Let F : (Sch)^{opp} —> (Sets) be a functor. Assume that

1. F is an fppf sheaf,
2. F is relatively representable, i.e., F —> F times F is representable,
3. F is limit preserving (i.e., locally of finite presentation),
4. F has effective versal deformations, and
5. F satisfies openness of versality.

Then, if S is an excellent base scheme, the functor F is an algebraic space. Originally Artin proved this over an excellent Dedekind domain. The excellent base scheme case is discussed for example in Approximation of versal deformations authored by Brian Conrad and myself.

What I want to know is this: Is it really necessary to assume that S is excellent? Can you start with a non-excellent Noetherian scheme and make a counter example? I now think sometimes you can.

Here is a related question. Suppose that X is a scheme and U —> X is a surjective morphism of schemes. Set R = U \times_X U so that we get a groupoid scheme (U, R, s, t, c). Let U/R denote the fppf quotient sheaf. Is it true that the canonical map U/R —> X is an isomorphism? The answer to this is: No! If you let U = Spec(Z[t]) be the normalization of X = Spec(Z[t^2, t^3]), then U/R does not have a good deformation theory. Namely, Spec(A)-valued points of U/R are given by equivalence classes of pairs (A —> B, b) where A —> B is faithfully flat of finite presentation and b in B is an element such that b^2 and b^3 are elements of A. The pairs (A —> B, b) and (A —> B’, b’) are equivalent if there exists a third pair (A —> B”, b”) and A-algebra maps B —> B”, B’ —> B” mapping b^2, b^3 to (b”)^2, (b”)^3 and (b’)^2, (b’)^3 to (b”)^2, (b”)^2. The transformation U/R —> X maps the pair (A —> B, b) to the ring map Z[t^2, t^3] —> A which maps t^2 to b^2 and t^3 to b^3. Let k be a field and let O = (k —> k, 0). We claim the tangent space of this point is zero. Namely, a first order deformation is given by a pair (k[e]/(e^2) —> B, b) where b is in eB. Hence b^2 = b^3 = 0 and so this pair is equivalent to the pair (k[e]/(e^2) —> B, 0) and so also (k[e]/(e^2) —> k[e]/(e^2), 0).

But what if we assume the following: (*) X is locally Noetherian and for every closed point x of X there exists a point u in U mapping to x such that the map on complete local rings O^_{X, x} —> O^_{U, u} has a section? I think in this case F = U/R has a good deformation theory at all closed points, which recovers the complete local ring O^_{X, x}. Moreover, by construction the quotient U/R is an fppf sheaf, and is limit preserving. It is also relatively representable (this is a general fact). I think openness of versality should be OK too (did not check this). So if Artin’s criterion applies then U/R is an algebraic space and the map U/R —> X is a morphism of algebraic spaces of finite type over the base which is bijective on field valued points and induces isomorphisms on complete local rings, which would force it to be an isomorphism.

If so, then this implies in particular that there exists a faithfully flat morphism of finite type X’ —> X which factors through the morphism U —> X!

There exists a Noetherian 1-dimensional local domain A whose residue field has characteristic 0 and whose completion is not reduced. There is a paper by Gabber where he proves that the completion A^ cannot be written as a directed limit of flat A-algebras of finite type. Let S = X = Spec(A). Let U = Spec(C), where C ⊂ A^ is a finite type A-sub algebra such that the map C —> A^ does not factor through any flat finite type A-algebra. Let R = U \times_X U as above. If the discussion above is correct, then the functor F = U/R satisfies all of Artin’s axioms but isn’t an algebraic space. [Edit July 3, 2011: This isn’t quite right, see this post!]

On the other hand, I have a feeling that Artin’s criterion may hold over Noetherian base schemes S such that for any local ring A which is essentially of finite type over S the completion A^ is a directed limit of flat finite type A-algebras. Is there an example to show that this is not the same as asking S to be excellent?

Yesterday I found this in a preprint by Brian Osserman and Sam Payne:

• If X —> S is locally of finite type, and x -> z is a specialization of points in X with z a closed point of its fibre, then there exist specializations x -> y, y -> z such that y is in the same fibre as x and is a closed point of it. Moreover, the set of all such y is dense in the closure of {x} in its fibre.

I was already planning to try to prove this and add it to the stacks project as I think that it could be quite useful.

To prove this statement you first reduce to the case where the base is a valuation ring and the morphism is flat. My idea was to use an argument a la Raynaud-Gruson to reduce to the case of a smooth morphism, where you can slice the map, i.e., argue by induction on the dimension. Brian and Sam’s argument is simpler: they show that you can do the slicing without reducing to a smooth morphism by showing that a locally principal closed subscheme which misses the generic fibre has to be “vertical”. This intermediate result is interesting by itself.

