = A^2 be affine two space. Let X = A^2 ∪ G_m be the disjoint union of a copy of S and a line minus a point. The map f : X —> S is the identity on A^2 and the inclusion of G_m into the line y = 0 with the origin the “missing” point of G_m. Then looking infinitesimally around the origin in A^2 we are led to think that the stratum containing 0 should have complete local ring equal to C[[x, y]]. But looking at the overall picture we see that f(G_m) has to be removed, i.e., we have to take V(y) – V(x, y) out of Spec(C[[x, y]]). This shows that a flattening stratification cannot exist in this case (not even by monomorphisms).

Of course, somehow the main result on flattening stratifications is that it exists if f is a projective morphism and S is Noetherian. You can prove it by applying result 1 above to the direct images of high twists of the structure sheaf of X. The examples above show that it is unlikely that there exists a proof of this fact which uses the flattening stratifications for affine morphisms, as these do not always exist.

Do you know a direct proof of existence of flattening stratifications for a coherent sheaf on a proper (finitely presented) algebraic space over S? One can deduce it “a posteriori” from Artin’s proof of representability of the Hilbert functor. But that is a bit ironic, since Grothendieck’s original proof of representability of the Hilbert functor used existence of flattening stratifications as a key step.

What is a reference for this original proof? I looked at Sem Bourbaki 221 for example. Is that what you mean? Mumford in GIT also mentions Altman and Kleiman and his own Lectures on curves on an algebraic surface.

I believe the reference in the Seminaire Bourbaki notes is Lemme 3.4 on pp. 221-14 and 221-15 of “Technique de descente … IV. Les schémas de Hilbert.” But it looks like Grothendieck doesn’t quite call this a “flattening stratification”. To be honest, I am more familiar with the argument in “Lectures on curves on an algebraic surface”.

In the above, “Hilbert functor” should be “Quot functor” (although, of course, for a flat morphism one can reduce the Quot functor to the Hilbert functor by considering quotients of the sheaf of O_X-algebras O_X + F, where F is a square-zero nilradical).

Dear Jason,

I am working on this now. I think it will work and it won’t be particularly hard, but you never know until you write out all the details. Right now I am trying to get the algebra figured out. I’ll make another post when the details are done.

Johan

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I do not think it is known whether there is a flattening stratification for non-projective proper morphisms X->S. Due to Artin (*) we know that there exists a universal bijective monomorphism S’->S with the property that X\times_S S’ is flat but it is not clear when S’ is a stratification (i.e., a disjoint union of locally closed subschemes). When X->S is projective this follows a posteriori using Hilbert polynomials but we do not have such for non-projective morphisms which makes me wonder whether there is a counter-example.

Rmk: In Olsson-Starr’s paper, it is conjectured that S’->S is a stratification when X is a proper DM-stack but as I wrote, I do not think that this is known even for a proper morphism of schemes.

(*) A simpler proof is given by Murre, Bourbaki seminar 294 (1965) \S3, Thm. 2. The main theorem of Murre’s paper is a special case of Artin’s algebraization theorem but his proof is simpler as he only considers unramified functors, thus necessarily ending up with schemes and it is enough to work Zariski-locally which avoids Artin approximation. It would be interesting to give an analogous simple proof of Artin’s algebraization theorem for quasi-finite functors. I suppose that the algebraization theorem for of a “monomorphic functor” (such as in the flatifying stratification) could be even simpler to prove (and perhaps this is what Johan had in mind in the comment above).

Ten days ago, Andrew Kresch sent me the following example of a proper non-projective morphism X->S of schemes such that the universal flattening (flatifying?) functor S’->S (always represented by a monomorphism) is not a stratification (i.e., a disjoint union of locally closed subschemes). The counter-example is done via Hironaka’s famous construction of non-projective schemes (cf. Hartshorne, B.3.4.1) as follows:

Let g:V->S be an étale covering of degree 2 between projective smooth threefolds. Choose a nodal curve D in S such that the preimage g^{-1}(D) is the union of two smooth curves C_1 and C_2 meeting transversally at two points P and Q (the preimages of the node in D).

Let X_P->V-Q be the blow-up of V-Q along C_1-Q followed by the blow-up of the strict transform of C_2-Q. Similarly let X_Q->V-P be the blow-up of V-P along C_2-P followed by the blow-up of the strict transform of C_1-P. We then let X be the gluing of X_P and X_Q along the restrictions to V-{P,Q}. Let p:X->V denote the induced map so that p|_{C_1} is flat over C_1-Q but not over P and p|_{C_2} is flat over C_2-P but not over Q.

Then f:X->V->S does not have a flattening stratification. Too see this, it is enough to show that f|_D:X|_D->D does not have a flattening stratification. Let D’->D denote the universal flattening functor of f|_D. Let E->D be the normalization of D and let P’,Q’ in E be the preimages of the node. Then the base change X x_S E -> E is flat at either P’ or Q’ but not at both (note that V x_S E is the disjoint union of C_1 and C_2 and that the fibers over P’ and Q’ in V x_S E are {P,Q} and {Q,P} respectively). It follows that D’=E-Q’ (or E-P’) is the flattening functor so that D’->D is a bijective monomorphism but not a stratification.

Footnote 1: It certainly looks like the flattening stratification of f:X->S is S’=S-D cup D’ but I suppose that this does requires some work to prove rigorously.

Footnote 2: The example given above is a counter-example to the conjecture by Olsson and Starr that the flattening functor always is a stratification (see Rmk at top of p. 4083 after Thm 3.2).

Cool!