Let i : Y —> X be a closed immersion of schemes. This gives rise to a morphism of topoi i_{big} : (Sch/Y)_{fppf} —> (Sch/X)_{fppf}. Question: Is the direct image functor i_{big, *} is exact on the category of abelian sheaves?

My guess is no. To find an example we can look for an Artinian local ring A with an ideal I and a finite flat local ring map A/I —> C such that there does not exist any finite flat ring map A —> B with the property that A/I —> B/IB factors through C. Namely, in this case the map of abelian sheaves

(Z/2Z)_{Spec(C)} —> Z/2Z

on Y = Spec(A/I) is fppf surjective because {Spec(C) —> Spec(A/I)} is an fppf covering. Here the first sheaf is the free Z/2Z-module on the fppf sheaf represented by Spec(C) over Y. But

i_*((Z/2Z)_{Spec(C)}) —-> i_*(Z/2Z)

is not surjective since the section 1 does not lift fppf locally on X = Spec(A) by our assumption on A/I —> C. To make an explicit example you probably can do something similar to Exercise Tag 02CV but I haven’t quite been able to make it work yet. Leave a comment if you have an example, or a reference, or if you think the answer to the question is yes.

In the stacks project a site is defined as in Artin’s notes on Grothendieck topologies, and not as in SGA4. Hence also our notion of a cocontinuous functor u : C —> D between sites C and D is a bit different (than Verdier’s original one). Namely, it means that, given any object U of C, and any covering {V_j —> u(U)}_j in D there should exist a covering {U_i —> U} in C such that the family of morphisms {u(U_i) —> u(U)}_i refines the given family {V_j —> u(U)}_j.

The reason this definition is convenient is twofold. On the one hand, it is easy to check that a functor is cocontinuous, and on the other hand, it is true that a cocontinuous functor u : C —> D gives rise to a morphism of topoi g : Sh(C) –> Sh(D). For example, for a sheaf G on D the sheaf g^{-1}(G) is the sheaf associated to the presheaf U |—> G(u(U)).

Here are two examples

• Let f : X —> Y be an open continuous map of topological spaces. Then the functor u(U) = f(U) is a cocontinuous functor between the site of opens of X and the site of opens of Y. The induced morphism of topoi Sh(X) —> Sh(Y) is the usual one.
• Let f : X —> Y be a morphism of schemes. The “forgetful” functor u : (Sch/X)_{fppf} —> (Sch/Y)_{fppf} is cocontinuous and the associated morphism of topoi is the usual morphism of big topoi f_{big} : Sh((Sch/X)_{fppf}) —> Sh((Sch/Y)_{fppf}).

A little less standard are the following examples, which are related to the discussion in the previous post. Suppose that i : X_0 —> X is a closed immersion of schemes defined by a sheaf of ideals of square zero. Consider the functor of sites u : X_{lisse-etale} —> (X_0)_{lisse-etale}, or u : (Sch/X)_{syntomic} —> (Sch/X_0)_{syntomic} given by the rule V |—> V_0 = X_0 \times_X V. Then you can check that u is cocontinuous (in both cases). Hence we obtain a morphisms of topoi

• g_{lisse-etale} : Sh(X_{lisse-etale}) —> Sh((X_0)_{lisse-etale})
• g_{syntomic} : Sh((Sch/X)_{syntomic}) —> Sh((Sch/X_0)_{syntomic})

These maps are somehow contracting the topos associated to X onto the topos associated to X_0. Now in the second case the functor u also gives rise to a morphism of topoi in the opposite direction, namely i_{big} (for the syntomic topology), but I think neither i_{big} nor g_{syntomic} is an equivalence of topoi. In the first case, even though u is continuous, it does not define a morphism of topoi in the other direction.

In any case, cocontinuous functors are very useful and often easier to deal with than the better known continuous ones. For more information see the chapter on Sites and Sheaves.

Let k be a field and let D = k[epsilon] be the ring of dual numbers. Suppose V is some geometric object over k. A geometric object U over D is called a deformation of V if it  (1) is flat over D and (2) has special fibre U_k = U \otimes_D k isomorphic to V. (This is intentionally vague.)

When V is a scheme, then U can be conveniently thought of as a locally ringed space whose underlying topological space is identical with the underlying space of V. In other words, you just change the sheaf of rings, and not the actual space.

However, some types of deformations in the literature do cause the underlying space or rather topos to change! And this is just one of the reasons why deformations of algebraic stacks are just a little more confusing than the case of schemes.

