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.