Here are some questions of the form: I don’t know this, do you? These questions showed up in the stacks project, but we did not take a lot of time to find them in the literature or try to solve them. Maybe you know how to do some of these or a reference? Here they are:

1. Does there exist a scheme which is connected, all of whose local rings are domains, but which is not irreducible?
2. Let X be a scheme over a field k. Let k ⊂ K be an extension of fields. Let T be a connected component of X_K. Is the image of T in X_k a connected component of X?
3. Is it true that a Noetherian ring all of whose local rings are Japanese is Japanese?
4. If f : Z —> X is a closed immersion of schemes, then is f_* exact on the category of abelian sheaves on (Sch/Z)_{fppf}? I think probably not in general, but I don’t have an example. Do you?
5. If X is an algebraic space (as defined in the stacks project, so not necessarily quasi-separated) does X satisfy the sheaf condition for fpqc coverings?

PS: Question 5 has a positive answer for quasi-separated spaces, see Lemma Tag 03WB. David Rydh has some more results on this. Maybe the real question here is if one can make some horrible counter example, or whether there is some straightforward argument covering all algebraic spaces that everybody has missed so far.

I think there is some confusion in the literature about what a band is, although likely this is just me. I just googled a bit and found, I think, at least two inequivalent definitions. I have also had at least one very confusing conversation with somebody (can’t remember whom), which I now think is due to us having different definitions. For the stacks project, I would prefer the band of a gerbe to be the finest possible invariant of the gerbe. I think this basically tells us what to do. Please don’t read the rest of this post if you already know how to do this.

First, let us make the following basic observation (and you are going to laugh at me for even pointing this out). For an element g of a group G let inn_g : G —> G be the map x |—> gxg^{-1}. Suppose that G, H are groups and that a, b : G —> H are homomorphisms of groups. Then the following are equivalent

• there exists an element h of H such that a = inn_h o b,
• there exist elements g of G and h of H such that a = inn_h o b o inn_g.

The reason is that b o inn_g = inn_{b(g)} o b and that inn_{hh’} = inn_h o inn_{h’}. If the equivalent conditions hold we say that a, b define the same outer homomorphism from G to H. You can compose outer homomorphisms because if you have a : G —> H and b : H —> F and g, h, h’, f in G, H, H, F, then we have (inn_f o b o inn_{h’}) o (inn_h o a o inn_g) = inn_{fb(h’)b(h)b(a(g))} o b o a. OK, so this gives us the category of exterior groups, sometimes called the category of outer groups. An automorphism in the category of exterior groups is often called an outer automorphism. It is clear how to generalize this to sheaves of groups over a site (you have to localize to get the correct notion of an outer homomorphism of sheaves of groups).

Let C be a site. Consider the fibred category PreBands over C whose category of sections over an object U is the category of exterior sheaves of groups over U, so objects are sheaves of groups on U and morphisms are outer homomorphisms. Stackify PreBands to get the stack of bands Bands over C. A band is then an object of the fibre category of Bands over a final object of C (and slightly more complicated if C does not have a final object).

What does such a band B look like? Let X be a final object of C. Then B is given by a system ({X_i —> X}, G_i, φ_{ij}) where {X_i —> X} is a covering, each G_i sheaf of groups on X_i, each φ_{ij} is an outer isomorphism of G_i|_{X_i \times_X X_j} —> G_j|_{X_i \times_X X_j} satisfying a cocycle condition. To get morphisms of bands ({X_i —> X}, G_i, φ_{ii’}) —> ({Y_j —> X}, H_j, ψ_{jj’}) consider the following two kinds of morphisms of systems

1. one given by a refinement of coverings {Y_j —> X} —> {X_i —> X} (note reversal arrow) where the H_j are the pullbacks of the G_i, and
2. one of the form ({X_i —> X}, G_i, φ_{ii’}) —> ({X_i —> X}, H_i, ψ_{ii’}) given by outer homomorphisms of sheaves of groups G_i —> H_i compatible with φ_{ii’} and ψ_{ii’}.

Then (roughly) you have to invert the ones of the first kind and the ones of the second kind where all the G_i —> H_i are outer isomorphisms to get morphisms of bands.

Finally, given a gerbe \cX over C we get a band B(\cX) by choosing a covering {X_i —> X} and objects x_i over the members of the covering in \cX. The associated band is ({X_i —> X}, Aut(x_i), φ_{ii’}), where the outer isomorphisms φ_{ij} come from the existence of local isomorphisms between the pullbacks of x_i and x_j over X_i \times_X X_j. This band is well defined up to unique isomorphism of bands.

Then, given a fixed band B, we say that a gerb \cX is banded by B if there exists a isomorphism θ: B —> B(\cX) to the band associated to the gerbe \cX. But be careful: If we try to classify all gerbes banded by B we could mean either of the following two things: Classify pairs (\cX, θ) or classify \cX’s such that a θ exists!

[Edit: I stole this description of bands from the paper by Max Lieblich and Brian Osserman, see arXiv:0807.4562. Unfortunately, there is a typo in their description of outer morphisms, they divide out by Aut(G) and Aut(H) but the should have used Inn(G) and Inn(H).]

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.]