**The fifth post is the October 23 version here. This fortnight’s reading is 6.5-7.5.** By the end of it, you will be able to work with morphisms of schemes. In case you have seen these ideas before, I’ve included two more advanced sections, 7.6 and 7.7 (starred or double-starred), as well as the first section of chapter 8.

I will continue to fall behind on responding to comments. I’ve chosen to post notes and move forward, but you’ll notice that I’m gradually catching up. (I expect to fully catch up in mid-December, and then not fall behind for any appreciable time thereafter.)

**For learners.**

*Exercises:* As usual, what you should do depends on who you are and where you (mentally) are. Likely you should do a subset of these, and then a selection of other exercises. But the following exercises include some key concepts: 6.5.B (meaning of associated prime), 6.5.D (commutative algebra to properties of associated points), 6.5.F (“being 0” is stalk-local), 7.2.A (morphisms of ringed spaces glue), 7.2.C (maps of stalks induced by maps of ringed spaces), 7.2.D (maps of rings induce maps of ringed spaces), 7.3.B (the previous construction is a map of locally ringed spaces), 7.3.F (maps to Spec A “are” maps of global sections in the other direction), 7.4.A (maps of graded rings give maps of Proj’s), 7.5.C (dominant maps are maps of function fields in the opposite direction), 7.5.E (categories of “varieties” and “finitely generated field extensions” are “basically the same”). The following exercises let you get your hands dirty: 6.5.A, 6.5.K, 7.2.E, 7.3.E, 7.4.C, 7.5.A, 7.5.H, 7.5.J (and 7.5.K and 7.5.L are fun). If you reading 7.6, try 7.6.B and 7.6.C

*Some questions for you:* Can you get insight out of section 6.5 on associated points? Are the problems do-able? Are there any hints I should have included to make them easier? (I want these problems to be as easy as possible, yet still be problems. I realize that many of them are *not* easy.) By the end, you should be perplexed by the pictures, but hopefully intrigued and not put off. These pictures are often secret knowledge, never mentioned in writing, because they are so hard to convey outside of a conversation at a blackboard. If you decided try to understand them only in the case of reduced schemes (which is reasonable, and which is even recommended in the text), did that turn out okay for you?

Does the hint make Exercise 7.5.D do-able? I can’t “unremember” what I now know to judge how reasonable this problem is.

**For experts.**

I had earlier hoped to duck associated points, and “star” section 6.5. But I realized that I later needed it, both for modules and for rings. Perhaps it should be starred on a first reading. I’ve tried to make the mathematical development as painless as possible, excluding all but that necessary, either for future use, or for the development of intuition. (I’ve changed the exposition from course notes in earlier years. Primary ideals appear only as an aside!) I’ve also taken the position that what matters most about associated points are their properties, and their definition should be in service of their properties, and not vice versa. (I do this a lot, e.g. for cohomology.) Associated points make sense for locally Noetherian and *integral* schemes, which makes for awkward exposition: we want both. I’m unhappy that rational functions are so hard to explain on locally Noetherian schemes, and don’t want to throw the reader. I introduce the total fraction ring Q(A), but have noticed that I never use it! (No, I don’t do Cartier divisors.)

I don’t define radiciel morphisms. If you find this a horrific omission, please say so, and make a case that they should be here (even in a double-starred exercise).

Any *pedagogical* improvements to the section on representable functors would be appreciated. As one of the topics where the main new content is a point of view (rather than explicitly mathematical content), it is an interesting challenge to explain this well. And yes, I know the initial definition of the Grassmannian is the wrong one (just as my initial definition of projective space is the wrong one). But I think the Grassmannian is unusual in that it tends not to be defined in any introductory source (what are the counterexamples?), yet more advanced sources assume that you know what it is. Somewhere you have to see it for the first time, and I like to introduce it as early as possible. I further think that the right first definition to see isn’t in terms of the quot functor, any more than projective space shouldn’t be first seen in terms of its functorial definition. Everyone “knows” what projective space is, and our job is to connect the “right” definition to that initial idea.

**For everyone.**

There are so many properties of morphisms of schemes that it is easy and natural to become lost. It still seems best (as someone explaining them) to introduce them in a huge torrent. Anything that can help to make them individually memorable is a good thing — ideas are appreciated!

October 23, 2010 at 6:11 pm

Exercise 7.5B is false without some “finiteness” hypothesis on Y. The paragraph below 7.5C gives a counterexample!

