The eighteenth version of the notes is the September 6 version in the usual place. The most important advance since the seventeenth version are a number of changes in response to many suggestions from various people (most recently some insightful ones from Jakub Byszewski). A chapter on the 27 lines is also added — more on this below. There will a pause in the posts for this reason:
I owe responses to a number of recent comments and emails.
Here is where the project currently stands. The year-long plan of putting most of the notes online is now declared complete, and is more complete than I’d hoped at the start. (Only) three substantive topics (chapters) remain to be added: smooth/etale/unramified; regular sequences; and formal functions (and related topics: Zariski’s main theorem, Stein factorization, etc.). A reasonable amount of topics need substantial editing in response to comments, and there are some small topics I still want to add. Other things will be added much later, including a proper introduction, bibliography, and index.
This academic year, I will teach a three-quarter graduate class based on these notes, so my focus will shift to where in the notes the class is. I will test my assertion that the important parts of this material can be covered in a year-long class that will be considered a success by a reasonable number of those students who last until the end, and that a reasonable number of students will last until the end. (I also want the course to be considered a success by students who choose to take only the first quarter, or the first two. It should be considered worthwhile — even if hard work — for students not intending to become hard-core algebraic geometers, and there should be a nontrivial number of these.) The webpage for the class will be on this “blog”, and I will welcome emails and comments from people whether or not they are enrolled in the class.
Advice sought.
In a course like this, it is essential that students solve a large number of problems, and get feedback on them. Unfortunately, we don’t have the resources for a grader, so I’m trying to think of some creative alternative. In quarters two and three of previous incarnations of this course, I’ve had people hand in problem sets, and have kept track of what problems people did, and read some of their solutions. It was far from ideal, but was better than nothing. This will be tricky for me, especially at the start of quarter one, as there will be more students (and I need to quickly convince those without sufficient background to take other classes instead), and the baby will be very small (and time and attention and sleep will be in short supply). One possible alternative is to require enrolled students to read and grade each others’ homework — say they have one week, during which they have to grade three of the problems that I pick. Students would hand in homework to me electronically (e.g. scanned, if their solutions were handwritten). I would glance at the solutions, keep track of who did what, pick the problems in each set to be graded, and then divide up the submissions among the enrolled students (so even unenroled students would get feedback), and give them a week. This would be a bit slow and cumbersome. Could this work? Has anyone tried this, or anything else (besides just letting students work on their own with no feedback — something which is standard, but which I think is far from ideal)?
Random questions
I’ve redone Chevalley and related things, including the proof that codimension is the difference of dimension for varieties. Any comments on this (or anything else) would be appreciated.
There is currently no definition of generically finite in the notes. This is because it isn’t clear to me what the accepted definition is, even though this phrase is tossed around in talks. I was pleased to recently find out that Johan de Jong was in a similar position; the section of “generically finite morphisms” in the stacks project had some theorems, but no definitions. We had a discussion on what the definition is, and the result is here. What do you think of these two definitions? Or is there some official definition in the literature?
I remain confused on the right definition of Hilbert function. It is one of two things. (i) If M is a finitely generated module over a graded ring, it can be the dimension of the various graded pieces. Special case: the graded ring of a projective variety. (ii) If X is a projective variety (with embedding into projective space), it can be the dimension of the restriction of degree n polynomials (in the projective coordinates) to X. I want to use (i), but fear that (ii) may be right, at least for some people. (The definitions disagree: consider 3 distinct points on a line, where the value of the first polynomial is 3, and the value of the second polynomial is 2.) Can anyone give an opinion, informed or otherwise? (This was discussed earlier here.)
The twenty-seven lines
I’ve thought through the 27 lines, and have tentatively decided that this is a worthwhile fun chapter to have near the end of the notes. I am tentatively hoping to end the 2011-12 course with this topic. It is beautiful, and also connects a number of ideas and themes. (On the other hand, I’m trying to hew to a tough line about what gets included, so I’m seriously considering removing it again.) As usual, comments would be very appreciated — if you’ve always wanted to learn why there are 27 lines on every cubic surface (not every standard source has a complete proof, although this sometimes is not clear), and want something readable, please take a look (and when you find it isn’t readable, complain). There are a few references to the chapters not yet public (Castelnuovo’s criterion; miracle flatness); please excuse them, and take them as black boxes.
After thinking this through, I’ve had some thoughts on this question, which I may as well record here. There are a number of different possible results one can prove. (a) One can show that every smooth cubic (over an algebraically closed field) has precisely 27 lines. (b) One can show that 
I decided to do something slightly longer, and prove (a) first, by showing the key result that the space of lines in a fixed smooth cubic surface is reduced of dimension 0. I find this enlightening for a number of reasons I won’t spell out here, even though it comes down to an explicit calculation. I’d be interested to hear what opinions people have on this; I may be convinced to save some space, and just directly find the two skew lines and five lines connecting them.
September 23, 2011 at 11:12 am
I agree with you on the definition of Hilbert function. And, as in any definitional issue, I prefer utilitarian arguments. Here, when one wants to compute the Hilbert function of a graded module, one frequently sticks it into an exact sequence of others, especially a free or injective resolution. There, your Hilbert functions behave in a completely obvious way, whereas the Hilbert functions you’re worried people might be thinking about don’t even mean anything as far as I can tell. So warn people up front about the three points on a line and tell them to Not Do That.
September 29, 2011 at 7:51 am
Thanks! I strongly agree with your reasoning, and your vote is one where I don’t feel the need for a second, so I’ll go with this. I’ll wait a month before implementing it, to give people a chance to bellow, or to give Hilbert a chance to claw his way out of his grave and slowly make his way across the Atlantic to wreak his revenge. (Update 1 minute later: I realize that I earlier brought this up, and there are no “no” votes, so I’ll do this soon. And I’ll try to sleep with one eye open in case Hilbert comes a-calling.)
Update March 25, 2012: at some point in the past, this was done.
