What should be learned in a first serious schemes course? - MathOverflow

What should be learned in a first serious schemes course?

2010-06-17T12:49:42Z

I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons (most of all the students, who were smart, hard-working, and from a variety of fields). I've particularly enjoyed talking with experts (some in nearby fields, many active on mathoverflow) about what one should (or must!) do in a first schemes course. I've been pleasantly surprised to find that those who have actually thought about teaching such a course (and hence who know how little can be covered) tend to agree on what is important, even if they are in very different parts of the subject. I want to raise this question here as well:

What topics/examples/ideas etc. really really should be learned in a year-long first serious course in schemes?

Here are some constraints. Certainly most excellent first courses ignore some or all of these constraints, but I include them to focus the answers. The first course in question should be purely algebraic. (The reason for this constraint: to avoid a debate on which is the royal road to algebraic geometry --- this is intended to be just one way in. But if the community thinks that a first course should be broader, this will be reflected in the voting.) The course should be intended for people in all parts of algebraic geometry. It should attract smart people in nearby areas. It should not get people as quickly as possible into your particular area of research. Preferences: It can (and, I believe, must) be hard. As much as possible, essential things must be proved, with no handwaving (e.g. "with a little more work, one can show that...", or using exercises which are unreasonably hard). Intuition should be given when possible.

Why I'm asking: I will likely edit the notes further, and hope to post them in chunks over the 2010-11 academic year to provoke further debate. Some hastily-written thoughts are here, if you are curious.

As usual for big-list questions: one topic per answer please. There is little point giving obvious answers (e.g. "definition of a scheme"), so I'm particularly interested in things you think others might forget or disagree with, or things often omitted, or things you wish someone had told you when you were younger. Or propose dropping traditional topics, or a nontraditional ordering of traditional topics. Responses addressing prerequisites such as "it shouldn't cover any commutative algebra, as participants should take a serious course in that subject as a prerequisite" are welcome too. As the most interesting responses might challenge (or defend) conventional wisdom, please give some argument or evidence in favor of your opinion.

Update later in 2010: I am posting the notes, after suitable editing, and trying to take into account the advice below, here. I hope to reach (near) the end some time in summer 2011. Update July 2011: I have indeed reached near the end some time in summer 2011.

Answer by Charles Siegel for What should be learned in a first serious schemes course?

2010-06-17T13:35:48Z

I actually think that the Hilbert scheme should be mentioned (and, if possible, proved to exist and discussed) as early as possible. It serves as a good example of a moduli space, and it exists! Plus, the infinitesimal study of the Hilbert scheme allows some deformation theory to be discussed (at least, the deformations of projective schemes inside projective space) which also helps explain, algebro-geometrically, what the normal sheaf really controls. Add to this the fact that a lot of research relies on moduli spaces these days (In particular, I know that people care about Hilbert schemes of points, and, if some GIT for PGL can be covered, it'll let you actually construct $\mathcal{M}_g$, which finishes the classification of curves that's given in chapter 1 of Hartshorne, though this is a bit more.)

Because you'll be wanting things fundamentally scheme theoretic, the first part of Kollár's "Rational Curves on Algebraic Varieties" might be a good reference for this stuff.

Answer by Georges Elencwajg for What should be learned in a first serious schemes course?

2010-06-17T14:45:33Z

Dear Ravi, here is a small suggestion. I think one might emphasize as soon as possible that the subschemes of an affine scheme $Spec A$ exactly correspond to the set of ideals of the ring $A$.( I don't know if this is deep or tautological: probably both.) This allows one to illustrate many of the strange and frightening features of scheme theory as compared to tamer geometric structures (that subschemes are not determined by subsets, that functions are not determined by their values, etc) without adding the complications due to sheaves and gluing. I remember it took me a long time to realize this and when I did I lost some of my fear of schemes.

Answer by David Speyer for What should be learned in a first serious schemes course?

2010-06-17T14:57:35Z

I found differentials hard to understand when I learned this material. Here are two things that helped me which I think are not in your notes:

(1) The description of the Zariski tangent space to $X$ at $x$ as those Hom's from $\mathrm{Spec} \ k[\epsilon]/\epsilon^2$ to $X$ which take $\mathrm{Spec} \ k$ to $x$. This is this is much closer to my physical intuition for a tangent space than the $(\mathfrak{m}/\mathfrak{m}^2)^{\vee}$ definition. It is also an early example of the power of using rings with nilpotents. Building the vector space structure from this definition is especially pretty.