Look at section 20 of EGA IV_4 for their discussion of “pseudo-morphisms”, which is “rational maps” — the key to avoiding noetherian or integrality conditions is to demand that the open set on the source be schematically dense, not just topologically dense (which you get at indirectly with associated points). There are some subtleties in general, and to get “real” results even EGA sometimes has to assume either loc. noetherian or reduced with locally finite set of irreducible component.

It should probably be emphasized that Prop. 7.5.3 is the reason that rational maps were so ubiquitous in pre-Grothendieck algebraic geometry whereas morphisms seemed very peculiar in those days (as Mumford notes in his book on complex projective varieties, I think).

In the paragraph after 7.5G, what’s up with the “rank 3” business? A typo?

In 7.6K, why not impress them with the elegance of Yoneda by having them prove that for a pair of group schemes, a scheme map respecting composition laws automatically respect identity sections and inversions, and then ask them to try to prove it without Yoneda. Ditto for the fact that the identity section and inversion are uniquely determined by the composition law.

October 23, 2010 at 8:57 pm

Dear Brian,

Isn’t “rank 3” just a way of saying “smooth conic”, necessary to eliminate the case of two lines crossing?

I agree with your point about emphasizing the historical significance of Prop. 7.5.3.

Maybe Ex. 7.5.B should just read “gives rise to” rather than “is the same as”; this would maintain its status as an “easy exercise”.

I second the Yoneda suggestions!

Best wishes,

Matt

October 24, 2010 at 3:46 pm

Matt, I was thinking about fiber rank, not rank of a matrix, which is why I got totally confused about “rank 3”. OK, so the text should be clearer what “rank” is referring to there.

December 8, 2010 at 2:38 pm

“Rank” now clarified there. I also noticed that my suggested argument for the well-definedness of rank was wrong — it is not true that every symmetric matrix is diagonalizable! That’s now fixed too.

October 28, 2010 at 5:57 am

Yeah, hmm… Is it really customary to use pseudo morphisms for rational morphisms nowadays? I decided in the stacks project to define rational maps as they are defined in EGA I, Section 7, i.e., just using dense opens. This is easier to define and often the thing you want.

Then you can have a separate section discussion pseudo morphisms where you use schematically dense opens.

Of course, then later when one discusses the sheaf of “meromorphic functions” you have to be more careful, and not call it the sheaf of rational functions unless the base scheme is reduced. Right?

October 29, 2010 at 8:10 pm

Johan, you’re right. I had in mind the issue of meromorphic functions via EGA IV_4 (with no reducedness assumptions), and only after posting my comment did I remember that there was the more pedestrian discussion of rational functions in EGA I.

October 23, 2010 at 6:23 pm

Ravi, the notion of radiciel very much illuminates the fact that surjectivity is much better for scheme morphisms than injectivity, so I think it should be included (somewhere…). Make 3.5.8 in EGA I an exercise (in field theory, essentially), and then pose two more exercises: a map between finite type schemes over an alg. closed field k is radiciel if and only if it is injective on k-points, and in the affine integrally closed domains case (say finite type over a field too) it is equivalent to the extension on function fields being purely inseparable (and show “integrally closed” cannot be dropped).

October 23, 2010 at 6:51 pm

In the final suggested exercise above I should have also imposed a module-finite condition on the extension of integrally closed domains.

March 17, 2011 at 9:34 am

I’ve put down on my to-do list to revisit radiciel morphism exercises in light of your comments. It will happen much later, but will not be forgotten!

October 23, 2010 at 9:15 pm

Dear Ravi,

I think that the end of the introductory sentence for point (1) on p.136 is missing. (It should probably read “… locally Noetherian scheme are associated points”, or something like this.)

[done -R]In the paragraph on p.136 beginning “We define a rational function …”, it would probably read better to have “even if there is more than one associated point” rather than “are more than one associated points”.

[done -R]Regarding associated points (say for affine varieties), the very definition is that these corresponds to those irreducible closed subschemes which support a section of the structure sheaf. I don’t think that this is particularly unintuitive. (It is truly the basic point I keep in my head about associated points, or more generally associated primes — after replacing the structure sheaf by a more general quasi-coherent sheaf.) I think that one could point this out explicitly, rather then burying it in the commutative algebra discussion as you have.

For example, rather than having the proof of property (2) (that reduced implies no embedded points) buried in a commutative algebra exercise (as it is currently), I think that it is possible to have the reader try to prove it without replacing the “geometric context” by a “commutative algebra” one. (Of course, what I’m saying here is a little vague, since it is about psychological rather than logical issues, but what I mean is that perhaps one could write the exercise in a way that encourages the reader to think about the support of a section of the structure sheaf, and try to keep the geometric picture in mind while arguing that a section not supported on a full component is necessarily nilpotent. If this seems too hard, perhaps one could actually include a proof of property (2) in such a framework, as an illustration of how to use the geometric intuition of supports and so on as a way of making arguments, rather than just shifting to abstract commutative algebra. In summary, since you’ve found yourself forced to introduce associated points in any event, I think that there is a teaching/learning opportunity here that can be exploited.)