October 24, 2011 at 11:12 am
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Peter Johnson from the Universidade Federal da Bahia (in Brazil) sent me two emails with lots of interesting comments. Because I expect to be able to deal with them only in slow motion, I thought it best to post (extracts of) them here, so others can see them.
—
Dear Ravi Vakil:
I am yet another reader of your notes, an older mathematician (mostly an algebraist) with NO special background in this area, although over the years I have read on and off about many things, including schemes, sheaves, etc. and more about categories. At least I am vaguely aware of what’s out there, going broadly rather than deeply.
Since I have been reading and enjoying your notes (slow and thorough bedside reading to distract me from my own research and help me sleep), the least I could do would be to send comments about many things I noticed. My plan was to send comments on all of Part I, but can see that my free time is about to evaporate (research in knot theory has finally turned up interesting stuff and I must now write a joint paper), so I will only send the part that has been typed, not even TeXed, then stop for quite some time.
Although I recently looked quickly through your blog, my comments have not been adapted to it, and are no doubt out of sync with where you are. Just saw you have a new version! However, feel free to do anything you like with these comments, maybe posting disputable opinions for discussion or just silently making easier changes and ignoring whatever you don’t agree with or don’t have
time/energy for. As we are both busy, questions I made can be treated as rhetorical. I expect to look at the blog at times but am unlikely to write more without some good reason.
– Peter Johnson
UFPE, Recife, Brazil
—
[his older comments are now cut, as they are trumped by a later digest, which are posted below]
November 3, 2011 at 12:58 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Revised comments from Peter Johnson (see previous comment for history):
—
Dear Ravi,
It was hard to stop so I just finished what should be the full set
of comments on Part I, once again a version behind (Oct 21), and am
sending a consolidated version — two earlier fragments can be deleted.
As always, feel free to adapt or ignore anything – these are all
my own opinions/ideas, some standard and others a bit stranger.
When you feel inclined to look back, you may find a few surprises.
Next I may skip forward, possibly thinking about comments in your blog.
I haven´t consulted older comments recently, but seem to remember seeing
something about this, one of many topics whose usual presentation may be
missing something. This fragment was written because I couldn´t locate
any similar remarks, and maybe it will be of some interest or use to you:
In the rush to adopt schemes, a useful intuitive idea from Weil’s
Foundations was lost. To explain more easily (could change), work with
equations over a field K, so we have varieties. To better understand
Mumford’s vaguely located fat points, fix some “universal” algebraically
closed field $\Omega$ that has infinite transcendence degree over K, and
indeed over all fields we intend to study. Use $\Omega$ only to supply
hosts and hosts of “semi-present” closed points.
—-
Consolidated comments on Part 1, all(?) from version of Oct 21, 2011 — Peter Johnson
p10.6 in *and around* Paris — cf. IH√âS.
p11.7 1.2.1 I’m not sure which of the two [KS] in your ref. list you mean, but there is
disappointingly little on foundations in “Categories and Sheaves ..”. McLarty’s
“Elementary Categories, …”, esp. Ch. 12, is a much more suitable reference.
p11.3 The axiom of choice will be needed if you ever define functors from universal properties.
p15ff Ch. 2, on Category Theory, is a nicely condensed exposition, with external references only
for minor technicalities. Could mention a few sources to guide those wanting more help.
McLarty’s text selects and expounds well fundamental ideas, omitting Abelian but treating
others introduced by Grothendieck. Clear and intelligent, good choices, not too long,
much relevant insight. Not for rank beginners.
p15ff Determine clearly that we usually say “the”, not “a”, for ANYTHING defined up to unique
isom. (cf. p23, 27,..)
p19 2.2.8: Defining posets with symbol \geq in place of \leq is too unconventional.
Was that for 2.2.9? This is done again in 2.4.6 (p33), still with no obvious need.
2.2.9: OK, delete “more”. The opposite may be more important: arrows as “restrictions”.
Could next bring in topology. If you name such categories, can make sheaf functors covariant.
p20 2.2.14: Won’t the contravariant case (2.2.20) come up even more?
p21.1 “Sometimes …” is an EXCELLENT idea: it is annoying to read co- contra-variant so often.
p21 2.2.20 (end): will meet .. again … *in 2.3.10* (Yoneda) and in …
p22 or You may not need this, but would if you decide to define certain functors:
2.2.22 To define functors such as lim, whose target objects are determined
only up unique isomorphism, one needs to deal with a detail that does
not arise in an object-by-object approach or in the frequent case that
some sort of “canonical” construction is available — localizations,
tensor products, etc. In general, in the usual set-theoretic setting,
the axiom of choice is invoked to save the day. (Example: Fundamental
groups of path-connected spaces; groupoids need no choice so are better.)
Choice is not strictly necessary: Makkai’s concept of an anafunctor,
using systems of all possible targets, is a satisfactory alternative.
[ Relevant places: 2.2.22, 2.3 (implicitly), adjoints (where?) ]
end p22 2.2.22. Name “a” for object seems strange – why not A? (also p38.7, p40.1).
Wrong font in id_A, id_B.
p29.5 Ex. 2.3.V may need expansion, as the converse could get confusing if the 2nd
piece of info (the isom) is not used first. The 1st info will give a fiber
product with a twist, also known as a kernel pair. The diagram with maps
f,f,g,h, where g,h: X -> X, leads to t: X -> X with tt = id, g = th, h = tg.
