Brian Conrad just emailed me to ask about gerbes in the stacks project. Unfortunately this is not yet in the stacks project (feel free to send your own write-up to stacks.project@gmail.com if you have one).

This is how I would define them:

(1) A gerbe over a site C is a stack in groupoids p : S —> C with the following properties:

• for every object U of C there exists a covering {U_i —> U} such that each fibre category S_{U_i} is nonempty, and
• for every object U of C and objects x, y in S_U there exists a covering {U_i —> U} such that for every i in I we have that x|_{U_i} is isomorphic to y|_{U_i} in the fibre category S_{U_i}.

Once this has been defined there should be a brief discussion of the “band” of a gerbe. In the case where the band is commutative it should be explained carefully that you get a sheaf of groups over the site. Actually, another important case is the case where you are given a sheaf of groups G on C and you consider gerbes whose band “is” G (this should be precisely defined). Also, it should be defined what is a “trivial” gerbe. I suggest we try to avoid cocycles as much as possible. For an informal discission for gerbes over topological spaces, see Lawrence Breen’s notes or Ieke Moerdijk’s notes. Another, less informal, reference would be the book by Giraud entitled “Cohomologie non abelienne”.

(2) Let \cX —> X be a morphism from an algebraic stack \cX to an algebraic space X. Then we say that \cX is a gerbe over X if and only \cX viewed as a stack in groupoids on (Sch/X)_{fppf} is a gerbe as defined above. Moreover, all the notions defined in the abstract setting can be used in this setting also.

This may not always correspond to the geometric picture of a gerbe, especially if the band (i.e., the automorphism group of an object) isn’t flat! But is it really always the case that gerbes in algebraic geometry have flat automorphism groups?

As usual comments are welcome.

[Edit: Brian adds that we could for instance prove (via erasable delta-functors) that gerbes describe H^2 with commutative coefficients (including its group structure *and* functoriality in both group and base space). And similarly give a torsor/gerbe description of a 7-term exact sequence of pointed sets associated to a central extension of group sheaves. Most of this can be done without using cocycles. On pp. 144-145 of Milne’s book on etale cohomology he gives a nice little summary of the highlights on this aspect of Giraud’s book.

Another great suggestion is: Explain as an example why Artin’s work shows that for an fppf group scheme (or algebraic space group) A, the A-gerbes are Artin stacks.]

As I mentioned earlier I have started to look into Hilbert and Quot schemes/spaces/stacks. This led me to think about the existence of a flattening stratification and then I started thinking about some of the results obtained in the paper [RG] by Raynaud and Gruson. For example, as I first mentioned here, I think that using Theorem 4.1.2 of their paper would be a good way to prove the existence of flattening stratifications (as formulated in this post) in the “correct” level of generality.

Thus it now seems to me that the basic “devissage” result of [RG] would be a worthwhile addition to the stacks project. Consequently, I have started adding material on geometric fibres in families which is needed for one of the key geometric lemmas in [RG].

I think there is a lot more, besides flattening stratifications, to gain from adding this to the stacks project. For example, there are several places in [RG] with directions for the reader to rewrite several parts of EGA using their methods, and I feel that others will show up as we go along.

I’d love to hear about any interesting foundational applications of [RG] you know about!

[RG] Raynaud, Michel; Gruson, Laurent
Critères de platitude et de projectivité. Techniques de “platification” d’un module. (French)
Invent. Math. 13 (1971), 1–89.

Lemma Tag 055J: Let R be a dvr. Let X be flat over Spec(R), with reduced special fibre, and connected total space. Then the generic fibre of the structure morphism f : X —> Spec(R) is connected.

You can find a version of this lemma as EGA IV, Lemma 15.5.6 where the hypotheses are that f is locally of finite type and open instead of flat. But in the proof of the lemma in EGA it is remarked that the hypotheses “loc. fin. type + open over dvr” imply “flat”, hence the lemma above implies the lemma in EGA. I urge you to try to prove the lemma above before looking it up, because it is fun when you find it!

Why is flatness necessary? If X is not flat over R, then a counter example is X = Spec(R[x]/(px(x-1))) where p ∈ R is a uniformizer. In words: X is a union of two copies of Spec(R) glued at 0, 1 of the affine line over the residue field of R.

Why is a reduced special fibre necessary? If not then a counter example is X = Spec(R[x]/(x(x – p))) where p is a uniformizer in R. In words: X is a union of two copies of Spec(R) glued at their special points.

Wie het kleine niet eert is het grote niet weerd!

This blog post is only for very nerdy people. I’ve just added an unstable git repository. The idea is that collaborators can push local experimental git branches of the stacks project to this unstable repository on the server. This may be useful if you are using multiple machines to work, for example a desktop at home and at work. This will also speed up communication with me. Namely, if your branch has some usable material in it, you can email me a pull-request to ask me to merge it into the stacks project.

If you’d like to try this, then email me your id_rsa.pub and I’ll email you instructions. This setup is not ideal since it is fairly easy to screw up using git, e.g. delete the main branch and/or delete branches of others. So I will not randomly let everybody have access. In fact, if you want to setup a remote location to store your changes or any project you are managing with git, then you might instead want to look at github. In particular, you could maintain your own clone of the stacks project there…

Random thoughts on material I added to the stacks project lately:

(I) Suppose you have a ring map φ: A —> B with the following properties: (1) Ker(φ) is locally nilpotent, and (2) for every x in B there exists a n > 0 such that x^n is in Im(φ). Then it is true that Y = Spec(B) —> Spec(A) = X is a homeomorphism, but it is not true in general that Y —> X is a universal homeomorphism. A counter example is where A is a non-algebraically closed field which is an algebraic extension of F_p and B is the algebraic closure of A.

(II) Let f : X —> Y be a morphism of finite type where Y is integral with generic point η. Suppose Z is a closed subscheme of X such that Z_\eta = X_\eta set theoretically. Then there exists a nonempty open V ⊂ Y such that Z_V = X_V set theoretically. (In the Noetherian case this is pretty straightforward.)

(III) A torsion free module over a valuation ring is flat. (If you don’t know how to prove this then it is a nice exercise for when you’re in the shower.)

(IV) Let f : X —> Y is a morphism of finite type where Y is integral with generic point η. If X_η is geometrically irreducible, then there exist a nonempty open V ⊂ Y such that all fibres X_y, y ∈ V are geometrically irreducible. Same with geometrically connected.

(V) Let f : X —> Y be a quasi-compact morphism of schemes. Suppose η ∈ Y is a generic point of an irreducible component of Y which is not in the image of f. Then there exists an open neighborhood V ⊂ Y of η such that f^{-1}(V) is empty.

Let me know if any of these assertions are wrong… thanks!

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…)