The first definition in “Grothendieck-style” algebraic geometry is the affine scheme 
for any ring 
. This is a topological space whose set of points in the set of prime ideals in . Then one defines a scheme to be a locally ringed space locally isomorphic to an affine scheme.
The definition of goes against intuition since it involves prime ideals, not just maximal ideals. Maximal ideals are more natural, since if ![R = k[x_1, \dots, x_n]/I](https://s0.wp.com/latex.php?latex=R+%3D+k%5Bx_1%2C+%5Cdots%2C+x_n%5D%2FI+&bg=ffffff&fg=444444&s=0&c=20201002)
for some alg closed field 
, then the set of maximal ideals of 
is in bijection with the vanishing set in the affine space 
of the ideal 
. (Of course one can give a geometric meaning to the prime ideals in terms of subvarieties, but it is less natural.)
However, in Daniel Perrin’s text Algebraic geometry, an introduction, he states/implies that one can define affine schemes just using maximal ideals (at least for finitely-generated algebras) and still get a good theory of schemes and varieties. Is this true? If so why don’t we all learn it this way? (One answer to the this latter question could be that some people are interested in non-algebraically closed fields.)
Let me be more specific. Fix an algebraically closed field . Suppose that is a finitely generated algebra. Then we define a locally ringed space 
whose points are the maximal ideals of . and we give it a sheaf of rings in the usual manner. Let us call these affine m-schemes over .
Then we define a “m-scheme over ” to be a locally ringed space which is locally isomorphic to an affine m-scheme over 
.
I believe we have a functor from finite type schemes over to m-schemes over . which on the level of topological spaces is taking closed points.
Is the following true:
This gives an equivalence of categories between the category of m-schemes over and the category of finite type schemes over .
Following a reference in Perrin, I looked at EGA IV, section 10.9 which discusses these issues. In particular, what I call “m-schemes” above seems to be called “ultra-schemes” there (or rather ultra-preschema). Proposition 10.9.6 shows that the functor from Jacobson schemes to ultra-schemes is an equivalence. (This differs in two ways from what I ask above, first I have no idea what a Jacobson scheme is, second this is not dealing with finite type schemes.)
One last question. One way to justify the usual is to say that 
is the adjoint functor to the global sections functor which goes from locally ringed spaces to rings. Is there a similar way to justify 
?
(By the way, I’m asking these questions since I’m teaching introductory algebraic geometry in the fall and I’m planning on using Perrin’s book.)
Secret Blogging Seminar 
Jacobson schemes are schemes that are locally the Spec of Jacobson rings. Jacobson rings are rings satisfying the Nullstellensatz (i.e. for any ideal I, the intersection of the maximal prime ideals containing I equals the intersection of the prime ideals containing I, and so equals the radical of I). Obviously any field k is Jacobson, and a general form of the Nullstellensatz says that if A is Jacobson, so is A[x]. Hence any finite type algebra over a field is Jacobson, and thus so is any finite type k-scheme.
The reason we don’t learn it this way is that for Grothendieck-style algebraic geometry (as to opposed to say Serre-style, as in FAC, where I think he did work with maximal spectra) it is important to pass to the Spec of local rings, say (which are not Jacobson), and many arguments use passage to generic points.
Also, if one is working in mixed characteristic (say taking a variety over Q and spreading out over Z, which is an important techinque in number theory, but also in parts of birational geometry, e.g. in proving bend-and-break theorems), a closed point on the variety over Q is no longer a closed point when one spreads out to a scheme over Z.
Another example: if one has a family f:X –> C, where C is a smooth irreducible curve, and the total space X is reduced, then f is flat provided that each generic point of each component of X is mapped by C to the generic point of C. Another way to say this, that avoids talking about generic points, is to say that f is flat if each component of X dominates C. But having the generic points around makes many arguments easier, because one now test flatness by checking something on certain points. I think there is an example of this worked out somewhere in Hartshorne’s section on flatness (which is where I first saw the above criterion).
By the way, there is a section of Hartshorne (in chapter II, I think) where he defines what he means by a variety over an algebraically closed field, and shows that no information is lost by forgetting non-closed points. (This is just a simpler version of the EGA result cited in your post.)
Hartshorne Proposition II.4.10, I think.
Matthew Emerton gives a great list of the reasons why one might want to work with nonclosed points. I’ll add two which have come up in my life: (1) “normal implies smooth in codimension 2” is incredibly useful, easy to prove using generic points, and a real pain otherwise. (2) Most Bertini-like theorems are best proved by showing that they hold at the generic point, and showing that they hold on an open set. The same is true of Grothendieck’s generic flatness theorem.
However, if you are doing an introductory course, I think sticking to closed points is a great way to reduce complexity. My first algebraic geometry course, taught by Brian Conrad, took this approach and it worked great.
Thanks Matt for explaining about the Jacobson schemes and thanks to Akhil for the Hartshorne reference which I haven’t had a chance to check out yet.
My point wasn’t so much why do we use schemes with nonclosed points, but rather why do we learn about the nonclosed points right away. When you are learning algebraic geometry (French style) for the first time, you have to learn about sheaves, locally ringed spaces, etc. Adding nonclosed points to the mix seems to make things unnecessarily complicated.
Anyway, as David points out, perhaps many people do learn algebraic geometry with only closed points in their introductory course on the subject. Perhaps my problem is that I never had an introductory course on the subject — after doing some reading on my own and taking commutative algebra, I jumped into a second semester class with Tom Graber which was all cohomology of sheaves, Riemann-Roch, etc (and was a great class).
The other point of my post is that I never knew all this schemes machinery worked fine with only using the closed points. I am wondering how many people out there were/are like me (so far based on the comments, not many …).
Incidentally, is the approach you mentioned by Perrin the same as or similar to what Mumford does in the first chapter of the Red Book or what James Milne does in his online notes?
“normal implies smooth in codimension 2″
a) Codimension 1, surely.
b) Since this post has so much to do with distinguishing between working over an algebraically closed field or not, don’t say “smooth” when you mean “regular”.
(If I remember the issue correctly, there are normal schemes defined over a non-algebraically-closed field of char p that become non-normal upon base extension.)
Joel, if you’re looking for other pedagogical suggestions re teaching schemes, I would suggest you point out to people that the three extensions
1) moving beyond affine (or projective over affine) schemes to general ones by gluing
2) allowing nilpotents in the structure sheaf
3) working over a non-algebraically-closed field, or indeed, over a ring that doesn’t contain a field
that are included in the definition of “scheme” are independent, and it’s only by historical accident that they are all sprung on the unwary at the same time. I was happy with just #2 for years, and have only comparatively recently allowed for #3. I still haven’t ever personally needed #1.
Joel, James Milne’s Algebraic Geometry course notes (200+ pages) might be useful if you consider using Specm.
http://www.jmilne.org/math/CourseNotes/math631.html
Allen’s comments underscore the fact that “algebraic
,
geometry” has become many subjects rolled into one.
It’s impossible to learn or teach or use algebraic geometry
without having some motivating problems and examples.
Much of the classical work can avoid schemes, but for me
the entry point was the study of linear algebraic groups
over algebraically closed fields (later arbitrary fields) of
any characteristic. You need some of the scheme
viewpoint to study homogeneous spaces
to compare rational points over a smaller field with those
over an algebraic closure, and especially to make sense
of the role of the Lie algebra in prime characteristic
where the notion of Frobenius kernel is needed. There
is no easy way to do all this, as seen in the texts by
Borel, Springer, and me as well as Jantzen’s big book on
representation theory (where some foundational results
must be quoted from Demazure-Gabriel’s optimistically
labelled Tome I). The Borel-Tits papers on reductive
groups over arbitrary fields require at least some scheme
ideas, but starting with SGAD or Demazure-Gabriel will
turn off anyone’s interest in the whole subject.
Dear Joel,
I think it’s a god idea to work just with MaxSpec (or Specm if you prefer) in an introductory course; I have done this before myself. (And Chapter I of Hartshorne does this, at least implicitly.) In fact, I like to begin just by working in affine space (and a little later on, in projective space), and consider solutions of affine (then homogeneous) equations. One can then introduce affine rings, and using the Nullstellensatz, prove that solutions match with maximal ideals in the affine ring. This then motivates the introduction of MaxSpec, and one can shift from the extrinsic viewpoint (where everything happens in an ambient affine or projective space, which provides the geometric glue that gives everything meaning) to a more intrinsic viewpoint in which the glue is supplied by the Zariski topology and sheaves of rings.
Working with MaxSpec (and reduced varieties) also has the advantage that the structure sheaf can just be defined as a certain subsheaf of the sheaf of all k-valued functions, which eliminates a lot of sheaf-theoretic machinery (which I think can be unnecessarily distracting in a beginning course).
Also, the Zariski topology can be motivated in the following way (closely tied to the idea of having a locally ringed space, rather than just a ringed space): it is the weakest topology on an affine variety with the property that if f is a regular function that is non-zero at some point P, then there is a n.h. of P on which f is invertible.
By the way, I think this kind of development of the foundations is extremely close (maybe identical to ?) Serre’s treatment in FAC, which is probably worth looking at.