Maybe this should be added to the Ex as part (a). End of Ex: Not clear where
in THESE notes one would use such a def. of mono.s – can you give a forward ref.?
p30 2.3.X (a) Say: Show that *the* i_c *are* induced …
p31.3 Maybe: .. index categories will *usually* be posets: cf. kernels on p34: could be a
mystery (parallel arrows not a poset) unless you refer to diagram on p39. cf. p32.8.
p31.5 The limit *of the diagram* is …
p32.4 cf. radically different treatment for remarks about colimits on p33 and, briefly,
in 2.4.8 (p34). I think true differences will have to stay vague, as you don’t want
to go into detail about categories with/without (co)equalizers (or (co)kernels),
nor the basic easy result about limits in terms of products and equalizers.
p32.8 [OK, researched use of names. The decision about what should be limits and what colimits
was controversial. In the end, some influential category-theorists insisted on their view
and algebraists had to go along. My opinion is that the other option had more merits; at
times you may get a false sensation that something is slightly out of kilter.]
p32.8 cf. 2.6.12 ff. Also give a forward reference to def. of (co)kernels.
p33 2.4.C OK, I guess a disjoint union of sets, and the notation for its elements,
are standard enough not to need definition.
p34 2.4.7 Grammar: an element .. that are .. <– a *family* (or system) of …
p34.8 2.5 Try to find a better word than "construction". "situation"?
p35.8 *semi*groupists/universal algebraists use "abelian" (mostly "Abelian") for a
DIFFERENT concept than commutative, related to commutator theory for congruences.
p36 Good place to define \NN. Your notation for that set (p48.5, p54.5, ..?..) is ugly.
p36 Ex 2.5.G: Pose exercise, ask to figure out meaning, then figure it out. A bit long.
p38.7 Here might add that even sparser foundations exist, where Ad1 arises as a non-obvious
consequence. A good reference (very detailed) is Freyd-Scedrov, Section 1.5. Probably
going too far, but could mention that there are now several good axiomatizations used
when Abelian cats are not general enough (I could provide references to recent research.)
p39.2 The kernel *of f* is … (used wrong font for f). Say clearly that here you also defined
an object ker f, which despite the name is NOT the kernel of f. Ditto for coker and im.
p39.6 Give warning/refs, as suggested exercises are not easy (cf. Mac Lane p198; Freyd, ..?..).
"It is unique …" B –> C …
p41.6 2.6.B “Recall” D is left exact iff its companion/dual (name?)
F: C^op -> D^op is right exact.
Unfortunate: Can’t even remark that (with Abelian) F left exact = F preserves finite limits
(cf. 2.5.4, 2.6.12). Is an analog of 2.6.13 (for Ab cats). (Would need def. of fin. lim.)
p42 etc. 2.6.7 When used as noun, not adjective, remove hyphen from “..-adjoint”. cf. “exact”, not
100% same: “adjoint is ..” vs “exact is ..” More: 2.6.12 (see below); 3.6.G (start); …
p42.7 I would delete “and a discussion … [W, 1.3]” – it just points to a rehash of your 2.6.4.
p43 2.6.G. Did this, forgot, did again. Both times needed only M f.g. (f.p. a tiny bit easier).
p43 At some point — in 2.6.12: “In particular, … are right exact”, then again just before
2.6.K (was that the promised proof?; maybe make it an Ex) — you will need that the first
two diagrams on p43 have a map that in fact *must* be a kernel of the other map. This
should be noted right away (an Ex.), with the R-Mod assumption to avoid Ab cat theory.
p44 2.6.12 lines 4, 7: dropped 2 hyphens: left exact. …. right exact. This seems correct,
but I can’t explain exactly why. However, in lines 3–6, drop 3 hyphens in “..-adjoints”.
2.6.12 Here limits (to be called left-exact) are being treated as functors. Explain! (cf.
Eisenbud, Appendix 6, for example). Although not wrong, it is perverse to hide a beautiful
unifying feature that explains these phenomena. Some students may sense something stirring
under the surface, or vagueness about what sort of concept a limit really is. I propose a
small but illuminating foray that will actually shorten things: Use any small \mathcal I to
introduce a category of diagrams over $\mathcal C$, where we suppose $\mathcal C$ has limits
of all these diagrams. Constant diagrams give a functor from \mathcal C to diagrams; it has
a right adjoint: the functor lim – great example, even without generalization! (cf. McLarty.)
Could add to table on p37. Now 2.6.13 (right adjoints commute with limits) gives much else
for free. Change p45.9.
p44 2.6.H Will puzzle some and violates your self-contained rule. Could be “From Here Hop Far”.
At end, “fourth” <– given your list, I would say "first".
p45 2.6.I Assume the limits exist.
p45 2.6.K Did not note that such categories (later on, O_X-Modules) DO have filtered colimits.
Also cf. 2.4.F.
p45 2.6.13 Formula with = may obscure the real conclusion: a limit exists, namely GA (with maps).
p46.9 2.7.1 Did you really intend p,q in ZZ rather than what I would call NN? cf. "some different
subset" on p47.3, "unless p,q .." on p48 — which would need to go AFTER what follows.
Could state clearly right now or that you intend to treat only the first quadrant case as
that's enough for what you have in mind.
p47.3 typos on right side of diagram: should be p+1,q (twice).
p49-50 The tip after (2.7.2.2.) seems unclear/misstated. Should the last "later" be "earlier"?
p50 Ex. 2.7.A Missing: "is exact."
p51 2.7.5 Wrong letters BCDGHI.
pp53-57 Here you are missing an opportunity for a TREMENDOUS simplification using Zeeman diagrams
instead of expounding dry technical ideas of Cartan-Eilenberg, which might still be useful in
more general cases, but NOT to explain something as simple as 1st-quadrant double sequences (cf.
remarks in B Mitchell (1969), most of which I disagree with). I failed to understand, even after
consulting over 40 of the most relevant books/articles/notes, and reading discussions on the web ,
why expositions almost always follow so closely those of C-E or Massey. Maybe because Zeeman
included so much unnecessary material, the simplicity of his main idea was lost, but how did he
fail to communicate it by other means? I see from Zeeman (1957) that his approach was intended
to replace both C-E and Massey, and he may well have been right. He tried again in 1993 (1 page
hand-written note), via a simple diagrammatic example that is extraordinarily illuminating but
went ignored, even on the web – it's at http://zakuski.utsa.edu/~gokhman/ecz/notes.html
I strongly urge you to work through that example, and will run over some points, the central
one being to ignore details that, surprisingly, are inessential and just complicate everything.
