Today I tried to figure out what a stratification is. Of course, I already knew the answer but I sort of got worried I had it wrong. Let me explain.

The “correct” (actually no, see Edit below) definition for general purpose applications in algebraic geometry is as follows. Let X be a topological space.

A partition of X is a decomposition of X into pairwise disjoint locally closed subsets. A stratification of X is a partition of X such that if X_j meets the closure of X_i, then X_j is contained in the closure of X_i.

For a stratification the index set gets a canonical partial ordering such that the closure of X_i is the union of the X_j for j ≤ i. [Side remark: at the moment of writing a stratification of a scheme isn’t formally defined in the stacks project, but is introduced in the text of a section on flattening stratifications, where it unfortunately conflicts with the notion as defined above. Argh! We’ll fix that soonish.]

Now, we are often interested in partitions and stratifications with additional properties. For example, it is nice if the index set is finite or at least if the decomposition is locally finite. If X is a spectral space (topological space underlying an affine scheme), then it is nice to require the parts and strata to be constructible locally closed subsets.

How is this used? For example in the following definition:

A sheaf of sets F on the etale site of an affine scheme X is called constructible if there is a finite constructible partition of X such that F is finite locally constant on each stratum.

The thing that got me worried is the following. Let X be a topological space with a closed subset Z and complementary open U. Then X = Z ∪ U is a partition of X. Now we ask ourselves: Can we refine this partition by a stratification? But wait, why is this even a question? Well, because Z’ = U̅ ∩ Z may be nonempty and strictly smaller than Z. Thus to get a stratification we need to split Z up into Z’ and Z \ Z’. However, then we need to do the same thing for the closure of the complement of Z’, etc, etc. The process potentially never stops.

Moreover, if we start with a finite constructible partition of an affine scheme X, then, in general, there does not exist a constructible stratification refining it. (Because it can happen that one has an irreducible quasi-compact open of X whose closure is not constructible.)

OK, and now it appears that we have two possible definitions for constructible sheaves. Namely, instead of the definition above we could require F is finite locally constant over the strata of a finite constructible stratification of X. But, happily, it turns out the second version isn’t the correct notion. Namely, we want pullback of constructible sheaves to be constructible. For a noetherian scheme the two notions give the same thing. But… the inverse image of a stratification by a continuous map is in general just a partition, not a stratification!

Thus for the Stacks project we should: (1) introduce formal definitions for partitions, refinments of partitions, and stratifications, and (2) search of all occurences of partitions and stratifications and make sure they match the given definitions. Moreover, we should have done this a long time ago.

Any comments on this are very welcome.

[Edit next day: I just realized that there is an intermediate notion, where one assumes that the index set comes with a partial ordering and the closure of X_i is contained in the union of X_j with j ≤ i. For example the stratification by Newton polygon has this property, but isn’t a stratification in the sense defined above. So probably that one is the “correct” one for algebraic geometry? So now I am tempted to take this weaker definition and then call the stronger one a good stratification.]

2 thoughts on “Stratifications

  1. I’m not a big fan of locally closed subsets, and usually axiomatize a stratification in terms of their closures, as follows: the intersection of any two stratum-closures should be a union of stratum-closures.

    This opens a can of delicious worms, because now we can think about the scheme-theoretic intersection. Should we ask the intersection be the _reduced_ union, or just ask that the underlying sets agree? The latter, if we want the (finitely many) orbits of SO(n) on GL(n)/B to be a stratification.

    Anyway, this is another goodness property one might ask of a stratification — that the intersection of stratum-closures be reduced. It holds for the set of compatibly split subvarieties of a Frobenius split variety.

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