Thanks for pointing me to Milne’s notes, which I had looked at briefly before. Indeed the approach taken by Milne is very similar to that taken by Perrin and very similar to that suggested by Matt above. Namely, they work just with MaxSpec and define the structure sheaf to be a subsheaf of the sheaf of k-valued functions.
To answer Allen’s comment: for me it is extremely “psychologically important” to define varieties in an abstract way (ie as a locally ringed space locally isomorphic to spec of a ring) rather than just thinking about affine or projective (or projective over affine) varieties. Even if one is ultimately interested in just affine variety, it seems me extremely important to communicate the idea of what a variety is and also understanding that varieties (schemes) are glued from local pieces motivates many later definitions (eg line bundles).
I have not yet had a serious reason to deal with a non-quasiprojective variety. But I constantly work in local coordinates, and compute the effects of changing coordinates. Once you’ve gotten that far, it seem silly to me to insist that all the coordinates come from some projective embedding.
Regarding Allen’s point of working with non-projective objects, it has become fairly common to prove that new objects (moduli spaces, etc.) actually are projective by first studying them as not-necessarily-projective objects (i.e., algebraic spaces or algebraic stacks). It is often easier to first construct them as not-necessarily-projective objects, then show that they have some nice properties as such objects, and finally use these properties to prove that the space is projective after all (e.g., Knudsen-Mumford, Kollar’s article, Cornalba’s article, etc.).
Is it completely unreasonable not to teach schemes as locally ringed spaces (either with Spec or MaxSpec) and just to think of functors of points? (I know how people will respond to this, but anyway..) That picture feels to me much closer to the intuitive idea of closed points – after all it just means solve the equations defining the scheme in various coefficient rings. (I never liked the locally ringed space POV and never use it — and I can, albeit unfairly, blame Matt Emerton, who taught me algebraic geometry!) This can be explained very intuitively as testing a space by points in it, lines in it, and other families, or as thinking of generalized functions as taking values on test functions, and is a point of view with much greater applicability. That seems to me much more natural than either 1. discarding crucial information you need to do geometry (the max-spec idea) or 2. working with a strange topological space with wierd points, rather than developing intuition from the point of view of machines for constructing solutions over different rings..
oh well. (luckily I’m not teaching intro algebraic geometry so no need to warn me of the irreparable damage I’ll do to impressionable minds.)
David, isn’t that pretty much the point of view Grothendieck was switching to for the second edition of EGA? I know he only updated the first book, but I recall spotting functors of points being featured prominently.
The “functor of points” approach is definitely the way the students of Grothendieck (or at least one of them, who I will only identify with his initials L. I.) teach (or used to teach, before retirement) introductory algebraic geometry…
Just to be clear, in the “functor of points” approach, do the basic definitions go like this:
– the category of rings with its Grothendieck topology
– sheaves (valued in sets) on the category of rings
– representable functors = affine schemes
– schemes = sheaves locally isomorphic to affine schemes
Is that logically coherent? (if not pedagogically coherent)
Joel – yes that seems coherent to me. An affine scheme is by definition a commutative ring (or k-algebra if you’d prefer) thought of contravariantly. all objects in algebraic geometry are things you can test on rings (by mapping affines to them), i.e. functors rings –> sets (say). the most basic thing is what are the points of your affine, and these come in flavors, you have K points for any algebraically closed field K, which you can think of as geometric points, and R points for various rings, which you can think of as families of geometric points. Then, nice functors on rings are ones coming from taking some quotient of an affine (representable functor) by an equivalence relation – the various Grothendieck topologies tell you how general a gluing you’re going to allow (giving rise to schemes, algebraic spaces, etc). Equivalently, geometric quantities can be patched together from local information, and this notion of locality is encoded in the topology. This point of view has innumerable advantages, but for example makes the idea of moduli problems very natural.. also it makes the construction of projective space pretty natural and coordinate free, as the moduli of lines in affine space (as opposed to defining it as a patching or as a whole new concept, that of Proj of a graded ring, as opposed to the geometric notion of quotient – after puncturing – by the multiplicative group).
I agree completely with David Ben-Zvi. It seems vastly preferable to me to *define* the category of schemes (or better, algebraic spaces) as a certain full subcategory of the category of functors from Rings to Sets. Then things simplify greatly. No silly topological spaces, no prime ideals, no axiom of choice, no local rings, no fields. All you need is all you ever used anyway: the category of rings, covers, and descent. Of course, you can use ideals, for instance prime and maximal ideals, if you want– they’re just no longer necessary to set up the theory.
I would love to teach a class from this point of view. From a pedagogical point of view, the key thing would be to explain how to visualize schemes defined in this way, without relying on their substrate of Zariski points. But first, you really ought to do that with scheme theory qua locally ringed point sets anyway. And second, it shouldn’t be that hard by fleshing out each concept in the case of finitely generated algebras over C, which can be accurately visualized by using their sets of C-valued points.
One objection might be that students often learn point-set topology before algebraic geometry, and so teaching them a second gluing formalism might seem wasted time. But in my opinion, continuing with the point-set-theoretic point of view is the pedagogical equivalent of throwing good money after bad, and is probably in the long run even worse.
I guess this is one of my pet peeves. It just seems that since the SGA seminars (which, being seminar notes, are not at all ideal introductions to the subject), each introductory text in algebraic geometry has taken another step in the wrong direction (with the notable exception of two books on group theory: Demazure-Gabriel and Waterhouse). The attempt to explain the theory in its simplest form is laudable and no doubt sincere, but I think these introductions make things more complicated in that, for them, simplicity equates to using the formalisms the authors learned as students, rather than the formalisms best suited for the job. The fundamental functorial concepts are really very natural and simple. We shouldn’t be thinking about how to vulgarize them; we should be thinking about how to explain them.
Are people seriously suggesting that we should teach beginners algebraic geometry via the “functor of points” approach? Frankly, I can’t imagine such a course not ending in complete disaster.
For my money, the best book for beginners (i.e. people who don’t yet know what a variety is) is Joe Harris’s “First Course”. I’m a firm believer in the philosophy that to understand a big machine like post-Grothendieck algebraic geometry, you first have to acquire a deep understanding a good number of classical examples (for instance : algebraic curves, linear algebraic groups, Grassmannians, etc.).
Maybe this is a function of the areas I work in, but my feeling is that most mathematicians today secretly understand their machines via these sorts of examples anyway, so why short-circuit the process?
While trying to learn algebraic geometry, I have found the first chunk of Eisenbud-Harris’ little book `Geometry of Schemes’ to be really helpful in getting my head around schemes and various ways of looking at them. You might suggest it as supplementary reading. I think it does a beautiful job of explaining the basic set-up, intricacies, and intuition in a very concise way – something I have not found in the standard algebraic geometry textbooks. (although I haven’t gotten a chance to look at Perrin’s book)
Andy: I wouldn’t jump right into the functors, but I would take the pedagogically best path towards them. So I’d probably first cover affine schemes, not as schemes per se (locally ringed spaces of prime ideals), but just as rings. I’d focus on finitely generated C-algebras, explaining how to visualize them and how ring-theoretic concepts (etaleness, flatness, surjectivity, etc) can be interpreted visually. You could also talk about how we can visualize other rings, like finitely generated Z-algebras, analogously (if perhaps less realistically). You could easily talk about affine algebraic groups and their functors of points here. You could also talk about other moduli problems, like those represented by affine cells in flag varieties.
Once the students have a good feel for how to think about commutative algebra geometrically, then I’d address the issue of gluing, pointing out that this can be handled with two different formalisms: locally ringed spaces of prime ideals, and functors of points. (I might provocatively call them the pre and post 1960 (?) ways.) Then I would explain why the second approach is better. The main reason is that any space X whose points have a “meaning” can be handled very easily using the functor of points. This includes Grassmannians, classical groups, moduli problems. On the other hand, the points of the topological space underlying a scheme and silly formal constructions, and they lead to false phenomena such as the fact that the set of points of a product of two schemes is not the product of the two sets of points.
To be sure, it would require some care, but no more, I think, than teaching any other class for which there is no adequate textbook. It seems half the people think this obvious, and the other half think it’s impossible. It would be great if someone who really believes it could be done would actually do it. (Or maybe someone could just post lecture notes from Prof. L.I.’s class…)
Andy – You make an important pedagogical point, but I believe
it’s somewhat orthogonal to James’ and my point – the assertion is not that we should shortchange examples, or have less familiarity with classical intuitions, or rush into machines, but rather that when you do get to introducing topics like schemes, the now-standard presentation of in terms of locally ringed spaces have serious drawbacks and that going directly to functors of points has advantages. In fact as James points out having a strong background in classical thinking will make the transition to functors easier, since many classical examples are very conducive to writing as functors.
Complex analytic spaces are also locally ringed spaces. The nicest formulations of GAGA use the comparison of a complex, proper scheme (or algebraic space) with its associated complex analytic space, as locally ringed spaces. So this is one point in favor of locally ringed spaces.
“David, isn’t that pretty much the point of view Grothendieck was switching to for the second edition of EGA? I know he only updated the first book, but I recall spotting functors of points being featured prominently.”