What pleases me most (and seems little known) is that the usual diagram-chasing is unnecessary.
Conventions about directions of arrows matter little, but if needed you could redraw Z diagrams.
An idea, implicit in Koszul (first article of 1947), explicit in Zeeman 1957: "On the Filtered
Differential Group", but maybe not fully explained there: Given a differential d on G (abelian
group, at least) and a filtration of G (double sequences give a choice: vertical or horizontal),
one obtains a "bifiltration" giving a Zeeman diagram (see Zeeman article or web source). I think
one should not go right away to quotient objects but regard the diagram as a "bipartition" of G-{0}
(allowing empty boxes anywhere; Zeeman did not handle that well), in which the definitions make it
clear that d maps some boxes ONTO others. Elements in boxes are in fact strips (as in your notes
or Bott and Tu), but don't bother with that. Here, with doubly-indexed direct sums, it is also
useless to study decompositions of elements in terms of gradings. Just think of d in a general G,
as a map between boxes. In the diagram, vertical stacks of boxes map onto horizontal ones or to 0.
All higher differentials are best split into restrictions of d to certain vertical unions of boxes
modulo other unions of boxes that are subgroups. Astonishingly simple when you get it! Just from
this example you see that the idea will work in general (requiring some sort of boundedness). It
will yield almost instantly all your goals on p54 (2.7.8) with very little notation, and illustrates
clearly what it means for differentials to converge to the cohomology filtration (2.7.2.2), p49.
What are the disadvantages? Maybe (here I'm just guessing) it hides too well details relevant for
computations. That should not worry us.
[There are a few typos in what you have at present, but I couldn't bear to read much.]
p59,72 While out of fashion for technical reasons, the espace étalé view of sheaves
is too beautiful to deserve 2nd class treatment here. Isn't it worth remarking
that it makes restriction to non-open sets trivially OK? (cf. p63.9). Perhaps it's
a matter of taste, but a continuous section approach to functorial properties of
sheafification (series of exercises on p72) seems slightly easier/more efficient,
and it can even identify Set_X (say) with a category of continuous maps "above" X
(admittedly with some more steps to verify — an optional topological exercise?).
On the other hand, Grothendieck [EGA 1, p25] was against such ideas as early as 1960.
Remark 3.4.7 (p72) makes dubious/unclear claims: "same"? "identical"? "Another"?
[EH p16] gives the étalé approach, without (as in 3.4.6) but then with the topology.
You seem to follow parts of [EH]; why downplay these, with their nice details?
p60 3.1.1 "shred": This important idea needs more intuitive motivation, along the
lines of: We can't assume there is an infinitely small neighborhood of p that
supports sections, but what can be squeezed in as a substitute?
Also, although obvious, say that any representative determines the germ.
p61.6 ".. any new complications." <– Actually AVOIDS problems. For general sheaves, it is
of course harmless to treat curly F(U) as a set of functions on U, and you imply that
you will always do this, even for presheaves. We have functions, and restriction is
just the usual concept. Could say that clearly. But I don't believe you can get away
unscathed with that simplification, as it excludes presheaves not satisfying Identity
and involves a slight fudge for res in Sheaf Hom. Your material works just fine with
res abstract. When you want functions (e.g. 3.7.E), say so. Could say "restriction"
is just a name ("so-called") else could confuse. I would prefer to define presheaf
as a functor, then expand on what this means and say the map is called "res".
p62 3.2.5 Another chance to say filtered colimit. After several appearances it will be no big deal.
p63 3.2.B OK, but similar ideas appear later, with more details (3.3.J, 3.4.9).
p63 Ref. to 3.7.4 is too advanced, especially without proper details.
Better to refer to Ex 3.4.O (cf. my remark on that).
p63 3.2.8 fun fact: Precise ref. is 3.6.2. But it is trivial with sheaves as continuous
functions (sections) to restrict to any subset of X, giving sheaves formed from certain
(not all) sections — could be an Ex. right here.
p64 3.2.9. p_*(S) is a harmless mathematical metaphor or notational extension, similar to
one on p42 of Weibel. To eliminate any trace of vagueness, could use 0 in place of e.
p64 3.2.E "Morphisms" is not good; maybe omit. The point is that these are sheaves
where res really is restriction, and continuity provides the glue. If of interest,
you could ask after 3.2.11 how to fit 3.2.E into the espace étalé setup that captures
all (!) examples of sheaves. The answer involves maps through graphs in XxY.
p65 3.2.13 Say loud and clear that curly O_X ALWAYS means some sheaf of rings.
[Qualify much later, for O_X(n).] Remarks about preferred notation such as O_U
would fit more comfortably here than in Ex. 3.6.G (p79).
p65 l1 In 3.2.11 Name Y is not good; try combining X (tilde?) and \cal F. (cf. diagram p61)
p65 l3 "section of s" <– That word can mislead here! Instead: each s in \mathcal F(U) ..
p65.3 Not Example 3.2.E but: the example in Exercise 3.2.E …
p65.7 Bottom arrow in diagram points wrong way. Probably should give labels.
p66 l5 3.3.1 .. values in any .. <– .. values in the same .. cf. my next remark on
an example (p67). The two sheaves will NOT in general have the same res maps.
De-emphasizing categorical ideas could be a mistake — they help point us in the
right direction. Why not state easy stuff without fuss before things get worse?
p66.2 "… forgotten anything." Set_U from which you may want to DEFINE the res in sheaf Hom. One easily
missed point (also not in Hartshorne) is that if I’m not mistaken the sheaf Hom at U
could contain maps not induced from morphisms between sheaves F, G at U (example with
2-point X; groups Z or 0, one res_G is x -> 2x).
OK, after that the idea of dropping the sheaves from curly Hom(F,G) is not too esoteric.