(2) A careful discussion of the relationship between the infinitesimal objects, i.e. the elements of the Zariski tangent and cotangent spaces, and the global objects, i.e. derivations and Kahler differentials.

Answer by David Speyer for What should be learned in a first serious schemes course?

2010-06-17T15:01:44Z

As you should, you prove the Nullstellensatz early on, as the statement that the closed points of $\mathbb{A}^n_k$ are in bijection with $k^n$, for $k$ an algebraically closed field. I wonder whether it is also a good idea to say that, for any $k$, the closed points of $\mathbb{A}^n_k$ are in bijection with the Galois orbits in $\overline{k}^n$. This might require too big a digression into Galois theory, but I remember a number of my grad school classmates having confusions about closed points over non-algebraically closed fields which could be immediately answered from this decsription.

Answer by Kevin Lin for What should be learned in a first serious schemes course?

2010-06-17T17:53:53Z

I'm not sure if this is the kind of answer you're looking for but...

There is a very useful and simple lemma on sheaves which is (I think) never explicitly stated in Hartshorne. It is Proposition I-12 of Eisenbud-Harris. I think you should definitely make sure to explicitly state this. Sheafification was very scary and mysterious to me until I learned this lemma.

Answer by Lars for What should be learned in a first serious schemes course?

2010-06-17T18:12:22Z

To prepare well for what comes after a first course, a more extensive discussion of étale morphisms than Hartshorne gives should be part of such a course, in my opinion.

Answer by Andrew L for What should be learned in a first serious schemes course?

2010-06-17T18:28:53Z

First off, I wanted to commend you on this whole project, Ravi. Algebraic geometry and the theory of schemes is a notoriously difficult subject to internalize for any advanced student and it's clear you've given a lot of serious thought on how to make it more digestible. I've browsed the old version of the notes and found them very readable and highly thought out. I firmly support this project and hope it goes through many revisions and drafts, evolving into a future classic. Algebraic geometry is a subject I haven't seriously begun broaching yet and I hope to use one of the newer versions when ready,

Secondly-I sympathize with your hesitancy to convert them into a book. What you might consider is creating an online text that will constantly be revised and will never be in "final" form. My old biochemistry professor Burton Tropp did this for many years and it worked out for him very well: The first edition WAS published, but all subsequent editions (and there was nearly a dozen before he retired last year) were online and subject to constant revision and improvement. I think this kind of format will work very well for you.

Thirdly -- history is so important in learning a new,conceptually difficult field. Some good historical notes would make the notes a lot more interesting to read no matter how good the exposition is. Students want to know how they came up with this crazy stuff -- if you know how the original source authors came up with these concepts and why, it'll make it a lot easier to not only internalize, but also to form thier own opinions on the subject.

Fourthly -- I think inserting references and research assignments relying on significant papers, such as Grothendiek's original schemes paper -- will give your students some much needed research experience in a very active field. These are advanced students and the more such experience they get,the better off they'll be.

Lastly -- I wanted to commend your humility and determination in asking other mathematicians and students for opinions and input on this project. It shows how committed you are to this project and experts should be chomping at the bit to give you thier feedback and opinions. I would, but my lack of expertise precludes that. Hopefully those with much more knowledge then I will jump at the chance to assist you with this wonderful project.

Good luck with this exciting project and looking forward to future versions!!!

Answer by Allen Knutson for What should be learned in a first serious schemes course?

2010-06-17T18:52:40Z

Toric varieties. They're so easy to define and work with, and to organize examples around. Like blowing up a scheme at a fat point, or blowing up in different orders, or big but not ample line bundles, ... Of course there's the danger that they'll give people the wrong idea about what general schemes are like, but a few curves-of-high-genus examples should help with that.

Answer by Allen Knutson for What should be learned in a first serious schemes course?

2010-06-17T18:54:53Z

Serre's criterion for normality, the valuative criterion for normality, normality vs. S2, maybe even seminormality.

Added by request: here's how I think about Serre's criterion. Call a rational function pretty good if it doesn't blow up in codim 1. Call it very good if it's actually well-defined in codim 1. Then a normal space is one for which pretty good rational functions are actually functions, whereas an S2 space only asks that very good rational functions are actually functions. To see the difference, look at x/(x+y) on {xy=0}, to see that the latter is not normal despite being S2. So how can normality fail -- how can f's value be ambiguous in codim 1? If there are 2 ways to approach some divisor -- non-R1ness.