Here is a silly example: Let’s look at the lisse-etale site of k, call it C_1, and the lisse-etale site of D, call it C_2. For simplicity (and because it doesn’t matter for the associated topoi), let’s assume we only look at affine schemes. So an object of C_1 is an smooth affine scheme V over k and an object of C_2 is a smooth affine scheme U over D. In fact the sets of isomorphism classes of objects of C_1 and C_2 are naturally bijective, via the rules V —> V \otimes_k D and U —> U_k (Hartshorne, Exercise II 8.7). Moreover, if U is such an object, then the categories of etale coverings of U and U_k are canonically identified (by topological invariance of etale morphisms, see Theorem Tag 039R). For every isomorphism class of objects pick a particular object U_i of C_2 and let V_i be the corresponding object of C_1. Then we can try to match a sheaf F on C_1 with a sheaf G on C_2 by the rule F(V_i) = G(U_i). Does this work?

It doesn’t! Given two objects U_i, U_j of C_2 the collection of morphisms in C_2 between U_i and U_j is drastically different from the collection of morphisms between V_i and V_j in C_1. For example the value G(Spec(k[epsilon, x])) is acted upon by all automorphisms of k[epsilon, x] not just the automorphisms of k[x]. And in fact there is no way of identifying the categories of sheaves on C_1 and C_2 in any reasonable way. (I have several ways of saying this precisely, but none that is completely satisfactory. If you have one, please leave a comment. In fact, I would love a direct argument showing that Sh(C_1) and Sh(C_2) are not isomorphic as abstract topoi.)

Maybe this is just another reason for thinking that the lisse-etale site was a bad idea in the first place?

Yesterday, David Rydh in this comment found a wonderful approach to proving Lemma Tag 03FM. In fact I had already more or less given up hope that the lemma could be fixed! I am so exited about David’s idea that I am going to try to explain it in this post. Of course any mistakes are mine.

First we are going to discuss a completely general construction. Namely, suppose that f : X —> Y is a morphism of schemes which is separated and locally of finite type. On the small etale site Y_{etale} of Y consider the following presheaf

(*) V/Y |—> {Z \subset X_V open, Z —> V is finite}

Note that as we assumed X —> Y is separated such a Z is also closed in X_V. Also note that the empty set is closed and finite over any scheme. Finite morphisms satisfies descent for the etale topology. This implies that the presheaf above is a sheaf on Y_{etale}. Now, it is a general fact that any sheaf on Y_{etale} is representable by an algebraic space etale over Y. Hence we obtain an algebraic space, which I am going to denote f_!X which comes equipped with an etale morphism f_!X —> Y and we get a universal open and closed subspace Z_{univ} \subset f_!X \times_Y X which is finite over f_!X.

Before we say more about this construction let us point out that f_!X is really an algebraic space and not a scheme. For example suppose that X = Spec(C) and Y = Spec(R[t]) and the morphism f : X —> Y maps the unique point of X to (t = 0) inside Y. In that case f_!X is a copy of Y with {0} replaced by {0_1, 0_2} with residue fields R, respectively C (but of course f_!X is still etale over Y). The complement of {0_1} in f_!X is an open subspace which is a copy of Example Tag 03FN. And this is the simplest example of an algebraic space etale over a scheme which is not a scheme.

Now a very important property of the construction above is that it commutes with arbitrary base change. This is the analogue of the fact that cohomology with proper supports commutes with any base change. Namely, suppose that Y’ —> Y is any morphism of schemes, and denote f’ : X’ —> Y’ the base change of f. By universality of f’_!X’ we obtain a morphism f_!X \times_Y Y’ —> f’_!X’ over Y’. To prove this map is an isomorphism it suffices to show that the stalks of the sheaf (*) can be computed from the geometric fibres of the morphism X —> Y. And this is almost exactly the content of Lemma Tag 02LN.

As in Keel-Mori we are going to use a slight modification of f_! in the case that the morphism f : X —> Y comes equipped with a section e : Y —> X. Namely, in that case we consider the sub presheaf of (*) consisting of those Z \subset X_V with e(V) contained in Z. In the same manner as above you show that this is representable by an algebraic space fe_!X etale over Y whose formation commutes with base change.