3.3.4, Presheaf kernel done early is nice. Later, with sheaves, can define another version
3.4 directly from stalks and the new sheaf of sections. New version also universal: 3.5.A
reformulated. Now can use the same stalk method to define cokernels, and verify its
universal property. Done that way, stalk characterizations of mono/epi/exact for
sheaves become *trivial*. Properties at stalks continue on small opens (in a base,
if one is given) – can do via stickiness (see below) in sheaf cokernel. AFTER all that,
global results/examples could be given. The philosophy is that if you want to run
stalk-based machinery, why not base key definitions on stalks, where all is simpler?
The focus is on algebraic objects and maps, with background organization via continuity.
Sheaves are cts sections, used only when you want (WHEN, really?) to get values at U.
p68.9 3.3.F: Better to condense/merge, maybe with ker first.
p70, Germ’s view of sheaves: The name “germe” (French) is meant to suggest a tiny shoot
cf. p59 sprouting from a seed (cf. germinate), but let’s forget that. Think of an element
of a sheaf at U (open set in X) as being an infection of the points of U by a film
(membrane) of germs, each point with a germ that cannot infect any other point.
A film on U restricts to (sub)films, one on each open subset of U, and compatible
films can fuse on unions. For films to share a germ, they are required to share a
subfilm containing the germ (they are sticky). The set of germs that can infect a
point p is called the stalk at p. This is a vivid and perhaps useful way to help
understand sheaves, obviously related to the espace étalé point of view.
p71 Ex 3.4.D. How is this tricky? No problem if the sheaf map at X is an iso. If not, it
is not monic or not epic. By defns of these (composing with parallel pairs of maps),
failure will show up on stalks at some point p. The result goes through for separated
presheaves with values of the usual kinds.
Ex 3.4.E … assuming it exists …
p72 Ex 3.4.M has a nice hint (indicator sheaves – stalks can be empty!) that works only
for Sets. Shame on you for quoting (after 3.5.B) exercises as if they had been
stated/proved more generally. We just need abelian structure, so ker does the job.
Can 3.4.M be made to work more generally? A general version of 3.4.N follows from
an idea very much like pushing forward a sheaf from U to X.
p73.9 3.4.O: Grammatical infelicity (maybe OK for *American* English): “not … on all ..”
Here a cokernel not satisfying Identity collapses under the strain of sheafification.
p74.2 “.. we must show ..” <– true, but still need to do a little more.
p74 3.5.A Yet another place to say filtered colimits (cf. 2.6.12)
p76 l1 Delete second "show that".
p76 3.5.H. Font in "(c) and (d)".
p76 3.5.J OK, but only after mentioning easier cases such as (arbitrary) products and
stating more details. I presume you have in mind not biliearity but a version of
the adjoint pair with tensor and Hom, which even with your need for cuts seems too
fundamental a fact not to mention explicitly, say here in an exercise few may do. Or
should readers work through tensor and stalks (or colimits), unaware of that insight?
p78 3.6.E, near end: Clearer without the last "Left-", e.g. "Exactness holds fully .."
p78.3 3.6.F. Good to give right now (Ex) the at first surprising result that supports are
closed (cf. germ-sharing). Related curious fact (could include in Ex): In sheaves
of sets, an alternate kind of support could be points where stalks are nonempty.
This gives an adjoint pair (and a sheaf straight away), but supports are now open!
p79 3.6.G starts with a rehash of 3.6.D. About def./new notation, see my note on 3.2.13.
p79 Bad line break just before 3.7.
p80 3.7.A and before: If examined closely, "recover" is a bit vague. It depends on
what sort of structures you take to be sheaves. By continuous sections is OK;
other defns may need values in a category with limits, to avoid anything stacky.
p80.3 "Things called …" is too much in the middle. Best earlier.
p80 3.7.1 Better in earlier version?: Assume values in a category with filtered
colimits, reminding us of examples.
p81 "Tricky" survived in 2 places.
p81 3.7.2 Sheaf elements are functions, as you seem to assume (a good choice), so the
functors WILL in fact be strict inverses.
p82 3.7.4 Something interesting and subtle here, worth indicating some of the details
(short starred subsection?). Used wrong fonts for F, G.
many You mention stalks at x or at p, almost at random.
Think about preferences. At least change end of 3.3.A.
Could well observe as an aside that what you write as "commutes with" is
often what others express in terms of a functor preserving something.
See 2.3.F, 2.3.4, 2.5.4, 2.6.8, 2.6.11, 2.6.12 and following Exs, 3.5.I, …
A matter of taste/habit/user pressure, but what's wrong with short names like epic?
How many times does "…morphism" need to appear?
Grouping the new points by which set of equations over K they define
(prime ideals) produces classes of indistinguishable "K-generic" points.
A fat point (maybe closed) is just a class, and we can more or less "see"
what its K-generic points range over, but it makes little sense to ask
about the location of the fat point itself. Inclusion of ideals defines
specialization (classes are mutually disjoint). Under field extension,
fat points just split. Example: A_1(Q).
Best,
Peter Johnson
November 7, 2011 at 3:12 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Due to a soon to be resolved email problem I am still posting from an old address (UFBA), but I am really at UFPE in Recife.
On what I earlier called fat points (trying to avoid “fuzzy” but accidentally usurping another whose usual meaning is more technical), I just want to correct a misconception about my separate suggestion (two paragraphs that got almost maximally separated when you posted them): it is above all a change of image, a very short way to give correct intuition without precise details (although they exist – cf. base change).
It boils down to how satisfied you are with vague statements about not being able to pin down generic points (cf. p88, 91, 98.9, …). My proposal is merely to alter these by a small shift to an image that has the advantage of making sense: “A generic point can be thought of as consisting internally of some kind of new points that range over the corresponding subscheme.” 4.2.G will of course corroborate this point of view.