Charlie, I think EGA I *defines* schemes in the second edition as ringed spaces which are locally isomorphic to affine schemes, although he talks about functors of points quite a bit.
I think A.O. taught from a functor of points perspective in 2004-2005 at Berkeley, but maybe it’s not surprising, since he’s been known to “pal around with” L.I.
I’m not sure where I stand on this question. Allcock once told me an idea he had for an “algebraic geometry from nature” class, basically teaching a large collection of examples, all drawn from real-life phenomena like shadows, reflections, configurations of linkages, etc., and it sounded rather appealing. On the other hand, modern toolsets can be both useful and compelling.
The functors associated to non-affine schemes like can be a bit more subtle than the locally ringed spaces, since you might have to do work to make sure you have a sheaf. At least, it took me a while to see why the “stupid quotient” doesn’t give projective space. I could see people arguing both ways about whether it is better to cover this sort of thing early.
Joel: This is a minor point, but if your functor is from rings, then you should have a cosheaf of sets.
I still haven’t ever personally needed #1.
Jason was kind enough in his comment not to point out that I had indeed needed some algebraic spaces (constructed, really from their functors of points) in a paper of mine that he generalized. I’m still hoping they’re projective, though!
I think the “functor of points” is particularly popular and helpful for algebraic group theory, because it is much easier to define the many homomorphisms required by group theory using their functorial incarnations on sets/groups of points — the extra points of schemes complicate the picture here. Similarly, the type of complications that schemes have been invented to deal with are often much milder for algebraic groups than for schemes (to say it differently, the difficulties only arise quite a bit deeper in the theory, or at least I’ve heard this from someone who is involved in the retyping/updating of SGA3). So for instance even such a book as Platonov and Rapinchuk’s on arithmetic groups — truly a research book, and not a textbook — deals with algebraic groups as sets of points over a “universal domain”, with some “rational structure” to deal with the questions of field of definition.
I’ll try to see if I can get some lecture notes of this course of “L. I”. I didn’t attend it myself, but a friend did and may have taken notes; that friend highly approves of a functorial presentation of algebraic geometry, so of course he liked it, but from what he said, it was not a universal feeling among the students…
I’m actually fairly interested in the pedagogical questions here because a pet project of mine is to write a book on exponential sums over algebraic varieties, mixing the algebraic-geometry and the applications to analytic number theory; since the goal is for the resulting text to be an accessible reference for analytic number theorists who are not acquainted deeply with algebraic geometry, I will have to find a way to present things which is suitable. Fortunately, most exponential sums of analytic interest live over affine schemes of finite type over the integers, so a fairly simple “functor of points” description is what I’m currently thinking of using (of course, the proofs or sketches thereof will involve more complicated things, but the statements of the important applicable results might not).
I would like to make an argument in favour of locally ringed spaces over functors of points. But I will say at the beginning that it may just reflect my own psychological weaknesses. In general in such a discussion, I think it’s important to remember that most of our preferences are based on habit and related psychological factors. It’s normally not very difficult to translate from one point of view to another if one tries, and once one becomes used to a certain view-point, it becomes easy to work with it. So the feeling that a certain view-point provides something indispensible that is missing in the a different view-point should probably not be taken so seriously.
This being said, I find a lot of utility to the topological space underlying a scheme, to the points that are part of it, to their various closure relations, and so on. I’ve never had to work with stacks or even algebraic spaces, and so I’ve never had to kick this habit.
As I said in an earlier comment, when one has a point of a scheme over Q, which one then spreads out over Z, the point which was (say) closed over Q becomes non-closed over Z, and has various specializations, into the different possible residue characteristics. Thinking about these points in a concrete way is something I find to be useful, and helps me understand what’s going on. I can localize sheaves around these various points, and this has various meanings related to interesting number-theoretic operations (inverting primes, p-adically completing, etc.).
Maybe more generally, I might say that one interpretation of geometry is that it is about arguing from pictures, and to me there is no question that a scheme with its Zariski topology is much more directly pictorial (and hence geometric, in this sense) then a functor on the category of rings. Speaking a little more technically, this might mean that something like intersection-theory is easier to explain when one works with point-sets than with functors.
Now of course, we could just do intersection theory by considering sheaves instead of the cycles that support them, and computing derived tensor products and so on, and in fact there are good arguments for doing this in certain contexts. This is analogous to the fact that a lot of what we might call geometric topology (and here, I don’t necessarily mean what is usually meant by that field, but just geometric ideas of intersecting, homotoping, cutting along curves, concrete obstruction theory, etc.) can be encoded in algebraic topology. But there are some intiutions to be gained from the geometric pictures that can be lost in the algebraic formulation, even if the latter is often more powerful and general, and so similarly I would say that there are some geometric intuitions that are lost in the the functorial formulation. (Of course, different intuitions, such as the importance of gluing and equivalence relations, are brought out, but to me these seem to be somewhat softer than the very geometric ideas in something like intersection theory. This is a fairly strong statement, and probably many people will disagree. Again, it may well just reflect my own hangups.)
Although (as Grothendieck pointed out, surprisingly) many constructions in algebraic geometry can be understood as quotient constructions of various kinds, starting from very simple objects, my own feeling is that this is not what should be emphasized at the beginning. When I introduce projective space, I explain it in classical terms as being obtained from affine space by adding points at infinity, so that we don’t miss any points (and hence any intersections). The description as affine space minus the origin modulo G_m I would mention only as a technical device, and (in a more subtle way) as a point of view that demonstrates the homogeneity of projective space. But again, this homogeneity, while obviously very important, wouldn’t be the first thing I emphasize.
I’m sure that even those who prefer the functor-of-points view-point, when they think of a space, imagine something geometric (in the literal, pictorial, sense). Beginning students have to learn to construct these pictures. Teaching them schemes (or at least, varieties) is one way for them to learn it.
David and James, feel free to critique this (and/or to subject me to psychoanalysis, which, in the spirit of my opening paragraph, is probably the same thing).
This is a very interesting discussion to read as someone who (very sporadically) uses a tiny little bit of algebraic geometry, of the extremely un-modern and complex analytic sort, and who sat uncomprehendingly through many, many lectures by a certain R. H. when I was a grad student at Berkeley.
As an ignoramus, I have a few questions for the cognoscenti.
I wonder if it’s fair to say that as a general rule, one wants the most functorial approach to a subject when one already knows in advance what kinds of things one wants to do with the objects. Going back later and adding “extra” flexibility to your objects can be a major headache; maybe this is one of the reasons why programming (or, more accurately, maintaining programs) in strongly typed languages can sometimes be very time-consuming. It is also, I think, one of the reasons why analysts typically have less use for functorial language than algebraists: every problem requires a slightly different estimation technique or function space, and book-keeping is less important than flexibility.
Does the functorial language in algebraic geometry obscure some ideas and constructions, and uncover others? I guess I’m specifically wondering whether Grothendieck-style algebraic geometers would have invented pseudo-holomorphic curves, and the application of “softer” (i.e. symplectic) methods to enumerative complex geometry. What about mirror symmetry? In fact, what actually is the history here? BZ?
For what it’s worth, the one time I taught (introductory) Algebraic Geometry, I used Mumford’s “Complex projective varieties” book, which was just right for me (I can’t speak for the students . . .)
Dear D,
I think that the functorial approach certainly obscures certain ideas and constructions (like any dogma, I would guess). In general, in taking a functorial approach to a subject, one sets boundaries from the beginning by specifying the category, which place a priori constraints on what one is allowed to talk about (and hence, in some subtle sense, to think about). My understanding is that symplectic methods came from outside of algebraic geometry in the beginning. On the other hand, once one knows to use them, they can be recast in purely algebro-geometrical terms, and the functorial methods make it easy to work with them (e.g. by making it easy to make spaces of stable maps, following Kontsevich). (Others can critique this analysis if my (implicit) history of the situation is wrong.)
I should add that working scheme theoretically is also a functorial approach, in the sense of the preceding paragraph, in that one has a certain prescribed category that limits the scope of the mathematics.
But, as Jason pointed out above, by virtue of being point-set topological spaces, schemes are closer to complex analytic spaces (and hence, to classical manifolds), then functors. So learning to work with them (or at least with classical varieties) might provide some intuition which is closer to other parts of geometry.
By the way, I think that Mumford’s “Complex projective varieties” is great.
I don’t think there’s any disagreement here that a strictly dogmatic mathematician, or one who ignores geometric intuition, is at a severe disadvantage in this subject. I think that’s irrelevant to the question at hand. I certainly think about algebraic geometry in a very pictorial and informal way (thanks in large part to having people like Matthew and James teach me enough intuition that I can get away without really working through the formalism). In fact I always visualize algebraic objects in terms of complex varieties (with the complex topology) – part of learning algebraic geometry (for me) is learning how to take that intuition and make it into rigorous math, in particular learning how to adapt one’s intuitions when your variety is over another field or ring, how nilpotents work, etc. I still claim however that this is independent of whether you think of schemes in terms of functors of points or locally ringed spaces. After all the functor of points POV is just a way to think of gluing affines — so you better have a good intuitive feel for affines, one way or another. And all the geometric information is encoded very pleasantly in the functorial POV.
