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?
I have copies of both the first (1965) and the third (1994) editions of GIT. There is a substantial and, I think, intentional difference in definition 0.6 (geometric quotient) between the two editions.
In the first edition condition iii) of defintion 0.6 reads: “phi is universally submersive” and it goes on to explain what this condition means. Morever, condition iii) is footnoted by Mumford and in the footnote he explains that he thinks universally submersive is a more topological concept than is merely submersive.
In the third edition condition iii) has been completely changed to read: “phi is submersive” and it goes on to explain what submersive means. The footnote to condition iii) that appeared in the first edition has been eliminated from the third edition.
One can only conclude that Mumford himself changed his preferred definition of geometric quotient sometime between the first and the third editions of GIT.
OK, interesting! Is the typesetting error already there in the third edition? [Edit: This was a nonsensical reply. I was confused about printings and editions. Sorry!]
It might be worth pointing out another difference between the first and third editions.
In the third edition, definition 0.7 defines “uniform categorical” and “uniform geometric” quotients. These differ from the universal categorical and geometric quotients in that they require that the specified conditions hold only for flat (instead of for all) morphisms.
In the first edition definition 0.7 does not define either sort of uniform quotient. It defines only the universal quotients and does this exactly as does the third edition.
I don’t know whether this difference is related to the difference seen in defintion 0.6 between the two editions.
My guess is that Fogarty made this change in the second edition. (Recall that the second edition is authored by Mumford-Fogarty.) Indeed, in the introduction to “Geometric Quotients Are Algebraic Schemes” Fogarty writes:
There is one important point concerning the hypotheses of our theorem which should be made here. In [GIT1], one of the conditions for a geometric quotient is that \varphi be a universally submersive morphism. In the second edition [GIT2] of this work, and in most of the recent literature, this has been relaxed to the requirement that \varphi be submersive. The proof given here uses only the weaker assumption.
(the second edition appears to be identical with the third edition when it comes to these definitions)
Perhaps most of the recent literature at that time did use the weaker assumption (but I haven’t seen that in any paper except for Fogarty’s!) but it is very awkward. Note that submersiveness is not stable under flat base change. Indeed uniformly submersive = universally submersive. So for example, (7) on p. 9 (geometric => uniformly geometric) does not hold with the weaker definition of geometric. On the other hand Prop 0.1 on p. 4 (geometric => categorical for schemes) does hold with the weaker definition.
Yes, Micheal Thaddeus and I had gotten to the same conclusion (i.e., that Fogarty must have been the one changing the defintion), but I was too lazy to actually look at the introduction to see if he mentioned this. Thanks David!
Note that it was not the introduction to GIT but the introduction to another paper of Fogarty that I quoted!
sorry i’m interesting in having a proof of the fact uniformly submersive =universally submersive or where i can find it. Thanks
This just follows from the fact that given any ring map A —> B then we can write B as the quotient of a polynomial algebra P = A[x_i] (infinitely many variables) which is flat over A. Hence, if X is a scheme over A, then the base change X_B is a closed subscheme of X_P and Spec(B) is a closed subscheme of Spec(P). You can easily deduce from this that if X_P —> Spec(P) is submersive, then X_B —> Spec(B) is submersive.
Thanks for your fast answer ^^