The issue here is not diagrams for schemes, a surprisingly deep topic (Mumford’s treasure map) — see “geometry and the absolute point” by Lieven Le Bruyn, at http://matrix.cmi.ua.ac.be/DATA3/ncg.pdf
OK, you may not agree to make this unconventional change, one I’m surprised not to have seen elsewhere, but the point of my suggestion should now be clear.
- Peter Johnson
November 28, 2011 at 4:34 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
More from Peter (in an email November 27, 2011):
Comments on Chs 4 and 5, based on version of Oct 30 2011. – Peter Johnson
pp 91.. Define “closed point” early: used in Fig. 4.3; around 4.2.F; 4.2.R.
4.2.9 and other places (4.2.1,…, 5.2:”invisible”, p136, …): I prefer
genuine functions with values in local rings, not “pseudo-functions”
in certain quotient fields. The once-scandalous idea may not have
made the other totally obsolete (cf. uses from Ch 14 on), but why
so much emphasis on an old way of thought that loses important
information: orders of zeros, power series expansions, ..?
4.2.R .. considers the “classical” (old-fashioned) … closed ..
Clumsy, and no clear def.
4.3 p98 Remarks on points as clumps via Galois conjugates may be too terse.
4.4.J A hint (maybe too clear!) is via 4.4.F, not 4.5.E.
4.6.2 Def. of connected (component is via Ax. Choice. [Too esoteric for
these notes, but could avoid AxC here, within usual (well-founded)
set theory if really desired: Consider the product of "all" (with a
bound on rank of sets involved) maps from X onto totally disconected
spaces. However, with irred. cpts (4.6.L) the analogous result is
*equivalent* to AxC, via tree-like (Alexandrov) spaces.]
4.6.3 Saw one easy way, did not see use for localization. But doesn’t all
this belong with 4.6.R (p 109)? My way: With X partitioned into
clopens Y, Z (sets of prime ideals of A), let I_Y = intersection of
the primes in Y = the later I(..). Can assume A reduced. Easily
see A-module A is direct sum I_Y+I_Z. 1 gives e. Etc.
OK, now I perhaps see your idea, improving on [Stacks, 7.15.3].
That is related but still too complicated.
4.6.5 Could state def. directly for any subset of X — avoids implicit
use of induced topology in places like Exs 4.6 G(b), H(b), O,
although readers clearly need some prior exposure to topology
(should STATE that on p 11; cf. 2.2.7, …). Can remind here that
topological definitions apply to any subset, via induced topology.
4.6.B moral <– Should say first that such topologies (not just A^2_C)
are in fact often irreducible.
4.6.F (b): Generalize this <– What is "this"? Just ask to prove we
still get an integral domain.
4.6.7 (just before): What _exactly_ is the connection with ultrafilters?
OK if |k|=2 — cf. 4.6.R. Omit or clarify.
4.6.8 Horrible line break. Used "bonus" only twice before (pp 91,93),
"new-fangled" never, "classical" (old-f.) once.
4.6.9 "nonsense" (and part lower down) <– Maybe — but cf. my suggestion
about giving internal structure to generic points.
4.6.J Matters little, but last assertion would be better if you swapped
both earlier instances of p and q.
4.6.L Def. just after may best be stated _first_ as ACC on open sets,
to match Noether's ACC on ideals or submodules.
4.6.P At end, define notation [..] — new? I failed to locate any
earlier def./use of quotient notation.
4.6.Q In fact, EVERY subset here is quasicompact.
4.6.R First part seems too obvious (from 4.6.L) and without point,
especially since (ii) says we won't use it. Not sure, but was it
really your intention to include the list of non-needed remarks
within the exercise? If so, clarify the spirit (what can/cannot
be used) — cf. 4.6.3: merge rather than hide such ideas/hints.
Last example clearer if also said that its conn. cpts are points.
4.7.E New kind of idea/technique, and important, so may need a hint.
p111 Before 4.7.G: Magic done in plain sight. Call attention to it.
—————————————————————–
Comments on Ch. 5, based on version of Oct 30 2011.
112.9 After 5.1.2: "the right way" <– too emphatic! Ways via
germs/stalks/local rings also give much insight.
5.1.2.1 Clearer if you define more explicitly the map between products
that involves sums.
5.1.3 Near end: better to replace ", then if" with "and the".
Maybe delete "and we will invoke this" or rewrite.
Fig 5.2 2nd diagram, with (x^2,y): As in text, fuzz should be a horizontal
line, not a thin box.
Fig 5.4 floated too far.
p119.4 5.3.1. adjointness <– refer back to 3.6.B. Gives the false
impression that ajoints take isos to isos. Holds here because the
adjoints happen to be mutually inverse. [Necessary and sufficient
for iso-iso bijection is that, in the unit and counit natural
transformations, all arrows are isomorphisms.]
before 5.3.2: Recall what is \Gamma (first use since its def. on p61!)
Grammar: confusion between \Gamma and its elements. These things are
better: they MAP to functions as in 4.2.1. Calling them functions
could be misleading, unless explained or after change suggested earlier.
5.3.A Just before, should be with {}: {f([m])} = V(m). What makes Ex. EASY,
bypassing ALL tricky details: the passage from A to Spec A produces a
unique (canonical) isom between A and the ring of sections of on D(1).
For extra precision (as just before Ex.): "identifying" the two rings
will of course "identify" their Specs.
5.3.B Two R at end should be A. As usual, I prefer to verify via stalks
rather than via distinguished opens.
5.4.1 "not enough … (x,y) not principal" <– May be too baffling without
a hint about radicals.
5.4.A At end of cocycle condition: swap order ik ij to agree with just above.
5.4.12 end p127: "can't be defined on" ?? <– "vanishes on". So, in what
follows for f/g, you are right to use f, not g. Seeing that no
polynomial defines the cone on l is doable by elementary algebra.
You could say that way is possible, but too messy. In "If .. then
D(f) is affine", the "if" is really a def. and the conclusion is
useless as there is no later D(f) — it should be V(f).