I think it’s also crucial what kind of applications you have in mind – Matthew, you say you usually deals with schemes, and so the topological picture is completely adequate (and perhaps has advantages) – though I’m not sure I believe this, since much of what you care about with schemes involves their (topological, not coherent) cohomology theories, which are all defined using the functor of points perspective (ie the etale or crystalline topologies or variants)…
On the other hand if your interests are in representation theory, noncommutative geometry, algebraic topology, for example I think it’s unassailably true that the functor of points IS the way to go. Representation theory for example is mostly concerned with stacks – stacks with very few C-points, like say the flag variety modulo the Borel. Tannakian theory naturally concerns the functorial POV and stacks. The theory of moduli spaces is concerned inevitably with algebraic spaces at least and usually stacks. The theory of classifying spaces for beginners, or the chromatic picture of stable homotopy theory, are about functors or stacks. Noncommutative spaces (both the algebraic and the C^* algebra-ic ones) can be represented very nicely in terms of stacks as well. Deformation theory is much more naturally a subject in functorial algebraic geometry. And so on… None of which is to say I would teach first year students stacks – but I would teach them a picture of algebraic geometry that is flexible enough to handle all of these contexts, which the theory of locally ringed spaces fails miserably at.
Danny, as for mirror symmetry and pseudoholomorphic curves, no algebraic geometer could have come up with them IMHO (nonfalsifiable statements are fun!) In particular the idea to think of a complex variety with its complex topology (which is necessary to make the leap you suggest) is equally natural in either of the points of view discussed (again the question is how you think of your building blocks, the affines.) When you speak of “Grothendieckian” algebraic geometer, though, it’s unfair to imply this means a less geometrically-intuitive one than a classical one. In fact the geometric intuition for moduli stacks (like stable maps) and their derived enhancements (like virtual fundamental classes) which are crucial for making sense of mirror symmetry are much more naturally “functorial” than “ringed-spaceish”..
wow that was quite a rant. not sure where it came from
– probably my current failures to make an abstract functorial construction concrete :-) at the end of the day I’ll probably end up teaching the way every one else does and wish I had James’ courage!
Once again, I agree completely with David. In particular, his first paragraph (in 32) expresses perfectly my thoughts on all this.
Two more comments…
Jason: I agree with your point about about holomorphic geometry. This is a good reason not to ban locally ringed spaces entirely. For a while, I have wanted to know if the functor of points approach can be done there. Presumably you can do things the same way once you’ve defined the category of Stein spaces with its Grothendieck topology. Does anyone know if there’s a direct way of doing this? (Maybe this is obvious. The only time I asked an expert on the foundations of these things, the answer was, “Who cares?”.)
Matt: I would probably agree that an abstract functor cannot really be visualize, but I also think that abstract locally ringed spaces can’t really be visualized, so I don’t think that’s so interesting. (For example, can you visualize the spectrum of an infinite product of fields as a locally ringed space?) So really the question is which point of view is better for visualizing affine schemes. I don’t think it really makes a difference. When I think about an elliptic curve, I see a torus. This is not because the locally ringed space or the functor it represents look like a torus in any real sense. It’s because one of the first things I figured out when learning algebraic geometry is that it’s always better to picture the analogous holomorphic space than to try to picture the locally ringed space. Now, if in my work, I visualize the holomorphic space but make it into real math by using the formalism of functors, is that any less geometric than keeping the same picture in mind but making it real by using the formalism of locally ringed spaces? I would say not.
I agree that geometry in its broadest sense is arguing from pictures. But I don’t see any reason why you can’t use the same pictures for the functors as for the locally ringed spaces. With the example of an integral model of a curve over Q, I probably have a very similar picture in mind to yours, but I just wouldn’t use the formalism of topological spaces to make it real.
One difference, though, might be in how we view spectra of fields (and hence scheme-theoretic points). In scheme theory proper, Spec of a field is a point. This is quite reasonable from the point of view of the Zariski topology, essentially module theory. But there are many other points of view from which it isn’t reasonable. For instance (as you well know), from the point of view of the etale topology, the fundamental group of Spec of a field is its absolute Galois group, and so a Spec of a field is not *really* a point unless it is separably closed. On the other hand, from the crystalline point of view, you’d want it to be perfect. For another example, should we view Spec C((z)) as a point or an infinitesimal punctured disk? So should Spec of a field “really” be viewed as a point or not? I would say the question is meaningless, it is not “really” anything. In particular, it is no more a point than some other figure. It should formally be just what it is (a certain object of the opposite of the category of rings), but we should have the flexibility to picture it however we find most helpful. For example, I like to view Spec C(t) as the limit of an increasing punctured copy of the complex numbers, like a some fly screen. (For instance, then it’s expected that its absolute Galois group is free on uncountably many generators.) So if the spectrum of a field is not really a point, why insist on making it one?
Anyway, I agree that it’s not hard to translate back and forth between the two formalisms (at least if you’re talking about schemes and you don’t mind invoking the axiom of choice). So the real question is whether one can teach a class or write a textbook from the functor of points point of view that will be as good or better, by general agreement, than one taught from the locally ringed spaces point of view. I think it would be possible, but it’s pretty clear that to convince other people, one would actually have to do it. Maybe when I’m old. (So much for courage.)
For the folks who are against non-closed points, how do you intend to talk about Hasse’s principle? How will you describe the Brauer-Manin obstruction? How will you define the Brown-Gersten-Quillen spectral sequence for K-theory? Regarding “pointless stacks”, do you realize how often arguments about BG reduce to classical algebraic varieties like P(V) for V a faithful representation (e.g., equivariant Chow theory)? Regarding algebraic spaces, do you realize how many theorems in Knutson’s book are reduced to the “classical” case of schemes via Chow’s theorem for algebraic spaces? I am all for algebraic spaces, stacks, and the functor of points — I use all of those. But the perspective of locally ringed spaces is also very useful. Why not teach both?
Coincidentally, today someone gave a talk about visualizing Spec Z[i] in a student seminar I organize. Of course, it was from the point of view of spaces of prime ideals, rather than functors of points. And it was clear that that approach has a big advantage in that it is possible to define Spec A and describe Spec Z[i] in a concrete sense as a cover of Spec Z, drawn as the usual two wiggly lines, all within one lecture. Even though the picture should not be taken too seriously, it’s better than no picture and is good for students to see at least once. Doing something similar with the functor of points would require much more discussion about analogies with Riemann surfaces, fibers of morphisms, and so on. I believe that in a class on algebraic geometry, all that should be done anyway, but doing it in one lecture is impossible.
So I actually probably would explain the prime spectrum of a ring in my ideal class on algebraic geometry, but I would emphasize that it is just a way of modeling certain functors using point-set topology or is one way of visualizing them. So it’s a point of view that can be psychologically helpful, but it’s not essential. I think there is a continuum of approaches between this one and the ones in the standard textbooks. I just prefer it a bit further along that continuum.
Jason,
You make good points, but I might suggest that a “barrage of questions” may not be the best way to maintain the appearance of a civil discussion (verbally or in writing).
Dear Scott,
No offense intended. If there is a clever way of presenting any of those topics using functor of points only, I would really like to learn about it. Over the 6 or so years I have been teaching algebraic geometry (certainly not as long as Matt, from whom I also learned much of algebraic geometry), I have had many discussions with students and fellow instructors over how to present the material. I have followed Shafarevich, Mumford’s Red Book, Hartshorne, Harris’s “First Course”, Cox-Little-O’Shea and my own notes (photocopied for students). Of course every approach has its advantages and disadvantages. But my experience is that Hartshorne’s book, whatever its flaws, gets students up-and-running and ready for more advanced material the most quickly.
Jason – Certainly the point was not to eliminate locally ringed spaces.
We can all agree that a mathematician ought to have as many tools as possible at their disposal and different pictures are better adapted to different goals. James and I were I believe combating the perception
(not shared by you obviously) that functors of points are these abstract unintuitive things and a first course is better off focusing only on the supposedly more accessible locally ringed spaces – i.e. we’re advocating, as are you, that a well informed algebraic geometer should be exposed to both. Or as Danny would say, should be exposed to much more, in particular should be aware of the potential impact of the great flexibility found in less rigid worlds than algebraic geometry. (Well ok we’re saying more, both of us strictly prefer one picture over the other, and in our favor is the point that one strictly contains the other, but that’s a personal decision in any case.)
As to your questions, I’m not sure I understand the complaint. Of course generic points are absolutely essential to understanding algebraic geometry, and are one of the most useful tools there are
(for example for formulating the things you mention). However that doesn’t mean one needs to think about them as points of a locally ringed space – they are after all (very special) k-points for k an
appropriate field. To put it another way, anything you can say in ring theory (like the field of fractions of an integral quotient of your ring) you can equally say with functors (by putting “op” in front – ie the notion of localization, field, and closed subvariety are equally accessible from the functorial point of view). So I’m not sure why say the Brown-Gersten resolution favors one picture over another
(but obviously it’s far closer to your expertise than mine). Again the point was that for certain kinds of algebraic geometry (say the kinds James or I practice) one picture is strictly better, and I’m happy to believe that for your purposes the converse is true.
