Encouraged by the success in studying finite flat modules, see the preceding post, let’s think a bit about flat, finite type ring extensions.

Question: For which rings R is every finite type flat ring map R —> S of finite presentation?

A Noetherian ring satisfies this property. In the paper by Raynaud and Gruson they prove that this holds if R is a domain. I recently added this result to the stacks project (with a purely algebraic proof), see Algebra, Proposition Tag 053G. If R is a local ring whose maximal ideal is nilpotent then the result is true as well. But I don’t know what happens if the maximal ideal is only assumed to be locally nilpotent, i.e., every element of the maximal ideal is nilpotent, i.e., the maximal ideal is √(0). Do you?

By the way, I still want more ideas about the question I posted here! [Edit: this question has now been answered.]

[Edit on August 23, 2010: As David Rydh points out in a comment below any ring which has finitely many associated primes satisfies the condition. This follows trivially from Raynaud-Gruson Theorem 3.4.6. Don’t know why I did not see this! Anyway, so a local ring whose maximal ideal is locally nilpotent is an example too.]

In my thesis, in the chapter on finite flat groupschemes, I made the mistake of thinking that a finite flat group scheme is the same thing as a finite locally free group scheme. In other words, I made the classic mistake of thinking that a finite flat module over a ring is finite locally free (or equivalently finitely presented). A counter example is given in the stacks project, see examples.pdf. Luckily I discovered this error (or maybe somebody else did and pointed it out to me) and the published version of my thesis does not have this mistake.

Why I made this mistake I am not sure, maybe because I read Matsumura’s Commutative Algebra, where you can find the result that a finite flat module over a local ring is finite free.

I have since learned that this is not as bad a mistake as one may think. Namely, it turns out that whether or not every finite flat R-module is finite locally free, is a property of R which depends only on the topology of X = Spec(R). The result is that every finite flat R-module is finite locally free if and only if every Z ⊂ X which is closed and closed under generalizations is also open. A similar result holds for schemes. (I found this in some paper a while back, but now I cannot remember which paper.)

I just added this to the stacks project this morning, see Algebra, Lemma Tag 052U and Morphisms, Lemma Tag 053N.

Here is a question I have been struggling with for the last couple of weeks.

Question: Let A be a Noetherian henselian local ring. Let A —> B be (a) local ring map of local rings, (b) essentially of finite type, (c) the residue field extension is trivial, and (d) injective. Is the map A^ —> B^ of completions injective?

If the answer is “yes”, then this somehow tells us that (very very roughly) “taking scheme theoretic image commutes with completion”.

The answer is yes if A is in addition excellent. But I would like to know if it is also true in general. It is very possible that there exists a simple counter example, it is also possible that it is true for trivial reasons. The most vexing aspect of this question to me is that I cannot even decide whether it should be true or not. Please leave a comment if you have any references, comments, or suggestions. Thanks!

[Edit on August 22, 2010: I finally figured out that this is wrong. Namely, take A to be the example of Ogoma. It is a normal henselian Noetherian local domain whose completion is k[[x, y, z, w]]/(yz, yw). So the completion is the union of a nonsingular 3 dimensional component and a nonsingular 2 dimensional component. Let C be an affine chart of the blow up of A at its maximal ideal. The special fiber of C has two irreducible components (a plane and a line). Let B be the localization of C at a maximal ideal which is a point on one of them but not the other. Then clearly the completion of B “picks out” one of the irreducible components of the completion of A.]

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…