Best wishes,

Matt

October 23, 2010 at 9:25 pm

Dear Ravi,

I’m sorry that this is getting a little long, but you’ve hit upon one of my favourite topics! One thing that can be emphasized (but often isn’t) is that Spec A stands for the spectrum of A, in the same sense that functional analysts use this word: the ring A can be thought of as a ring of commuting operators, and Spec A is the simultaneous spectrum of these operators. So when we look at the support of a section of the structure sheaf, we are looking at how much we are allowed to vary the eigenvalues of this operator before they simply degenerate to zero (which is what happens outside the support).

This is the point of view that I have in mind when talking about giving a geometric proof of “reduced implies no embedded points”.

Later, when one introduces quasi-coherent sheaves (or let’s just say coherent sheaves, to avoid problems coming from non-finitely generated modules), one now has explicit objects (the module) on which the ring A acts, and localizing is just the process of spreading out the eigenspaces (or, better, eigenquotients) of the module over the spectrum. (So Spec A is all the simultaneous eigenvalues that are a priori possible, and the support of M is all the simultaneous eigenvalues that acutally occur.)

The associated points are again very natural in this context: they describe how various particular elements of M can be specialized into different eigenquotients, without degenerating to zero.

This is a point of view which I’ve found very useful, but which (at least in my experience) isn’t always spelt out in texts, and it’s what I would hope to see in my fantasy exposition of support of a module, associated primes, etc.

Best wishes,

Matt

October 27, 2010 at 5:17 am

Dear Matt,

Thanks very much for this! I strongly agree with you that this topic looks very algebraic, and should be explained and understood to be really very geometric. Your explanations are very helpful (to me, not just the reader). I’m going to take the time to digest them, and not just tweak this section. I realize that my digestion time may not come until late December (and for the rest of the academic year I expect to have digestion time on a more regular basis), so this may take some time.

To others: Matt wrote more to me in an email of Oct. 24, 2010, that I will also later fold into the notes. Excerpted paragraphs (excerpted without his permission!): “My point is that this is a simple geometric argument that

illustrates how the Spec can be used to prove even statements

of pure algebra in a natural way, but is the kind of argument

that a novice reader is unlikely to discover themselves without

being carefully set up for it.

Just because the topic of associated primes is more sophisticated

than some more basic parts of commutative algebra, I think it is

a good place to demonstrate the power and beauty of geometric

reasoning, rather than punting to a series of fairly unmotivated

commutative algebra theorems and exercises.”

best,

Ravi

p.s. to everyone, repeated elsewhere: until the end of 2010, my goal is to get draft notes out on a regular basis, and to deal with small edits when I find small chunks of time. Larger suggestions are very valuable to me, and will be dealt with later! So anything not responded to right now is

notbeing ignored.October 28, 2010 at 10:38 am

There’s another point I would like to add here: One can actually define integral schemes corresponding to the associated points, and obtain a surjection from the disjoint union of these integral schemes to X. If X is affine, you can replace these integral schemes with “coprimary schemes” corresponding to a primary decomposition of 0, giving a rigorous version of Figure 6.3.

October 30, 2010 at 10:24 am

I don’t think I expressed myself very well here, so I’m going to elaborate a bit. If R has minimal primes p_1,…,p_n, then R/p_1,…,R/p_n are the coordinate rings for the irreducible components X_1,…,X_n for X=Spec R. Since the map from the disjoint union of the X_i to X is surjective, we understand X set-theoretically once we understand the X_i and “how they fit together.” However, unless X is reduced, the scheme-theoretic structure is not yet captured. That is, the ring morphism from R into the product of the R_i need not be injective. However, we can make it injective if, rather than requiring the R_i to be integral, we require only that they be coprimary, i.e., having a unique minimal prime and having no non-nilpotent zero divisors. With this requirement, there exists a decomposition of X into “coprimary components” given by Spec R/q_i, where q_1 \cap \dotsc \cap q_n = 0. Unfortunately, the q_i cannot be chosen canonically (except for those of codimension 1), but their radicals can be; these are the associated primes.

March 17, 2011 at 9:36 am

Thanks Matt and Charles! I have in my to-do list to revisit associated points following your comments (unfortunately much later). Matt, your “eigenvalue” comments were new to me.