I think it might be interesting to isolate at least one question which is underlying some of this discussion. I’m not sure how to phrase it pithily, unfortunately, but I will try to explain it as best I can.
An illustration of what I have in mind is given in (I think) one of the appendices to Miles Reid’s book “Undergraduate algebraic geometry” (or, as it was apparently translated in Russian, “Algebraic geometry for everyone”), where Reid describes the case of a thesis about cubic surfaces (or something similar) which was derided by the examiners because the natural setting was “an arbitrary ringed topos” (or something similar).
Probably there is agreement among most of the commenters here that many problems in geometry (and especially moduli-type problems) are about making gluing constructions, or fibre products, beginning with fairly basic objects. And this is probably a fairly general principle, at least in algebra. (The kind of thing I have mind is, for example, the way many constructions that one makes with modules in algebra are (perhaps secretly, at least at first) just special cases of either a Hom or a tensor product, and hence are subsumed in the general theory of those two operations.)
Hence it might well be that the solution to some question posed about a cubic surface might well actually involve a framework whose natural setting is a general ringed topos.
The question I want to ask now (since I think it might be underlying some part of the preceding discussion) is: how does one effectively teach this principle?
It’s easy to imagine that if one spends all ones time teaching Hartshorne Chapter I type stuff, then students will have a good feeling for a lot of examples of different varieties, but won’t have a sense of how to make general arguments with them, or to work with important constructions involving them. (The basic theory of individual curves is quite a bit simpler than the theory of the moduli spaces of curves.) On the other hand, most of us have probably had the experience of teaching general machinery in a course, and having the unpleasant feeling that it is completely meaningless to the students.
I think one merit of Hartshorne’s book is just that it is a very substantial book. Taken as a whole, it develops a fair bit of machinery, but also actually does a lot of concrete geometry, including a lot of the theory of curves and surfaces. If you study the whole book carefully, then you can get a sense of how the theory and examples fit together, and how you can actually use the theory to make particular arguments and computations. Many of the other texts available may not be flawed in their approach, but simply don’t go as far as Hartshorne , and so in the end, don’t serve as well as self-contained texts.
As one possible counterexample to this, I would propose Mumford’s “Lectures on curves on an algebraic surface”, which I think is one of the really great algebraic geometry texts. It deals not just with foundations, but with really substantial, and concrete, geomeric questions, whose solutions depend on all the subtleties of the foundations.
It would be nice to have something like this in the spirit that James and David are proposing: a text that worked with the functor of points approach, perhaps entering into stacks and algebraic spaces, but which also illustrated everything with substantial applications, which could be used as a way to lead students into this point of view.
Maybe there are some reasonable texts out there that I just don’t know about. E.g. is there a textbook treatment of Deligne and Mumford’s paper on moduli of curves, which really exposes it in the same accessible way that Hartshorne exposes the theory of surfaces?
Or is it still the case that students just read the original paper?
Dear BZ,
Our posts just crossed, I saw.
I think that one thing Jason was getting at was that in the technical theory of algebraic spaces and stacks, the proofs of several results are reduced to the scheme case by Chow-type lemmas (just as in the theory of proper maps, several results are reduced to the projective case in the same way). The question (at least implicitly, and maybe made explicit in his reply to Scott) is whether one can deal with these kind of question directly from the functor of points view-point, without at some stage having to reduce to the scheme case and get ones hands dirty with the locally ringed space.
I am not an expert on this kind of question, but I wouldn’t be surprised if that the answer was that “yes”, one can find other ways to argue that more intrinisically adopt a functor of points view-point, but I also wouldn’t be surprised if a lot of such arguments aren’t yet developed.
I think the comparison with proper maps might be a good one. My understanding is that it is a fairly recent development (the last 10 years or so?) that people have known how to prove things like “pushforward of coherent is coherent” for proper maps in general settings, without having to reduce to the projective case. (I remember hearing about a theorem of Faltings of this type for proper maps of stacks in some generality roughly ten years ago. Hopefully I’m not totally mistaken in what I’m saying.)
It seems that there are imperatives from non-commutative geometry, and derived algebraic geometry, which are stimulating people to find new characterizations of various geometric properties, e.g. in terms of properties of derived categories of coherent sheaves, which are leading to greater flexibility of working with these notions in a more stacky/functor of points, rather than in a traditional scheme-theoretic way. BZ and Jim, is this right? How far can one go in developing the full geometric theory of stacks (including proper maps and the like) without ever using the crutch of (non-affine) schemes as locally ringed spaces?
Best wishes,
Matthew
Probably there is agreement among most of the commenters here that many problems in geometry (and especially moduli-type problems) are about making gluing constructions
Can you give me an example where you actually glue, say along an open set? My limited experience with moduli spaces involves overparametrizing, then dividing by a group.
BTW most commenters here know that the “Knutson’s book” that Jason refers to is not by me, but my father Donald Knutson; just thought I should mention that in case anyone else is still reading.
I guess by gluing I meant “dividing out by an equivalence relation”; but in many questions one does break things up into open sets, work on these individually (perhaps taking a quotient, among other things), and then glue. For example, I would think this happens when one makes the Picard scheme of a variety. (At least, on the few occasions when I’ve tried to think this through for myself, this was one of the steps.)
Dear David,
You and I might mean different things by “Brown-Gersten-Quillen” (there seems to be disagreement, for instance, between “BGQ” in McCleary’s book and in Srinivas’s book). I mean Theorem 5.20 on p. 65 of Srinivas’s “Algebraic K-theory”. This is a spectral sequence for the higher K-theory of a Noetherian scheme X in which the (p,q)-part of the 1st page is a product over all codimension p points of the scheme of the K-theory of the residue field of that point. For me this is strong motivation to include the codimension p points as part of the “primary definition” of a scheme rather than a “secondary definition”.
Hi Jason – I certainly agree, that’s a beautiful picture taking great advantage of generic points of subvarieties, and makes a good case for their centrality (another very similar example is the Beilinson adelic Cousin complex). It might be worth pointing out though that the spectral sequence can be defined without mentioning nonclosed points – it comes from the filtration on the category of sheaves on a space by codimension of support, i.e. it’s a fancy version of the spectral sequence calculating the cohomology of a stratified space, in which you don’t fix a stratification. I think you can define the same for the cohomology of any sheaf (the BGQ case being the sheaf of K-theory spectra) – though if you’re not in a Noetherian space (or in the presence of some similar finiteness, as in categories of constructible sheaves) this likely gives intractable nonsense.
This discussion seems to be fraying into more than one issue: MaxSpec vs Spec (both as locally ringed spaces, the original issue), functors of points vs locally ringed spaces, algebraic spaces vs schemes vs projective schemes, Hartshorne’s book vs someone else’s, and maybe some more. I don’t think these are all orthogonal issues, but they are definitely independent, and it seems to me that Matt and Jason are at times identifying them.
So, for example, I would not be surprised at all if one needs (in some sense) projective schemes to prove basic finiteness theorems about proper algebraic spaces. But this doesn’t mean that we have to model the gluing of affines that produces projective space using the formalism of locally ringed spaces. It’s easy enough from the functor of points POV to make the following real: take affine n-space, delete the origin, and then quotient out by the action of G_m. I’ve been told before that you need general schemes (rather than just projective schemes) to prove basic facts about algebraic spaces. Although I’m a bit skeptical, no problem: just define a scheme to be a certain kind of functor (e.g., an algebraic space, which is very easily defined from the functor of points POV, with a covering family of open immersions from affine schemes). It would great if, as Matt said, one could prove such statements without first reducing to the case of schemes or projective schemes, but the functor of points POV doesn’t require that at all. On the other hand, as Matt said, probably many of the arguments needed to do this from the functor of points POV aren’t yet written down, which is to me the main reason to use the ringed spaces point of view.
Similarly, as David pointed out, you can still talk about generic points if you want to. They’re just certain maps from Spec of certain fields. (However, from this point of view, there is less of a reason to call them points, which I actually like, as I wrote before.)
Finally, let me say something about Matt’s question of writing a textbook at needs the formalism of the functor of points, for example with some treatment of Deligne-Mumford. I now think something implicit in this question is really what’s at the heart of this discussion. That is that the functor of points, being a harder formalism than ringed spaces, needs more justification than, say, the justification for the formalism of schemes found in Hartshorne. I, and I think David, would disagree that the functor of points is a harder formalism than ringed spaces. In fact, I think it’s an easier one, and I think that’s really what’s going on in this discussion. Why do I think this? For example, suppose you had to explain to the analyst down the hall what the group scheme GL_2 is. I would say this: In algebraic geometry, a space is a functor from Rings to Sets, i.e. a rule that gives a set for every ring in a natural way. Then we define GL_2 to be the rule that assigns the group GL_2(R) to the ring R. What could be easier and more natural? That’s exactly what GL_2 *should* be, once you accept the very natural point that we want to look at all rings. I wouldn’t even have to define what an ideal is, much less a prime ideal, the spectrum of a ring, a locally ringed space, and so on.