Answer by Kevin Lin for What should be learned in a first serious schemes course?

2010-06-17T19:08:40Z

I think analytification and GAGA should be mentioned.

Subsequently I think that it should be mentioned that Serre duality can be viewed as a refinement of Poincaré duality.

Answer by Georges Elencwajg for What should be learned in a first serious schemes course?

2010-06-17T22:02:27Z

Since in 2007-2008 you evoked [ Class 24, §1.8, The problem with locally free sheaves] the equivalence between locally free sheaves and vector bundles on a scheme, the following point, potentially confusing for a beginner, could be mentioned.

A locally free sheaf $\mathcal E$ has a sheaf fibre $\mathcal E_x$ at $x$ but also a vector fibre $\mathcal E[x]=\mathcal E_x \otimes _ {\mathcal O_x} k(x)$. The fact that tensoring is not exact explains the paradox that a locally free subsheaf of a locally free sheaf does not yield a sub-vector bundle of a vector bundle in the above equivalence. The contrasting notation $\mathcal E[x]$ versus $\mathcal E_x$ (that I learned from German mathematicians) may help clarify this subtle point .

I am quite aware that there is nothing grandiose in this technical suggestion, but little points like those can be quite frustrating when learning a new subject

Answer by Ben Wieland for What should be learned in a first serious schemes course?

2010-06-17T23:05:36Z

Resolution of singularities.

That isn't really an answer to the question - I don't think it's necessary in a first course, but I do think resolution should be rotated in on a regular basis, which requires the annual core to be small enough to make room for it. I think it's valuable not just to teach the material somewhere in the curriculum, but to put it in an introductory course, to emphasize that it is elementary and not impossibly difficult. Also, to contrast with the Grothendieck-flavored majority.

Answer by Johnson-Leung for What should be learned in a first serious schemes course?

2010-06-18T00:05:55Z

I think that base change is a very important and subtle idea which should certainly be included in a first course. In particular, one should discuss properties that are stable under base change and those that are not.

In a similar vein, in discussing cohomology, the difference between the coefficients of the motive and the base should be emphasized. This was confusing to me as I learned the subject.

Answer by Allen Knutson for What should be learned in a first serious schemes course?

2010-06-18T02:41:34Z

One of the wholly unnecessary reasons that schemes are regarded with such fear by so many mathematicians in other fields is that three, largely orthogonal, generalizations are made simultaneously.

Considering a "variety" to be Spec or Proj of a domain finitely generated over an algebraically closed field, the generalizations are basically

Allowing nilpotents in the ring
Gluing affine schemes together
Working over a base ring that isn't an algebraically closed field (or even a field at all).

For many years I got by with only #1. More recently I've been interested in #1 + #3. Presumably someday I'll care about #2, but not yet. Anyway I think it's crazy to give the impression that the three are a package deal that one must buy all of simultaneously, rather than in much easier installments.

I think it could be useful to explain which subfield of mathematics, or which important example, motivates which of #1,#2,#3 is really a necessary generalization.

Answer by David Speyer for What should be learned in a first serious schemes course?

2010-06-18T10:51:16Z

Victor Protsak suggests this, and I'll endorse it: the careful construction of the Grassmannian. This is a good example for 3 reasons. (1) It is extremely important. (2) It is a situation where it is both natural to work in local coordinates and with a global projective embedding, so students can practice transforming between the two perspectives. (3) It is small enough to do in full detail, but it usually isn't.

Ideally, this would include proving that the Grassmannian represents the functor of flat families of subspaces of a vector space. (Or quotient spaces, whichever you prefer.)

Answer by Georges Elencwajg for What should be learned in a first serious schemes course?

2010-06-18T11:39:35Z

If you decide to teach a more arithmetically flavoured algebraic geometry, students should be made aware that schemes over a ring $A$ are stranger than they might think.

For example $A$-rational points of $\mathbb P^n_A$ are far from being given by non-zero $(n+1)$-tuples of n elements of $A$ modulo tuples of invertible elements, but are described by rank-$n$ projective summands of $A^{n+1}$. More generally morphisms to projective space are described in terms of line bundles and their sections and might be seen as an interesting illustration of these concepts.