October 24, 2010 at 4:03 am

Dear professor Vakil,

Here are some typos I found(+ my guess for correct ones) or points I do not understand.

3.2.F. Exercise 3.2.E(b) -> Exercise 3.2.E

4.2.K,L. I guess you would change it when you have enough time to do.

5.3.A. my question is somewhat weird, but what is the strangely confusing point?

p.114 line 6 : O_X ~= f^*O_Y -> O_Y~=f_*O_X

5.4.9. §s:visualizingschemesII.

5.5.B. line 4 : each homogeneous primes

line 7 : if and only -> if and only if

5.5.7. line 5 : V -> V dual

p.124 line 1 : There are better reasons to

6.1. line 2 : generalization -> generization

Prop 6.3.3. line -1 : no period at the end of sentence

6.4.2. (a) A integrally closed -> A is integrally closed

6.4.H. line -4 : (g^2-h^2f)T -> (g^2-h^2f)

6.4.6. line 3 : uw-v^2 = ??

6.5.1.(c) ?

7.1.(b)geometric motivation, … which like schemes -> like which schemes

All the best,

PhilSang

October 26, 2010 at 8:30 am

Dear PhilSang,

Many thanks! (And to others: as promised I’m far behind on responding, and I’ll respond in bits and pieces as I have time, and will be faster around mid-December.) I’ve made all the changes, except for the following.

5.3.A If you’re not confused, that’s fine; it’s hard to explain what is confusing!

p. 114 l. 6: I think this is okay (and what you wrote is equivalent). Or perhaps I’m misunderstanding your point?

5.5.7 l. 5 I think the original is right.

Prop 6.3.3 I don’t see the period-less sentence.

6.5.1 I suspect your point is that I’m already assuming that 0 is a zero-divisor (which I mentioned here). So I’ve fixed this. But if you meant something else, please let me know.

7.1(b) I think the original is fine (and means something slightly different than what you wrote).

best,

Ravi

October 28, 2010 at 10:43 am

A couple of minor grammar issues:

After (6.5.0.1), when you write “Equivalently, it is the colimit of…,” the word “it” appears to refer to “rational function,” although I assume you mean the set of all rational functions.

In the line right before Easy Exercise 6.5.A, “associated prime of I” should be plural “associated primes”.

November 13, 2010 at 6:56 am

Thanks Charles, fixed!

November 9, 2010 at 10:51 am

Regarding 7.5.D – I was very confused with this exercise because I didn’t know what a “finitely generated field extension” is (but I thought I did know!). I assumed it was a finite degree extension, in which case, Spec K seems like it solves the problem (e.g., the map from k[t_1,…,t_n] -> K is surjective, and so K is a k-variety)

Once someone here explained to me what a finitely generated field extension was, the problem wasn’t too bad.

I suspect that this is an elementary definition I missed somewhere along the way, but if it is something that comes up less frequently, perhaps adding “elements of K are rational functions in a finite set of generators from k” would help people in the same boat as me.

November 13, 2010 at 6:55 am

I’m happy to add sentences that get rid of natural bumps in the road like this. Now fixed!

October 30, 2011 at 3:29 pm

[…] over a year ago, Matthew Emerton vigorously argued in this comment that there was a much better geometric point of view. My experience in the past is that when Matt […]

June 27, 2012 at 8:20 pm

Dear Prof. Vakil,

In section 7.5 you introduce the concept of rational maps X —-> Y, where X is a reduced scheme. In this setup, how do you compose a dominant map and a rational map? It seems that some assumption of irreducibility, i.e., the existence of generic points is made somewhere, but it isn’t clear from the text. Am I missing something, or can you compose dominant and rational maps using just the reducedness assumption?

Regards,

Rankeya.

July 13, 2012 at 12:52 pm

Dear Rankeya,

That is a good point — you are certainly correct. The expositional problem arose because I first defined rational maps from integral schemes, and later realized I need to be more general. I’ve now attempted to patch this by defining dominant map from an irreducible scheme. But if you realize that there are other problems elsewhere (or that this patch will cause other problems elsewhere), please let me know!

best,

Ravi

July 16, 2012 at 12:45 pm

Thank you for your response. I will reread the section on rational maps once you post an updated copy of your notes, and will let you know if changing reducedness to irreducible causes problems elsewhere.

Regards,

Rankeya.

November 27, 2013 at 4:26 pm

The proof Proposition 6.5.5 is a mess, and it will be rewritten. To tide you over until that happens: Daniel Barter sent me a short argument this summer, which he has kindly allowed me to post

here.