5.5 l 10: Did you mean "quite hard" (not "even harder")? This seems
to mean less hard (qualified) than "hard".
5.5.1 Prudent to give some external ref. with diagrams, for those not
already familiar with these fundamental ideas.
On 5th line, could say opens are OK if you remove the origin.
Equations cutting out V in P^n and unions of P^1s?? Too premature
and confusing; even diagram with projective cone may be a bit early.
Probably start preparing the way in 5.4.9. At least you need to
define "homogeneous" before its first use in 5.5.B.
5.5.3 Preliminaries … preliminary(!) <– Clumsy/uneccesary.
5.5.4 Say right away that that the ring S is a direct sum of abelian
groups S_n, else many will think subrings. At end, clearer with
"(in other words, …)" for S_0 subring … is S_0-algebra.
5.5.C Better to say “ideal I” and “if and only if”, and to remove
parentheses from last sentence.
5.5.5 A very minor point, but “good reason”? Is this nonstandard even
among works that focus on algebraic geometry?
5.5.5 “Base ring” was not actually defined — is it always S_0? Could
say early that we usually write A for S_0, except that at times
(cf. following example) it seems S_0 may be a _quotient_ of the
base ring A. Need to be careful in all that follows. Main point
is probably that all graded rings being studied are in the same
class (category) of A-algebras.
5.5.D Just before: Maybe OK, but I would prefer to see S_0 (= A?)
somewhere in the formula with Sym.
5.5.D (a) Algebras appear (pp 24, 34, 44, 93, 97, ..) but were never
defined. You implicitly require generated objects to contain 1,
and in fact A. Here you need that, so best to actually say it.
5.5.6 3rd line: where f is homogeneous. 7th line: considerING
p131 We obtain _a copy_ [or isomorphic copy] of the ring …
Could fix formula break.
Not clear how could jump to Exs 5.5.JKL and do WHAT? without
the important Ex. Aiming here only at P^n? Explain.
A smoother path to all this: Proj is a subset of Spec, so
already has induced topology, with associated defns and base
of induced D(f)s. Homogeneous f clearly suffice in Proj.
Use certain D(fg) to also assume deg(f) positive (or large).
[If S is concentrated in deg 0, Proj is empty and OK trivially.]
To get structure sheaf for P^n you used functions. This can
motivate the harder ideas/Ex. given now, not at the beginning.
5.5.E Remove (b), which is not really an Ex.
Right after: “with” <– "as".
A set of ideals is a subset of Proj, and in 2nd para.: a homog.
prime ideal as a subset of Spec. Confusing? Clearer if you say
(maybe give earlier ref.) ideal p, subset [p] bar, and remind
about non-closed points. Confusing: more on cones, right after
ideas involving f. Could move that, or else add something to
dispel any thought that the quadratic (all that's in sight) is
what will play the role of f.
5.5.F containing T *but not S_+.* Or else: "V(T) to be … ideals
in Proj … that contain T."
A few lines later: subscheme — from context, of Proj?: too early.
Subscheme is OK if FIRST say D(f) "is" the given subset of Spec.
p132 "make these narrower" <– better: "stick to the narrower"
Nice example of non-affine D(f)? Say f=1 in P^1.
5.5.H [(a) The question is motivated by homogeneity, but that is
irrelevant for the solution.]
(b) I(.) .. it .. <– I(Z) Could use any index set in Ex.
5.5.K No need to exclude zero, but cases without points are pointless.
5.5.O (a) f as section of line bundle <– ref., probably to 17.4.1.
5.5.9 Why "But"?
5.5.P Result is about *closed* points — also below, for Spec.
Another reason for using duals: construction then gives a
*covariant* so "natural" functor from Vec_k to PVec_k.
————————————————————————-
December 29, 2011 at 3:29 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
More from Peter Johnson, from Dec. 28, 2011 email:
Miscellaneous supplementary remarks, based on version of Dec 20, 2011.
— Peter Johnson
General: McLarty’s article “The rising sea …” is a very interesting history
of the development of theory related to schemes.
A more advanced ref. to keep in mind: notes of Dieudonn√©’s lectures,
U Maryland, 1962, say the 2nd of the two parts: “Fondements de la
g√©om√©trie alg√©brique moderne”, U Montreal Press, 2nd ed. 1966, 151p.
This is different from Dieudonné fleshing out a skeleton provided by
Grothendieck; it is his own digest of how to present concisely some
principal ideas and techniques. Especially relevant for ideas/facts
like how to drop Noetherian, and around your Ch 8 or Sec 10.4.
6.5 If out of curiosity you want other relevant material and refs, see
“Associated prime submodules of finitely generated modules”, Kamran
Divaani-Aazar and Mohammad Ali Esmkhani, arXiv:math/07492v1, 9p.,
28 Jul 2004. Quotes nice theorem of Anderson (1994) near the end.
Ch 7 Say here and/or elesewhere that “pullback” is often used with no (?)
category-theoretic connotation.
General There is a slight overuse of f that in the long term you might want
to address: two conventions f \in A and f: X -> Y. You can I think
get away with this, but should at least alert readers. You also at
times shift to name \pi for things like X -> Y. Be more consistent.
3.2.6 Just before: “value of section doesn’t make sense” <– Any
*natural* example in mind? DOES make sense with germ as value.
4.6.4 "other words": Still need to repeat that X is nonempty.
Could contrast/explain how to decide on defs for empty set:
connected yes? (matters little), irred. no (for generic pt).
4.6.8 "new-fangled" is a joke, right? You do say "classical" on p95
and continue below (3 times in 4.6.K), and with "traditional".
Maybe change last C (in A_C^n) to k or k bar.
Personally, I hate that we all must say "point" even for not
very pointlike points in schemes, which really have a lot of
semi-visible internal structure. I prefer point *represents* ..
but know few will care about a now-entrenched bad terminology.
[6.5.2 My "unfortunate" remark sent before was due to these opinions.]