When I first learned scheme theory, there were lots of geometrically minded students in the class who just couldn’t get past the crazy topological spaces. The simplest varieties, such as n-dimensional affine space over the complex numbers, were these unvisualizable things. The problem was that they took the Spec construction too seriously. Completely understandably, they really wanted to understand Spec A (A=C-algebra, say), when in fact the right approach is to view it as a formal gadget and imagine it the high school way as spaces cut out by equations. Now I believe that the functor of points POV is, not just formally more powerful, but easier and closer to the best intuitive (=high school) way of looking at varieties. What is the variety cut out by equations f_1,…,f_m? It’s the set of simultaneous solutions to these polynomials in R^n, for R a variable ring. That’s exactly what the definition should be, and you could explain it to anyone who knows what a ring is.
Becoming comfortable with the whole functor of points formalism of course requires more effort, but I think if it were properly taught, all that would actually require less effort than with the ringed spaces formalism.
Dear James,
I agree that there are several related questions. Regarding “locally ringed spaces” versus some other way of introducing schemes (i.e., as locally representable functors on the category of rings with the Zariski topology), I still consider the “Brown-Gersten-Quillen” spectral sequence as very good motivation. Whichever approach to schemes we choose, and whichever approach to K-theory we choose, I think we would agree about the higher K-theory groups of a Noetherian scheme, as Abelian groups. The BGQ spectral sequence has for (p,q)-term a product over the codimension-p points of the K-theory of the residue field at that point. So the spectral sequence already encodes a lot about the locally ringed space of the scheme: the point-set, the codimensions of the points, and their residue fields. As David points out, this spectral sequence is an instance of a more general construction. But I still don’t see how one would naturally introduce this spectral sequence without first introducing the locally ringed space of a scheme (of course, I would love to learn if there is a way to do it). There are some similar spectral sequences in Grothendieck’s Brauer exposes (and I think also in SGA 2 somewhere).
I mentioned Hartshorne’s book because, if one decides that locally ringed spaces are the way to go, then I think Hartshorne is the textbook which most quickly gets students up-to-speed with that approach.
Of course that is just my opinion, heavily influenced from first learning algebraic geometry from Matt and Brian Conrad. But I also teach my students about the functor of points at the earliest possible moment. And I try to teach about algebraic spaces and algebraic stacks as well, although with limited success. Johan is running a student seminar on stacks for the second time this semester. Maybe my opinion will completely change after that.
Best regards,
Jason
Dear James,
It’s not that I think that the functor of points POV is intrinsically harder (although I think your “analyst down the hall” scenario is slightly rosy, just because, although the operation of forming GL_2(R) is very simple, the idea that the object itself *is* this operation is something that I think could be hard to get your head around if you hadn’t thought about it before). What I am curious about is the following:
Properness is usually defined for schemes in terms of the concept “universally closed”, which uses the topological spaces that you want to avoid. Now one can rephrase this via the valuative criterion, which is pure functor of points. But can all results about properness be proved just using the valuative criterion?
Another concept that comes to mind which superficially uses the underlying topological space of the scheme is arguments via specialization or generalization. Presumably, though, these are not hard to rephrase functorially.
My point is that if there are arguments along the way that really use the topological spaces associated to non-affine schemes, then one has to develop this theory as well as the functor of points POV. But if one can really make all the arguments without this, I think it would interesting to record this fact, and to record some of the arguments.
Incidentally, I hadn’t realized that non-closed points were such a pedagogical issue before taking part in this thread; despite the sentiment of Joel’s original post, they always seemed quite natural to me.
Cheers,
Matt
P.S. The fact that a variety or scheme is just the high-school notion of a system of equations is something that I try to emphasize from the beginning. But actually, I think this is a difficult thing to convince students of (and not just because of any particular choice of formalism — I think that there is a tendency to psychologically separate high school and early undergraduate mathematics like calculus, linear algebra, and analytic geometry, from the “sophisticated” mathematics of upperclass undergraduate and graduate courses, which is not easily overcome).
I had a question stimulated by the K-theory discussion: Is there a notion of Krull dimension at a k-valued point of a functor F: (k-alg) -> (Sets) for an algebraically closed field k, say when F is a locally finitely presented fppf sheaf (but not necessarily formally smooth)? Ideally, I’d like a notion which spits out the Krull dimension of F at the image point when F is a scheme.
“Properness is usually defined for schemes in terms of the concept “universally closed”, which uses the topological spaces that you want to avoid. Now one can rephrase this via the valuative criterion, which is pure functor of points. But can all results about properness be proved just using the valuative criterion?”
I was also curious (and sorry if this is irrelevant): Is it possible to define a notion such as quasi-compactness of morphisms, which is quite natural via topological spaces, easily using the “functor of points” approach?
This has been an enormously fun discussion to read (or, more honestly, skim). I lament only that Danny Calegari has hijacked the “ignoramus” moniker — for if he’s that, what am I? Let’s go with “doofus.”
This doofus never got a proper foundational grounding in algebraic geometry (or any other math subject) and can appreciate all the thought that is being put into what the students might need down the road. But… what do you want to *do* in your course? Why not just work backward from there? If the answer to this question truly is, “set the foundations for algebraic geometry in its most broadly applicable setting,” then go slowly and start with categories/functors/representability (or whatever). If the answer is, “explain the Kodaira embedding theorem” then start with manifolds and complex structures — then maybe reprove it from another perspective for students who will advance.
I guess I’m just saying introduce the “hard” stuff only if you’ll need it (I know this repeats many previous thoughts in the thread).
No calculus student needs to learn the vectors-as-differentials point of view. Further, every student familiar with multivariable calculus can prove the equivalence between vectors and differentials whenever s/he needs to. Worldviews can change over time, but learning must needs be connected to familiar objects, whatever those may be.
Doofusfully yours,
-Eric
p.s. Mirror symmetry comment. The flexibility of symplectic techniques in mirror symmetry is really cool. But the cat got loose and became feral. Now there are all kinds of efforts to tame it through the *algebraic* structures of the Fukaya category. Algebra always wins in the end.
Still thinking about how I will try to present algebraic geometry for my intended exponential sums book, here’s another concrete example of question where one formulation seems to me certainly clearer to present, but it’s rather the usual topological/ringed space one: in families of exponential sums over finite fields (and in other things like Katz-Sarnak-type studies of L-functions over finite fields) it is essential to be able to speak of the Zariski closure of a subset of some GL(n,k), where k is a ell-adic field. How easy would it be to do this as a “functor of points” construction? (Note that the subsets in question have no structure a priori, except being subgroups of GL(n,k)).
Emmanuel – I imagine this is not helpful (ie just a tautological reformulation), but I think this can be said functorially as follows: given a subset S of X(k), the k-points of a variety (X=GLn in your case), its Zariski closure is the initial object of the category of closed subvarieties of X with an inclusion of S in their k-points. Or equivalently, Zariski closure is the left adjoint of the functor from closed subvarieties of X (or varieties with a closed embedding into X) to subsets of X(k) (or sets with an embedding into X(k)), so that inclusions of sets S into Y(k) are the same as embeddings of closed subvarieties Closure(S) into Y. (Of course this presupposes we know closed subvarieties in the functorial language, but that’s immediate from the ring-theoretic definition of a closed subsets or Zariski opens, which is part of the functor-of-points package..).
Regarding quasi-compactness and properness, I believe that some properties cannot be made sense of on the category of all functors, though as you point out, Matt, properness can. But in general, I think we need to look at certain subcategories. So I should probably be more specific about what I have in mind. I pretty much think of this as Grothendieck’s point of view, perhaps as interpreted by others, such as Artin. Maybe some flourishes are due to me, but I doubt it.
The basic point is that there are three categories: the category of functors from Rings to Sets, the category of sheaves contained in it, and the category of algebraic spaces contained in that. I think of these as loosely corresponding to set theory, topology, and geometry. Here’s what I mean:
1. Let Aff denote the opposite of the category of rings. Call the objects “affine schemes”.
2. Put the etale topology on Aff. For many purposes, you could probably also use the Zariski topology. (It’s not necessary at this point, but note that etaleness can be easily defined for any functor X from Rings to Sets. This is using the nilpotent lifting criterion for formal etaleness and the commutation with filtered colimits criterion for locally of finite presentation.) Note that we have to make sense of an etale cover here. I’m not sure what the best way of doing this is. You could use prime ideals of rings, but that goes against the spirit of what I’m proposing. You could also use faithful flatness. Since we’re just talking about the surjectivity of etale maps, there might be a nice way of defining cover that I haven’t thought of.
3. Define the category of spaces to be the category of sheaves of sets on Aff. (Actually, one would probably want some set-theoretic smallness conditions on such sheaves so we don’t have to work with larger universes. This is alas not addressed in SGA 4.) Any affine scheme represents a functor (its functor of points) which is in fact a sheaf, so we can view Aff as a (full) subcategory of Spaces.