Incidentally, a sufficient reason for introducing a little arithmetic geometry is to have the pleasure of reproducing Mumford's incredibly enlightening drawing of the arithmetic surface $\mathbb A^1_{\mathbb Z}$ (in his Red Book), with its points having each a diferent personality and its curves. ( I concede that although Mumford's picture is beautiful, the artistic competition was not so great when he wrote his notes: the EGA's strongest point is not its illustrations...)

Answer by Emerton for What should be learned in a first serious schemes course?

2010-06-18T14:27:04Z

Let me begin with something essentially obvious: students should learn to work with non-closed points. In practice, this means learning how to use them to simplify life.

Here are some suggestions as to how to do that:

(a) Explain that coherent sheaves are generically free, and use this to prove things like generic smoothness of varieties (by applying it to the tangent sheaf).

(b) Explain carefully the proof of Chevalley's theorem that the image of constructible is contstructible. (Note that this latter result has the advantage of being extremely useful, and also has likely not been covered in any form in a previous varieties course.)

Note also that one can deduce the Nullstellensatz from this result, which kills two birds with one stone. (See the discussion in this answer, and the notes of Mumford and Oda that are linked there.)

(c) one can beef up (a) by looking at say a fibration $X \to Y,$ and then looking at fibres over a generic point of $Y$, and then extending information to a n.h. of that point. Incidentally, it was the desirability of this kind of argument that first led Zariski to point out the importance of studying algebraic geometry over non-algebraically closed fields. For him, these non-algebraically closed fields were not $\mathbb Q$ or $\mathbb F_p$, but rather function fields of varieties (with the initial ground field being a good old fashioned algebraically closed field).

Examples like this last one can really help demystify not just the role of generic points, but also the role of non-algebraically closed fields. (In particular, they show that the latter are not just of interest in number theory. Zariski was certainly not a number theorist!)

Answer by Steven Gubkin for What should be learned in a first serious schemes course?

2010-06-18T16:01:44Z

I expressed my frustration with Hartshorne's book a bit here:

http://mathoverflow.net/questions/12436/motivation-for-concepts-in-algebraic-geometry

The point is that many definitions in algebraic geometry are basically obtained by taking definitions from topology or algebra, translating them into "purely category theoretic language" and then using that definition as a substitute in the category of schemes.

In particular I unravel the definition of a separated morphism:

"A seperated morphism of schemes is one where the image of the diagonal is closed."

If we just replace "schemes" with "topological spaces", then this property for spaces says (after a little definition chasing)

"Any two distinct points which are identified by the morphism can be separated by disjoint open sets in the domain"

Thus a space is Hausdorff as a topological space iff the unique map to the one point space is separated. Before I worked through this I had no real reason to believe that separated morphisms were a natural concept. Why don't people ever talk about the topological analogue?

Another point of much confusion for me was the definition of derived functor cohomology. Why should we care about injective resolutions? Anton gives a great answer here:

http://mathoverflow.net/questions/1151/sheaf-cohomology-and-injective-resolutions/1165#1165

Anton's line of thought is also beautifully developed in Gunter Harder's book "Lectures on Algebraic Geometry 1". The quick and dirty version is that cohomology should have nice properties (ses gives rise to les, etc) and acyclic resolutions compute cohomology. Hey! Injective objects are always acyclic (this is reasonable because they make ses's split). Thus injective resolutions are a nice generic thing to use.

Answer by Andrew Critch for What should be learned in a first serious schemes course?

2010-06-18T20:19:11Z

Stalk-local detection of irreducibility on locally Noetherian schemes, which I prove directly here with no primary decomposition tricks. It helps with a lot of exercises, and intuition.

Sheafification of base-presheaves (presheaves defined only on a base of open sets). I see from your TOC that you cover the unique extension of base sheaves to sheaves as per Kevin Lin's answer (E-H's Proposition I-12).

When I took Arthur Ogus' algebraic geometry class, he was very insistent about teaching us this, and it really paid off for the remainder of the course, particularly in exercises. It categorically exclaims (pun intended) the credo always start with the affine opens, so one sees explicitly how special and critical they are to the theory.