OK, so how do we apply this construction to a groupoid scheme (U, R, s, t, c, e, i) with s, t separated and locally of finite type? Well, consider U’ = se_!R and the universal closed sub space Z \subset R \times_{U, s} U’. The inverse map of the groupoid scheme shows that U’ is canonically isomorphic to te_!R which gives that i(Z) \subset U’ \times_{U, t} R is universal. Compatibility with base change and the fact that R \times_{s, U, t} R is isomorphic to R \times_{s, U, s} R over (the second) R shows that there is a canonical isomorphism U’ \times_{U, t} R = R \times_{s, U} U’ over R. Then you check by chasing some diagrams that this isomorphism maps the universal closed subscheme Z into i(Z). Hence if

R’ = U’ \times_{U, t} R \times_{s, U} U’

is the restriction of R to U’, then we conclude that Z is an open sub space of R’ which is finite under both s’ : R’ \to U’ and t’ : R’ \to U’. (Here we use that U’  \times_{U, t} R = R \times_{s, U} U’ can be identified with an open in R’ as these spaces are etale over R.) Because we have e’ inside of Z by construction some more diagram chasing shows that Z is a sub groupoid space! Awesome.

Of course it is possible that U’ is empty. But, as in the statement of Lemma Tag 03FM if we have a point u of U where the set of points {r_1, …, r_n} of R mapping to u under both s and t is finite, and if s, t are quasi-finite at each of those points, then we get a canonical point u’ of U’ whose residue field is the same as the residue field of u, and which corresponds to an open and closed sub scheme of the fibre of R —> U whose points are exactly the set {r_1, …, r_n}.

OK, so this utterly general construction allows us to find an open subgroupoid which is a finite groupoid, at the cost of replacing U by an algebraic space etale over U. And, as pointed out by David Rydh in his further comments we can then use further properties of the original groupoid (U, R, s, t, c, e, i) to prove that U’ has additional properties. For example, in case the morphisms s, t are also of finite presentation and flat, then (I think) U’ agrees with an open of the Hilbert scheme used by Keel and Mori and it is a quasi-projective scheme over U. Wonderful.

So I was asked today if all of the “omitted” proofs and explanations in the stacks project are supposed to be filled in. The answer is: Yes, they are!

Since my goal was to build enough theory to get to the more advanced material on algebraic spaces and algebraic stacks, I have not always filled in all the proofs in the earlier parts. On the other hand, I claim that what I omitted is consistently of a relatively low difficulty (provided you read the definitions, and understand the material generally speaking). Of course my judgment could have been off and if you find a more difficult omitted proof or something you don’t understand, then please email stacks.project@gmail.com.

Now if you are an undergraduate or graduate student, then this is a perfect opportunity to hone your algebraic geometry skills. The key word to search for in the documents or in the book is “omit” (case insensitive). Find a spot where the proof is missing and you feel comfortable filling it in. Write up the proof (with references only to other results in the stacks project) and email the tex file to the address mentioned above. Your writeup will be reviewed and perhaps edited, and then added to the stacks project.

A quick search through the tex files in the stacks project shows there are 740 occurrences of the stem “omit”. So there are many things you can do!

And before somebody asks: All errors are supposed to be fixed. So email any typos, misspellings, mistakes, arrows pointing the wrong way, etc to the email address above. Don’t wait, just send any you find right away.

Just a short post to point out that I just defined the notions of “coarse quotient”, “good quotient”, and “geometric quotient” in the stacks project. To see the details see the chapter on Quotients of Groupoids. Please let me know if you think the definitions aren’t what you think they should be.

Really the idea for this chapter is to have silly things like: “U reduced => X reduced if X is a categorical quotient of s, t : R —> U” (from GIT, paragraph 2, Chapter 0) proved in excruciating detail and in ridiculous generality. It seems to me this is somewhat worthwhile since the first thing you always read about this material is that chapter of GIT and it can be confusing.

Also, the level of generality where R is not necessarily a groupoid can be useful, for example when forming the MRC quotient of a variety U you look at families T <--- C ---> U of rational curves C_t on U and the pre-relation you consider is R = C \times_T C (which has two maps to U). There is absolutely no reason that this should define a transitive relation and of course in general it really doesn’t.

Let X be a reduced algebraic space with 1 point. Then is it true that X is representable by the spectrum of a field?

The answer is no in general. The space [Spec(\bar{Q})/Gal(\bar{Q}/Q)] of Example Tag 02Z6 in the stacks project is a counter example.

Currently the best positive result (in the stacks project) is Lemma Tag 047Z which says that this holds when X is a decent algebraic space.

An algebraic space X is called decent if every point of X corresponds to a quasi-compact monomorphism Spec(k) —> X with k a field. This is not currently the definition in the stacks project, but it can be shown to be equivalent. It turns out that this is a convenient class of algebraic spaces to work with. It contains all schemes, all quasi-separated algebraic spaces, and likely all locally separated algebraic spaces (David Rydh, private communication). On the other hand a decent algebraic space is a bit like a scheme and has “enough points” in some sense.