The category of spaces forms a topos, and so we can probably define most properties of schemes we think of as topological at this stage. For example, a space X is defined to be quasi-compact if every covering family has a finite subcover. This answers Akhil Mathew’s question. There is a lot about such topological properties in the first two volumes of SGA 4. For quasi-compactness and quasi-separatedness, see the Grothendieck-Verdier expose VI. In particular open and closed subspaces are defined (exp IV, I think). I would imagine you can define what it means for a map of spaces to be closed.
4. Let AlgSp, the category of algebraic spaces, denote the closure of Aff under disjoint unions and quotienting (in Spaces) by etale equivalence relations. (Every algebraic space in the sense of D Knutson is an algebraic space in the sense here, and the converse is true if and only if the space is quasi-separated. Also, note that once you add all disjoint unions, the quotienting procedure terminates in two steps, I think. So they really are accessible. After the first step, you get the algebraic spaces with affine diagonal.)
5. Define an open immersion of algebraic spaces to be an etale monomorphism. (This definition can be stated for general functors, but I don’t know if it’s reasonable in that generality.) Define a scheme to be an algebraic spaces with a covering by open immersions from affine schemes.
Thus we have schemes and open and closed subsets, so I would expect that any topological property in scheme theory could be defined from this point of view, though I’d have to think about closed morphisms. One nice thing about this approach is that because our underlying “set theory” (i.e. functors from Rings to Sets) is in some ways quite rich, the passage from set theory to topology to geometry is made by adding *properties* to our “sets” (=functors), rather than structure, as is the case with differential geometry in the usual sense. So defining GL_2 is easy, but the more sophisticated the questions you ask, the deeper you’ll have to go into the theory. So the scenario of the analyst down the hall is a bit rosy, I admit, but I think there is some genuine truth to it.
Now, you might be tempted to read this as a satire on the categorical point of view. A thousand pages of SGA 4 to eliminate prime spectra of rings! A few comments on that: First, just as you don’t need a definitive tome on general topology to teach scheme theory, you could probably introduce the general sheaf theory using only a fraction of the account in SGA 4. That said, I could easily imagine that if I were to teach such an algebraic geometry class, I would retreat quite a bit because students know topological spaces and I wouldn’t want to spend the whole semester on sheaf theory (actually, I might…).
But I do believe that, in principle, this is the right way of thinking about the foundations of the subject, and so one should try to teach as close to this point of view as is reasonable. But some people who know the details of the big arguments in EGA much better than I do disagree with me. Probably Jason and Matt know them better than I do. Certainly Brian Conrad does, and I don’t think he’ll mind me saying that he thinks I’m crazy (presumably only about this). I suppose the only way I’d be convinced otherwise is to try to teach a class and fail, the flip side to the remark that the only way to convince the unbelievers is to teach a successful class.
Finally, another thing I like about this approach is that it would clearly work for categories much more general than the category of commutative rings. Indeed, Toen-Vaquie have a paper “Au-dessous de Spec Z” about these things. In particular, I think it would be interesting to look at semi-algebraic geometry from this point of view. Also, this approach doesn’t use the axiom of choice, because we don’t need to have prime ideals to make our underlying space: we just need the ring itself. Now, at a practical level, I don’t actually care about these two issues, but it is my experience that the way of understanding things that generalizes most cleanly, that uses no “recondite axioms of set theory”, is usually the right way and that understanding things in general improves your point of view, even if you only want to work in a special context.
Dear Jim,
Thanks for the nice survey of the category of points set-up.
Regarding Akhil Mathew’s question, let me take one more step, just to be completely explicit:
a morphism X –> Y is quasi-compact if and only if the preimage of any affine is quasi-compact, if and only if its fibre product with any map S –> Y, where S is affine, has affine domain. Since we can define fibre product in the category of functors (the categorical definition of fibre product just means that we define the fibre product of functors in a “point-wise” fashion), and since Jim above defined what it means for a given space to be quasi-compact, we can define what it means for a morphism to be quasi-compact.
(More generally, since fibre products are, by their very nature, something that fits very well in the functor of points POV, once you can define an absolute notion (e.g. quasi-compactness or affine), the corresponding relative notion is very easily defined in the functor of points POV.)
Also, it is worth pointing out for any non-experts still reading that the exercise of making definitions (and proofs) in the functor of points set-up is an important one (as far as I understand) whether or not one wants to retain the locally ringed space POV on schemes, because often this is what is needed to extend definitions and theorems to the setting of stacks, or non-commutative spaces, or (perhaps?) the setting of derived algebraic geometry.
Cheers,
Matt
There’s still non-experts reading, I think that learning the different ways that experts think about the things they know is not too much less important than learning the details of what they know, and this kind of discussion is perfect for that. I’m glad this post has generated such vigorous discussion, since I’d like to learn more about both points of view :-) It seems (to a non-expert) like there’s quite a lot of machinery required for the locally-ringed spaces POV also. Eisenbud’s “commutative algebra, with a view towards algebraic geometry” is 800+ pages, and one of the goals was to prove all the results assumed in Hartshorne. Of course, not all of it is necessary, but I’d assume not all of SGA 4 is needed to give the needed results for the ‘functor of points’ POV.
Peter, I think there wouldn’t be much difference with the commutative algebra requirements. The functorial POV might require a little bit less to get the general categories defined (you probably only need some basic facts about nilpotent elements to prove a few basic facts about etale ring maps). But most of the commutative algebra is for doing real work with finite-dimensional varieties, and I don’t think that would be any different from the functorial POV. You might save what you need to prove that the opposite of the category of rings is equivalent to a full subcategory of the category of locally ringed spaces, but I doubt much more than that.
It seems to be very hard to find a good way to teach algebraic geometry. The main problem is that it is impossible to get very far into the material, even in a year long course. Every time I teach it I use a different approach. I have had most success by teaching a first semester course on varieties over algebraically closed fields with closed points only (usually a la the red book), and to follow this up with a second semester on schemes.
Note that Hartshorne’s book on algebraic geometry has the same structure. I claim we would not have such a successful and active field of research had Hartshorne not written his book. It is a marvellous book.
I have recently taught an algebraic geometry course where the first semester was spent teaching commutative algebra, and the second semester was a course on schemes. This did not work as well, perhaps because I was not able to get as far as with the other method.
In any course on schemes you will mention Grothendieck and the functor of points, fibre products, the notion of base change, etc. In fact, this is how most students are introduced to the “functorial POV”. You can start convincing them that this is useful in the discussion of separated and proper morphisms (where you show that certain diagrams are fibre product diagrams by thinking in terms of the functor of points).
The next layer in the story are algebraic spaces. Depending on their current topic of research an algebraic geometer may not need to use, or know about these at all.
Why are algebraic spaces important? My favorite example is the following: Take a degree d > 3 surface X in P^3 with a single ordinary double point and otherwise smooth. Let X’ be the resolution of singularities of X. Then there exists a flat proper family Y –> T of algebraic spaces over T = Spec(dvr) with Y_0 = X’ and Y_\eta a surface of degree d in P^3 with general moduli (in particular with Picard number 1). However, the algebraic space Y is not a scheme.
After algebraic spaces one introduces algebraic stacks, in order to coherently think about moduli problems, such as moduli of curves, surfaces, etc. But note that for certain questions, involving moduli you can find substitutes for arguments that would seem to require knowing about algebraic stacks, for example if you want to prove that there exist nontrigonal curves of a given genus, then you can get by with naively counting moduli. The key is to think about the collection of all families of curves — which is also how the algebraic stack M_g is defined theoretically.
After having introduced algebraic stacks you can start to think about higher algebraic stacks, noncommutative spaces, etc.
It seems madness to try and teach algebraic geometry starting with the category of rings, and then introducing functors, sheaves, etc. For example, without introducing varieties or schemes, the Zariski topology may seem artificial. Why consider etale maps if you do not know that a variety of dimension d is in reality a 2d-dimensional gadget? In fact, introducing etale ring maps and proving even the most trivial properties is difficult and takes lots of lectures. Explaining what those properties mean will be hard if you have not previously introduced any geometry. Finally, I think the devil is in the details and that it would be very hard to actually prove anything geometrically interesting even in a year long course using the “functorial POV”.
What I am really trying to say is this. If you are an algebraic geometer you probably enjoy the layering and abstraction that we do in our field — roughly in the order I sketched above. You likely enjoy the fact that there is a lot of commutative algebra underpinning algebraic geometry, and also that there are spaces, sheaves, and cohomology. Once you learn about Artin’s work on algebraic spaces you are amazed at how a simple list of conditions on a functor encapsulates the notion of an algebraic space. You enjoy reading Schlessinger’s article. You admire how Deligne and Mumford introduced algebraic stacks as a good way of thinking about moduli of curves, and you love how it fits with your notion of a family of curves. And so on.
Getting rid of one of the layers is not a good idea IMHO, especially when teaching students. I often find myself telling students: “You have to know everything!”
Thanks to David (Comment 52) for ways of describing the Zariski closure in functorial form. I like it in an abstract way, but it seems quite clear to me that this will not be the right way to present things optimally for a book for analytic number theory…
Thanks to James Borger and Matthew Emerton for describing how quasi-compactness can be expressed functorially. Perhaps this is a better approach for generalizability.
I will also take a look at SGA 4.