Does anybody have a reference for this, or similar, results? (I looked in EGA…)

Since the last update about 3 weeks ago I have added a small amount of material on morphisms of algebraic stacks and some more material on generic flatness and the flattening stratification. I will discuss some thoughts related to this in another post.

Today, I got an email from Christian Kappen mentioning a mistake in Lemma Tag 002X. You can figure out what the mistake was by looking at the development log. It was easy to fix the lemma by adding an extra condition — but it was used in a number of places. It actually took quite a bit of work to repair. Here is the list of lemmas which were affected: 002X, 004X, 002Y, 04B0, 00XS, 04BH, 04BB, 020W, 020Y, 0210, 021E, 021F, 021H, 04HC, 04HD, 021V, 021W, 04CC. I’ve fixed all those places, and hopefully that is it.

One thing I observed while I was doing this is that actually always referencing the exact result that is being used (which is one of the goals of the stacks project) is extremely useful. Namely, you can grep the tex files for the latex label and quickly find all the spots where a given result is being used. Moreover, if every lemma/proposition/theorem states all its assumptions (another goal of the stacks project) then it becomes relatively straightforward to check whether other lemmas/propositions/theorems that are being used in the proof are applicable. Thus such a repair can often be completed in a short time.

Let f : X —> S be a separated morphism of finite presentation. Consider the functor F : (Sch)^{opp} —> (Sets) which to a scheme T associates all pairs (a, Z) where a : T —> S and Z is a closed sub scheme of the base change X_T such that the projection Z —> T is an open immersion. In other words, this is the functor of flat families of closed sub schemes of degree <= 1 on X/S, as we discussed briefly in this post. As we saw there it is not true in general that F is an algebraic space. If X = A^1_S then F is (probably) a directed colimit of schemes. But if X has higher dimension I’m not sure how to “compute” F.

Here are some general properties of this construction. There is a canonical morphism

j : X —> F

which is an open immersion by construction. Moreover, there is a canonical morphism

∞ : S —> F

which associates to a : T —> S the pair (a, ∅). And of course on points we have F = j(X) ∪ ∞(S). The structure morphism p : F —> S is locally of finite presentation and satisfies the valuative criterion (both existence and uniqueness). These properties tell us p is “proper”. Thus F is morally speaking the one point compactification of X/S.

When I was discussing this with Bhargav Bhatt he suggested we think about the etale cohomology of F. Now that I have had some time to think about his suggestion, I think this is a splendid idea. Namely, it seems to me that we could try to use Rp_*Rj_! to define Rf_! for the morphism f : X —> S…

This is a continuation of previous post on flattening stratifications. The experts reading this blog could probably tell that I hadn’t really understood what is going on at all. I still haven’t mastered the subject but I think I know a little bit more now.

Let f : X —> S be a morphism of schemes. Consider the functor F : (Sch)^{opp} —> (Sets) which to a scheme T associates the set of morphisms T —> S such that the base change X_T is flat over T. Clearly the map F —> S is a monomorphism. We propose to introduce the following

Definition: We say the flattening stratification of f exists if F is an algebraic space.

What I added to the stacks project last Friday is the following: Assume S is the spectrum of a Noetherian complete local ring and f is of finite type. Then there exists a biggest closed subscheme Z of S such that X_Z —> Z is flat at all the points of the closed fibre. Moreover, Z satisfies a universal property which is formulated in terms of local morphisms of local schemes and flatness at points of the special fibre. If in addition X —> S is closed, then it follows that X_Z —> Z is flat as the set of points where X_Z —> Z is flat is an open set.

Assume S Noetherian and f of finite type and proper. In terms of Artin’s axioms for F the result in the previous paragraph takes care of the existence of a formal versal deformation. I think there is a straightforward little argument which takes care of openness of versality (but I did not write this out completely). Since f is of finite presentation, it follows that F is of finite presentation by the usual arguments on limits and flatness. Relative representability is OK too. Hence, if S is excellent then F is an algebraic space by Artin’s theorem. But of course we can descend X —> S to a situation of finite type over Z and hence we get the result in general (with same hypotheses). In fact, using limit arguments we may be able to prove the same thing when S is arbitrary and f proper and of finite presentation.

Still, my answer to Jason’s question here was a bit premature. Some of the above may work exactly as stated in the generality of Jason’s question. But I was trying to prove flattening stratifications exist without using Artin’s theorem. In particular, it should be possible to avoid using general N\’eron desingularization.

The reason I started looking at flattening stratifications was to construct Quot and Hilbert schemes/spaces/stacks. And the reason to discuss those was that Artin’s trick uses Hilbert spaces. However, it only uses the Hilbert space parametrizing closed subschemes of length n on a space. Of course I could take the easy way out and just use one of the explicit constructions of Hilb^n. But once I started looking at the problem of constructing flattening stratifications (which is related to descent of flat modules) I just couldn’t stop myself.