Because there are non-representable 1-point spaces it turns out that the notions of “radicial” and “universally injective” are not the same for morphisms of algebraic spaces. Namely, the morphism [Spec(\bar{Q})/Gal(\bar{Q}/Q)] —> Spec(Q) is universally injective but not radicial (with any reasonable definition of radicial I can think of). Again for decent morphisms this does not happen, see Lemma Tag 0484.

Suppose you have two morphisms a, b : X —> Y and you want to know whether a = b. If

1. X, Y are schemes,
2. X is reduced,
3. a(x) = b(x) for all x in X, and
4. the induced maps on residue fields are the same too,

then a = b. If

1. X, Y are algebraic spaces,
2. Y is locally separated,
3. X is reduced, and
4. a(x) = b(x) in Y(K) for every x in X(K) and any field K,

then a = b. But the last statement does not hold if we replace condition 2 by the condition that Y is quasi-separated. Recall that quasi-separated algebraic spaces are the “usual” algebraic spaces, i.e., the ones in Knutson’s book, not some bizarre ultra general class of algebraic spaces.

This comes up when you consider quotient maps for groupoids in algebraic spaces, and it is just the first small sign that things get a little more confusing when dealing with algebraic spaces. Namely, the above means that if we have a groupoid in algebraic spaces (U, R, s, t, c) and a morphism U —> X then even if all of U,R,X are reduced to see whether U —> X is R-invariant (i.e. a quotient map), it is not enough to check that this holds on field valued points.

My student Yanhong Yang today noticed this typo in the definition of a geometric quotient in GIT (Definition 0.6).It states that the morphism \phi should be submersive and then has a blank line, after which it says

U’ \subset Y’ is open if and only if \phi’^{-1}(U’) is open in X’.

without any further explanation. This is present in all the editions that I have been able to find, including the latest one. Kollar, Keel and Mori, and Rydh have concluded from this that what is meant is that \phi should be universally submersive. (Just look at their papers. Maybe they even called up Mumford to ask him — another possibility is that they looked at the proof of Remark (4) of Paragraph 2 of Chapter 0 where Mumford seems to be using the fact that it is universally submersive.) There are also plenty of places in the literature where authors do not use universally submersive, only submersive.

Not only that, there is also a huge variance in the literature as to what a geometric quotient is. In the two papers of Kollar dealing with quotients (on by actions of group schemes, the other by finite equivalence relations) his definitions are adapted to the problem at hand, and in particular include the condition that the quotient map is locally of finite type, or finite — either of which seems like the wrong thing to require when considering the problem in general. In Rydh’s paper he includes the condition that the quotient map is universally submersive in the constructible topology (although this is in almost all cases implied by the other conditions). In a paper of Sheshadri he requires the quotient morphism to be affine. And so on. (I even found an article where they use Remark (4) of Paragraph 2 in Chapter 0 but do not require universally submersive…)

Let (U, R, s, t, c) be a groupoid in algebraic spaces (so R = G \times U if we are talking about an action of a group algebraic space). Let \phi : U —> X be an R-invariant map of algebraic spaces. It seems to me, but I may be wrong, that in each of these references, except for those where the author misread the typo, everybody always at least requires the following:

• X is an orbit space, i.e., the maps U —> X and R —> U \times_X U are surjective,
• \phi is universally submersive, and
• the structure sheaf O_X of X is the sheaf of R-invariant functions.

(The last condition is thrown in to attempt to make the quotient unique, but that only holds if you work in the category of schemes.) I think that in analogy with the introduction of algebraic spaces, we should use these three conditions to define the notion of a geometric quotient in the stacks project. Then we can have fun and add adjectives to describe additional properties of geometric quotients. For example, I particularly like the condition, introduced in the paper of Rydh, that a quotient is strong if it has the property that R —> U \times_X U is universally submersive.

By the way, I really really dislike the numbering scheme in GIT. Don’t you?

In the last couple of days I have added a few results on group schemes over fields to the stacks project. I mainly wanted to add the result that group schemes locally of finite type over a characteristic zero field are smooth which I hope to use later in an idea I have relating to finite groupoids in characteristic zero.

The sheaf of differentials of a group scheme over a field  is free (this holds in any characteristic). But actually I am not sure that a scheme over a field of characteristic zero whose sheaf of differentials is free is even necessarily reduced. In fact, in a paper entitled “Algebraic group schemes in characteristic zero are reduced” (1966) Frans Oort asks: Is every group scheme over a field of characteristic zero reduced? I googled and tried mathscinet but this question seems to be still open.

Another question I have is: Does any group scheme over a field have an open subgroup scheme which is quasi-compact? It seems that this could be true… but maybe I simply do not know any of the truly enormous group schemes that exist out there?

Leave a comment if you have an idea about either of these questions.