The sheaf of meromorphic functions $\mathcal{K}_X$ on $X$ can be defined by sheafifying the naive base-presheaf $\mathcal{K'}(U)=Frac(\mathcal{O}(U))$ on the base of open affines. This formula doesn't define a base-sheaf on affines, and as Georges Elencwajg and BCnrd explain here, it doesn't even define a presheaf when applied to arbitrary opens. I suggest at least mentioning these three facts, to save people from re-wasting the time that I and many others have in wondering what the resulting sheaf looks like.

Locally representable means representable, i.e. if $F:Sch^{op}\to Set$ is a sheaf when restricted to a base of (Zariski) opens on every scheme, and $F$ has a covering by representable open subfunctors $F_i$, then $F$ is representable (very much along the lines of EGA 1 (1971), Chapter 0, Proposition 4.5.4). I advocate this because the work that goes into the proof is essentially the same work we inevitably do to prove fibered products of schemes exist, so it gives fibre products as a special case, but also offers up a rigorous-but-quick route to other constructions like global Spec and global Proj.

The general definition of quasicoherence and coherence for modules on local ringed spaces / non- locally Noetherian schemes... not as a gratuitous generality, but as a foreshadowing/reminder that presentations, not just surjections, are what make coherence work.

Basic Dedekind domain theory, along the lines of Lang's Algebraic Number Theory, chapter 1. I found curves and their divisors — even in characteristic 0 — impossible to understand until I read that.

Quasiseparatedness is something I'm glad to see you including, because using it explicitly is the key to a lot of proofs, so having it in mind as a word helps me remember how to do them.

Your affine communication lemma is a must-have, for anyone else reading this answer!

Answer by Allen Knutson for What should be learned in a first serious schemes course?

2010-06-18T22:50:42Z

Generic fiber vs. general fiber vs. geometric generic fiber.

Answer by Peter Bruin for What should be learned in a first serious schemes course?

2010-06-19T10:58:48Z

Why the Spec functor is a natural thing; this is not so clear (at least to me) from the definition in Hartshorne. Bas Edixhoven made me see the light by saying that Spec is adjoint to the global sections functor from locally ringed spaces to commutative rings: $\mathrm{Hom}_{\mathrm{Rings}}(A,\Gamma(X,{\cal O}_X))\cong\mathrm{Hom}_{\mathrm{LRS}}(X,\mathrm{Spec}(A))$ . Exercise II.2.4 of Hartshorne asks you to prove this with locally ringed spaces replaced by schemes, but this is less clarifying.

Answer by S. Carnahan for What should be learned in a first serious schemes course?

2010-06-19T17:33:48Z

Perhaps this should be attached to Charles Siegel's answer about the Hilbert scheme, but some concrete examples of degenerating flat families could be helpful. Some easy examples include conics turning into a fat line, skew lines colliding to produce an embedded point, and pairs of points on a line colliding to become fat. There are some nice relationships between these objects and families of constant coefficient linear differential equations via spectral schemes, e.g., the colliding points example says something about the behavior of solutions to $(\frac{d}{dz} - a)^2 - \lambda^2 = 0$ as $\lambda$ hits zero.

Answer by Emerton for What should be learned in a first serious schemes course?

2010-06-19T19:19:21Z

I haven't seen it mentioned yet, so let me suggest it (and I'll be curious to hear people's responses): the theorem on formal functions.

In suggesting this, I am certainly taking full advantage of the fact that we are supposed to be discussing a year long course.

Let me now give justification (in case it is needed; I don't know how others will feel about this suggestion).

First, my own philosophy is that an algebraic geometry course, even one focusing on the theory of schemes, should be about geometry. So I think that it is important to discuss some geometry, including the basic theory of curves (which is very pretty from the schemes point of view, since one gets the interaction between a more geometric picture, and the more valuation-theoretic function-field picture, by studying the interaction between the generic point and the closed points).

But the theory of curves is not enough concrete geometry for one year; I think some discussion of surfaces adds an enormous level of geometric understanding, just because the theory of surfaces is much closer to the theory of arbitrary dimensional varieties than the theory of curves is. At the same time, by doing some stuff with surfaces, one does a valuable service for many students in the class: pure geometers will certainly need to know this, but so will arithmetic geometers/number theorists, because a curve over a Dedekind domain behaves like a surface, and one studies bad reduction of curves using ideas from the theory of surfaces (blowing up, minimimal models, etc.). So even if one doesn't touch directly on the particularities of degenerations of curves (which, however, is also a topic of very general interest and importance!), by saying something about surfaces, one prepares the way.