Consider the functor F which to a scheme T associates the set of closed subschemes Z of T \times A^1 such that the projection Z —> T is an open immersion. In other words F is the functor of flat families of closed sub schemes of degree <= 1 on A^1, whence the title of this post. We note that F is a sheaf for the fppf topology.

What is fun about this functor  is that it is a natural candidate for a 1-point compactification of A^1, as the following discussion shows.

Namely, consider for each integer n >= 1 the scheme P_n which is P^1 but crimped at infinity to order n. What I mean is this: If y = x^{-1} denotes the usual coordinate on the standard affine of P^1 which contains infinity, then the local ring of P_n at infinity is the Z-algebra generated by y^n, y^{n + 1}, y^{n + 2}, … Note that there are morphisms

P_1 —> P_2 —> P_3 —> …

and that for each n there is a natural map P_n —> F compatible with the transition maps of the system. Hence we obtain a transformation of sheaves

colim P_n —> F.

It seems likely that this map is an isomorphism (we take the colimit in the category of fppf sheaves), but I did not write out all the details.

Does anybody have a reference? What about the same thing for A^2?

[Edit 18:57 July 31 2010: Original definition of F omitted the condition that the fibres are a point or empty. Replaced by the open immersion condition. This make sense because a morphism Z —> T which is flat and locally of finite presentation whose fibres Z_t are either empty or Z_t = t (scheme theoretically) is an open immersion.]

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:

1. 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.
2. 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.

Certain results have a variant for generic points, and a variant which works over a dense open. As an example let’s discuss “generically finite morphisms” of schemes.

The first variant is Lemma Tag 02NW: If f : X —> Y is of finite type and quasi-separated, η is a generic point of an irreducible component of Y with f^{-1}(η) finite, then there exists an affine open V of Y containing η such that f^{-1}(V) —> V is finite.

The second variant is Lemma Tag 03I1: If f : X —> Y is a quasi-finite morphism, then there exists a dense open V of Y such that f^{-1}(V) —> V is finite.

Comments: (a) In the second variant it isn’t necessarily the case that every generic point of every irreducible component of Y is contained in the open V, although this would follow from the first variant if we assumed f quasi-separated. (b) The proof of the first variant in the stacks project is basically elementary; the proof of the second variant currently uses (a technical version of) Zariski’s main theorem.

The point I am trying to make (badly) is that you can often get around making any separation assumptions by trying to prove a variant “over a dense open”. Maybe the archetype is the following result (Lemma Tag 03J1): Every quasi-compact scheme has a dense open subscheme which is separated.

I just added some generic flatness results to the stacks project (only for morphisms of schemes so far). There are two interesting features of the presentation in the stacks project:

1. Assuming only the morphism is of finite type the conclusion is that the morphism is flat and finite presentation over a dense open of the base, and
2. it suffices to assume the base is reduced.

Using these results we can discuss “flattening stratifications”. But I want to discuss this in a maximally general setting. Reader beware!

Let f : X —> S be a morphism of schemes of finite type. I want to find a stratification of S by reduced locally closed subschemes S_i such that X_{S_i} —> S_i is flat. If f is of finite presentation we can reduce to S Noetherian and there is (locally) a finite stratification that does the job; so what I am interested in here is the case where S is not Noetherian.

Step 0: Find the open stratum. Just replace S by its reduction S_{red} and let S_0 be the open dense U ⊂ S you get from generic flatness. Step 1: Let S_1 be the dense open of (S – S_0)_{red} you get from generic flatness. Step 2: Let S_2 be the dense open of (S – S_0 – S_1)_{red} you get from generic flatness. Etc.

Now we get S_0, S_1, … but it may not be the case that S = \bigcup S_i. For example the last post contains an example. So then you start all over again. Namely, note that the complement of S_0 ∪ S_1 ∪ S_2 ∪… is closed in S hence a scheme. So we restrict our family to this closed subset and we continue. Doesn’t it feel like we can just continue forever using transfinite induction? And moreover, the process does really have to stop as S has an underlying topological space which has a finite cofinality. Thus we do get our desired stratification of S.

But this is madness! Surely there are at most countably many strata…!?!

Here is a fun example. Take U = Spec(k[x_0, x_1, x_2, …]) and let G_m act by t(x_0, x_1, x_2, …) = (tx_0, t^px_1, t^{p^2}x_2, …) where p is a prime number. Let X = [U/G_m]. This is an algebraic stack. There is a stratification of X by strata

• X_0 is where x_0 is not zero,
• X_1 is where x_0 is zero but x_1 is not zero,
• X_2 is where x_0, x_1 are zero, but x_2 is not zero,
• and so on…
• X_{infty} is where all the x_i are zero

Each stratum is a gerb over a scheme with group \mu_{p^i} for X_i and G_m for X_{infty}. The strata are reduced locally closed substacks. There is no coarser stratification with the same properties.

So clearly, in order to prove a very general result as in the title of this post then we need to allow infinite stratifications…