I claim we would not have such a successful and active field of research had Hartshorne not written his book.
(Off topic)
My father once described to me what it was like doing algebraic geometry before [Hartshorne] (and before Xerox machines). He would go to conferences, tell people he was at MIT, and see naked envy in their eyes that he didn’t have pore through 6th-generation mimeographed notes — he could learn algebraic geometry by just breathing the Boston air.
Hey, if y’all are going to *insist* on continuing this conversation – which, as I remember it, started at least as far back as that party at BZ’s place, in Berkeley back in ’98 – well, then we should at least be at Rivoli or something. I may have to pore over all these posts one more time, but I’m pretty sure Jim said he’d buy (as long as Danny springs for the wine…)
Well, this blog was intended to be a continuation of drunken conversation by other means….
When studying a new subject, the first thing that (a majority of) people learn are the formal definitions and how to manipulate them. As time progresses, we start to develop an intuitive feel for the material, and finally, we create our own internal stories for what is “really” going on. But these internal stories don’t make any sense without the hard work that went into creating them. It’s tempting to imagine that with the help of a few well placed examples we can impart all our hard won secrets about algebraic geometry (or any other subject), but I think this is misguided. I’m not quite prepared to say that teaching an algebraic geometry course from the functor of points POV is falling for this trap, on the other hand, I think we can all agree that a certain book by “Geometrix the Gaul” (or R.H. for those unfamiliar with the work of Goscinny) has lots of great exercises.
The first time I thought about a scheme was while looking at Mazur’s Eisenstein Ideal paper. He draws a “picture” of Spec(T), where T is the Hecke Algebra. (T is finite over Z, and one may as well imagine it to be Z[x]/(x^2-px) for this discussion.) For me, the cartoon of two copies of Spec(Z) intersecting transversely at p was illuminating and instructive, and it’s also a picture that is ultimately relevant to the decomposition (up to isogeny) of the Jacobian of a modular curve.
Later on, I learnt about finite group schemes (over affine bases, so we are still talking about affine schemes here). It is somehow obvious in this context to think about such objects as functors — even though the underlying rings T may be very similar to the ones considered above. All of which is to say that context is surely everything when deciding what perspective to consider, and understanding every approach is useful.
In response to Dipankar, _any_ conversation would be better at Rivoli, my friend.
OK, a real post: why don’t the Shafarevich books come up at all in these discussions? We all know a few Russians who learned this stuff reasonably well – did they start with Shafarevich, or did their adviser also tell them “first do all the problems in that R.H. book, and then we’ll talk” (something i never did, btw, which is why my knowledge of AG never budged beyond G&H). Or did they all read EGA/SGA on their mother’s knee like we were supposed to?
I always fantasized about spending a year doing the problems in Harris’s book, followed by going thru Shafarevich’s books, but that fantasy never got airtime during waking hours. Of course, this is along the lines of my other fantasy from those days, which was to go thru C.L. Siegel’s three-volume series… which is I guess the antipodal stance from the Arakelov/Lang thing. Ah well…
Hmm, I see now that Jason Starr did indeed mention Shafarevich in passing…
Dipankar! I’ve never looked at Shafarevich’s books, so I’m afraid I don’t have anything to say. (And I think you might be mistaken about the origins of this conversation, because as far as I can remember, no one talked about anything then but hair.)
I think that both Johan and Anonymous have made thoughtful comments about the pedagogy of algebraic geometry. It’s a little depressing (from the point of view of a teacher of algebraic geometry) that both of them conclude with some version of “you have to know everything”.
On the other hand, despite the comment of Anonymous that it is “somehow obvious” to think of (say) finite flat group schemes as functors, I’ve had the experience of watching people battle with Hopf algebras to prove statement that were fairly evident from the functorial point of view. I think that there is something to be taught about the effective use of the functorial viewpoint, and this viewpoint is underrepresented in Hartshorne. (This is not a criticism of Hartshorne — it is already quite a tome, and it can’t do everything.) I learnt the functor of points viewpoint myself by reading Grothendieck’s Bourbaki seminar about faithfully flat descent. He states that this is obvious when the faithfully flat map has a section, and then gives a simple argument via the functor of points. I remember spending a long time trying to resist this argument, and instead battling with legions of diagrams in the category of schemes, until I became convinced that the Yoneda viewpoint was the best one to adopt.
It would be nice to have a more text-bookish discussion of the functorial point of view, with examples and exercises, not (at least to my mind) as any kind of replacement for Hartshorne, but as another resource for students. (My experience with algebraic geometry students is that they’re grateful for whatever resources they can get.)
After all, if they have to learn everything, the more resources that are available, the better.
P.S. When I wrote that “the Yoneda viewpoint was the best one to adopt”, I meant for the particular issue at hand. (Effectiveness of faithfully flat descent data in the presence of a section.)
Regarding Matt’s “text-bookish discussion of the functorial point of view”, what do people think of Waterhouse’s book Introduction to Affine Group Schemes? He starts right away with functors of points. It’s a book on algebraic groups, so doesn’t really deal with varieties much, but it is at least approximately the group-theoretic analogue of the kind of introduction I’ve had in mind. Has anyone tried to teach a class using it? While it might be hard for students on their own, it might be reasonable if they have access to someone who is comfortable with that point of view.
Like James, I’m curious as to what people think of the book
by Waterhouse and especially its pedagogical value. There
is no doubt that the book outlines an attractive spectrum
of ideas in relatively few pages, but with few big theorems
proved completely along the way. I’ve always been reluctant to recommend it to graduate students, partly because of my own interest in characteristic p and
the role of quotient varieties such as flag varieties. The
somewhat impenetrable book by Demazure-Gabriel is
longer and more detailed but also fails to get very far
toward current research. (It costs a fortune, too.)
So Danny Calegary is the ignoarmus and Eric Zaslow is the doofus. Then what I am? I shudder just to think about it.
As someone who remembers vividly what I felt when I tried to learn algebraic geometry, I will just mention that I, for one, like prime ideals (which are not maximal), because they are so important in my trade. Had I learned algebraic geometry the Yoneda way, or even the spec max way, I wouldn’t have found a lot a what motivated me to learn in the first place.
But this emphatically doesn”t mean that either ways are bad; it is just a reminder that different students with different tastes, backgrounds and especially learning habits will react differently to different material. I think algebraic geometry might be particularly hard to teach in that respect because students will typically expect something different from the course, depending on their main research interests.
If someone already ancient can intrude, I’d like to make a couple of points.
(a) In my view one should first learn algebraic geometry in the context of algebraic varieties over algebraically closed fields. Someone with a very strong background in commutative algebra can read Hartshorne without much difficulty, but it is possible to do this without acquiring much geometric intuition.
Also I think algebraic varieties should be defined as ringed spaces from the start (i.e., maxspecs) for a number of reasons: it’s actually easier; someone who has learned to think of manifolds as ringed spaces will find that he already understands much of the theory of algebraic varieties; it makes the transition to schemes easier; etc..
In fact, many algebraic geometers write “scheme” but think “closed points” — for example Lazarsfeld in his two volume work “Positivity in Algebraic Geometry” writes in Notations and Conventions: A scheme is an algebraic scheme of finite type over C… We deal exclusively with closed points.
However, for other algebraic geometers, and all arithmetic geometers, schemes (including nonclosed points) are vital.
(b) For a while in the 1960s, true believers *knew* that the *only* way to introduce schemes was as functors, and there are several horror stories from students who attended such a course.
Nevertheless, I do think that the correct way to define affine (algebraic) groups is as functors. For example, what *is* SLn? A commutative Hopf algebra with antipode? A ringed space whose underlying space has nonclosed points and is not even a group? I think not. It is something that, when applied to a commutative ring R gives the group of nxn matrices of determinant one, in other words, it is a functor.
Waterhouse’s book is a fine work, but it doesn’t get very far. The heart of the theory of algebraic groups after all is the study of semisimple algebraic groups (roots and weights), which isn’t covered. Incidentally, I have taught a course on algebraic groups using the functorial point of view, which (I think) went quite well.
Regarding algebraic groups as functors allows you to go a long way without using much algebraic geometry. In the book I’m working on, I plan to assume little algebraic geometry in the first three Chapters (only what’s in my notes on Commutative Algebra), but then use whatever algebraic geometry (including schemes) I want to in the last three.
Incidentally, some of the links to my website in the above discussion have already changed — you should always go to the top.
James Milne
In my opinion, one main benefit of working with varieties (reduced, and finite type over an algebraically closed field) is that morphisms between them are completely determined by the set-theoretic maps of their closed points. So being a morphism is a property of a set map rather than supplemented structure, which is the case in rings, groups, topological spaces, differential manifolds, complex manifolds, etc. Because of the fully faithful functor to schemes, one doesn’t have to worry about non-closed points or specifying a pull-back map of structure sheaves. So both pedagogically and in practice, varieties have an important role.
Joel, as for your question of Spec-m as a adjoint. I think that if you restrict yourself to locally ringed spaces where the structure sheaf is a sheaf of finite-type k-algebras, and all residue fields are k, then Spec-m will be adjoint to the global section functor.