Hartshorne Ch. V gives a really nice treatment of many of the basics of surface theory, and the main tool he uses, beyond all the generalities of cohomology and sheaves, is the theorem on formal functions: both in its application to Zariski's main theorem, and to the proof of Castelnuovo's criterion. And these are both beautiful results, the kind of results that would make a good capstone to a one year course. (And they are also basic algebraic geometry knowledge --- the kind of things that you would hope students know after taking a year of the stuff!)

Answer by Anweshi for What should be learned in a first serious schemes course?

2010-07-23T20:12:09Z

I am surprised that no one mentioned this so far; I am only imagining that everyone thought it so natural that it escaped their mind.

Most "standard courses" would be following Hartshorne's book, I assume. It is a great loss that this book does not mention the "functor of points" view at all. It would maybe take 10 or 15 minutes to state and prove the Yoneda's lemma, and a little more time to mention the functor of points and the advantage of this point of view for applications to arithmetic geometry(points with values in a certain ring, base change, etc.), and more importantly for moduli problems. One could also give a definition of a fine moduli space and coarse moduli space, and as examples just mention the the moduli space of curves with marked points(but without proofs, of course).

Answer by Charles Staats for What should be learned in a first serious schemes course?

2010-07-23T23:20:01Z

1.$ $ This is really about commutative algebra more than algebraic geometry as such, but something I found incredibly frustrating for a while was what to do when I need to compare $M \otimes_A N$ with $M \otimes_B N$. I finally discovered the following illuminating lemma:

If $M$ and $N$ are $B$ modules, then for every ring homomorphism $A \to B$, there is a natural map $M \otimes_A N \to M \otimes_B N$. Moreover, this map is an isomorphism for all $M, N$ iff it is an isomorphism for $M = N = B$ iff $A \to B$ is an epimorphism of rings.

In particular, the last condition holds if $B$ is obtained from $A$ by some combination of localization and taking a quotient ring, or if $\operatorname{Spec} B \to \operatorname{Spec} A$ is any kind of immersion. The same "abstract nonsense" shows that if $Z \to Z'$ is a monomorphism of schemes (in particular, any kind of immersion), then the product of two $Z$-schemes over $Z$ is naturally isomorphic to their product over $Z'$.

2.$ $ I found the usual description of gluing schemes and morphisms (i.e., requiring things to agree on $U_i \cap U_j$) frustrating to use sometimes, because in general, $U_i \cap U_j$ might not be affine even if $U_i$ and $U_j$ both are. To glue morphisms only, one can require that the morphism be defined on every set of an open cover, such that whenever $x \in U \cap V$, then $x$ has a neigborhood $W \subset U \cap V$ such that $f_V|W = f_U|W$.

For gluing schemes, one can use a commuting poset of open immersions. Given such a diagram, with objects ${U_i}$, there exists a scheme $W$, together with open immersions $U_i \to W$ commuting with the diagram, such that the $U_i$ cover $W$, and $x \in U_i$, $y \in U_j$ map to the same point in $W$ iff they may to the same point in some $U_k$. When this is combined with the statement on gluing morphisms, one sees that $W$ is actually the colimit of the diagram; and, in fact, the statement that "any such diagram has a colimit" more or less encapsulates both glueing schemes and glueing morphisms. Realizing this was also the first time I felt like I understood colimits.

For a more streamlined, if less general, version of the above, one can use a version of the cocycle condition with $U_i \cap U_j$ replaced by a cover of $U_i \cap U_j$ by simultaneously distinguished affines, assuming the cover ${U_i}$ is by open affines.

In either formulation, this combines with the previous point to give a very quick construction of the fibre product: simply take the colimit of the diagram consisting of maps $$\operatorname{Spec} (A \otimes_C B)_{f \otimes g} \to \operatorname{Spec} A \otimes_C B$$ such that the images of $\operatorname{Spec} A$ and $\operatorname{Spec} B$ lie in $\operatorname{Spec} C$. (If these images in fact lie in a distinguished open subset $C_h$, we get the tensor product over that for free by point 1.) Of course, one still has to verify that this colimit behaves as desired; but this is not hard using the more general "gluing morphims" to show existence and uniqueness.

Note: if it's not clear already, my perspective is that of a student rather than an expert.

Answer by ABayer for What should be learned in a first serious schemes course?

2010-07-23T23:56:21Z

(Maybe this is a standard thing to do already but I think it's still worth mentioning:) A proof of Bezout's theorem via Hilbert polynomials of subschemes of $\mathbb P^N$.

Of course, this isn't fundamentally different than the proof in Hartshorne I.7, but in scheme language it is much much more natural, and might be the best motivation for allowing nilpotents in the structure sheaf.
This is extremely vague, but it's something I wish someone should have told me 5 years earlier: Since algebraic geometry is so rigid (few polynomials compared to many differentiable functions), we often have to deal with singularities. E.g. in many cases we can't make intersections transversal, or all interesting families (of certain types) have singular fibers. But since algebraic geometry is so rigid, we also have fairly good tools dealing with singularities, or with degenerate cases.

Answer by Simon Pepin Lehalleur for What should be learned in a first serious schemes course?

2010-07-27T09:58:07Z

A small suggestion : the deformation to the normal cone is a nice construction that I would have liked to see in a first course. It illustrate the use of blow-ups, the degeneration of a family with constant fibers (an highly non-obvious concept the first times you see it) and how important intuitions from differential geometry - tubular neighbourhoods - have a non-trivial translation to algebraic geometry.

Answer by roy smith for What should be learned in a first serious schemes course?

2010-10-31T03:58:10Z

In reference to why the spec functor is a natural thing, (low tech answer): isn't this essentially what the nullstellensatz says? Or rather it generalizes the nullstellensatz. I.e. spec is a good thing because it lets you make a construction that gives you some "geometry" associated to a given ring.

Perhaps the main thing beginners should learn about schemes is that they are needed. I.e. schemes should be motivated. In books which try to restrict to varieties such as Shafarevich's BAG, schemes still raise their heads sometimes unnoticed. E.g. Shafarevich states in chapter I sections 4.4 and 6.4 that the set of hypersurfaces of given degree in a given projective space are parametrized by a projective space, which is not true unless one considers more than the variety defined by a polynomial.

If one is guided on what to include by the section headings of chapter 2 of Mumford's red book, in addition to fields of definition and the functor of points, one finds there a section called specializations, which also contains one of his exotic illustrations.

Even in a classical book like Walker's algebraic curves, schemes arise when studying singularities. The tangent cone to a cuspidal plane curve requires more structure than a variety. Even the fundamental theorem of algebra does not count the roots of a polynomial correctly unless multiplicities are considered.

Some of these examples require only cycles or divisors rather than schemes, but more general tangent cones should provide more general schemes. One can also consider the problem of varieties varying in families and try to fill in something over the limit point of the parameter space. Sometimes non reduced objects will force themselves on us.

The best motivation for differentials may be learning the classical Riemann Roch theorem for curves.

Of course this is probably obvious and taken for granted by most people, but it seemed worth mentioning as a guide to choosing first examples of schemes. I.e. we should not take schemes for granted and choose what to teach based solely on the needs of experts, but we should assume that schemes may be quite strange to beginners and spend some effort showing that they are natural.

Answer by Joël for What should be learned in a first serious schemes course?

2010-12-22T02:36:09Z

At a relatively late point in the course, I believe that the idea of descent should be explained, with two examples: Zariski-descent, or gluing, and faithfully flat descent. The latter should then be applied, for example to prove that some examples of functor of points are representable.

Answer by Charles for What should be learned in a first serious schemes course?

2010-12-22T09:02:06Z

As David and Anweshi told before, think it could be very interesting to deal with functor of points, with main example being subfunctors of Grassmannians. I would make some general statements on functor of points (Yoneda lemma, definition of functor of points, vector bundles) and then begin to study as soon as possible classical examples, such as Grassmanians, Severi-Brauer varieties and their tautological vector bundle, varieties of flag of subspaces...

Finally it would lead to a glimpse on group schemes and algebraic groups.

Answer by Jason Starr for What should be learned in a first serious schemes course?

2011-07-28T13:49:44Z

Dear Ravi,

Here is my suggestion for a list of topics in an intro schemes course: open the table of contents of Hartshorne's book. That is the list of topics that should be covered in a first schemes course. And although the many alternatives to Hartshorne's book all have their selling points, I still feel the best choice for an introductory text is Hartshorne's book. I am not writing this to be contrary to your goal of writing a better foundations book. But I honestly and earnestly believe AG students must learn Hartshorne, and (usually) the earlier the better.