Exs 4.6.J-K are indeed improved (via hints), better-positioned.
4.6.J Remove d from "recognized".
4.6.K Say X is Spec A, with functions on X, not A.
4.6.11 Strange to refer to 4.6.11 JUST before! Mention also 4.6.P.
I don't believe it was a good idea to defer/downplay (?)
comparison with those familiar things, connected components,
which I think will be slightly relevant in quite a few places.
In general, it really helps readers to compare/contrast ideas,
to more easily cement a shaky understanding. It is safer to
write a little that can be skipped or believed, than to let
some go astray and maybe get stuck on something unimportant.
p109 Forgot to delete lines before 4.6.Q that are also in 4.6.16.
4.6.X First part best with iff. Maybe drop/alter tiny Ex 4.6.W.
To clarify better (needs easy def.): 4.6.U,V OK(?) Then:
A-module M is Noetherian iff M f.g.and A/Ann(M) Noetherian.
5.4.4 Near end: just as easy to glue any sheaves (can use germs).
p167.2 so longer as <– so long as
One can make a strong case for deferring until much later Secs
2.6, 2.7, 3.6 on Abelian categories and homological algebra, say
keeping some examples with modules, to get to schemes faster.
What is the reason for the present organization?
First quote mark in " " always comes out badly. Don't you care?
December 29, 2011 at 3:31 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
And more from Peter Johnson, from email Dec. 28, 2011. -R.
Remarks on beginning of Ch 7 (some touching on Ch 3). Version of Dec 20, 2011.
— Peter Johnson
7.2.1 It’s good to be in an adjoint situation, but for an important
definition I would use something as simple as possible, which
is via the diagram with U,V at end of 3.6.B that does not even
mention functors. THAT is the right place for the def. Number
the diagram. Looks better with arrows going down and left, like
F(U) Y: morphisms are compositions with pushforward;
V determines U and the situation seems less rich.] Ex 3.6.B is
essential for absorbing a more sophisticated approach. Trying
to use germs will just lead through the inverse image functor.
In 7.2.1 the ONLY new detail is that we want sheaves of rings.
All up to 7.2.D could have been (was?) done generally, before.
3.3.A The germy view of a sheaf morphism F -> G above X: a family of
maps phi_p: F_p -> G_p such that for any germ-valued section s
of F above open U the composition p -> s_p -> phi_p(s_p) is cts.
Being a morphism is a continuity/local compatibility requirement.
Ch.3 Extra details related to f: X -> Y : I like defining pushforwards
f_*(F) via composing sections on opens of X with function induced
by f: {opens of Y} -> {opens of X}. This clearly gives a sheaf.
3.6 Nice way to understand inverse image presheaf at U: via sections on
f(U) that are restrictions of a section on an open containing f(U).
Get germs, but need to redo locally to go from presheaf to sheaf.
Alternate restatement: Any sheaf G on Y “is” a space √©taƒ∫√© Y tilde
above Y, giving functions (sections) on opens and *by restrictions*
on ALL subsets of Y. Via composition with f, G immediately induces
a presheaf on X, with sheafification called f^-1 G. This way, it
seems a bit easier to check Hom(f^⁻1 G,F) Hom(G,f_*F) with
less worry about opens (although can’t extend some properties).
[Minor detail for the curious: ANY section to Y tilde (germs) on
a subset of Y extends to a section on an open subset, using Zorn.
Without Zorn/AxC this gets tricky and I suspect it could fail.]
Probably all with X->Y should go AFTER 3.3 on morphisms over X
and after actions. In Ch 7 (sheaves of rings), could just refer
back. But as this is so important, with potentially confusing
directions of arrows, best to revise details and give diagram.
New ingredient here is that we start with a ring homomorphism.
This is not a good place to be too terse.
Notation: A little awkward to briefly use \pi here in place of
what until now has been called f. You used g, already avoiding
a clash (mentioned above) with two uses of f. But it would be
better to start with the ring hom \phi: B -> A, so can use
name \phi^* or even f, not \pi, for map X=Spec A -> Y=Spec B.
Name phi_U,V of 3.6.B will fit well into these conventions.
Uses of \pi where usually say f: 4.4.4, 7.3.B, 7.3.3, …
7.3.1 Give easy example?: non-scheme loc. ringed space; identity map.
Could summarize the series of specializations to get schemes:
sheaves .. of rings .. stalks are local rings .. morphisms must
respect them .. base of affine opens. [Last seems a big step.]
7.3.2 The ring hom f sharp from (canonical copies of) B to A gives
a morphism of locally ringed spaces Spec A -> Spec B. Once
you have proved that this, as a point map, is f, there is
little more worth doing: f sharp: B=O(Y) -> A= O(X) is
specified by f and since compatibility is part of def. it
then determines the germ maps B_q -> A_p and hence also all
maps O_Y(V) -> O_X(U) (sections), exactly as given by f.
Thus I believe germs give a simpler argument than via D(b),
although I admit this is also a matter of preferred methods.
7.3.4 2 lines after “structure morphism”: on each open set *of X *
Twice mentioned case S = Spec A. Maybe try merging.
7.3.5 Adjoints for contravariant functors; never contemplated before.
February 2, 2012 at 6:43 am
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
On the newly-added (Jan 14) non-Noetherian 6.5.8: It is misleading to
give the impression of having a result about minimal prime ideals when
it is really about more general weakly associated prime ideals, those
defined by (A) on p150, best understood in terms of annihilators (defined
implicitly in (D), p154). The methods needed are not really geometric but
are hardly different from those used for the special case 6.5.8, where M=A.
(*) For any A-module M, the elements of A lying in associated primes of M
are precisely those that annihilate some nonzero m in M.
Given nonzero m in M, s in A with s.m = 0, let p be a prime ideal minimal above
Ann m (these DO exist, by Zorn). Then p is an associated prime in which s lies.
Conversely, given s in a prime ideal p minimal above Ann m, just localize at p,
much as in 6.5.8, to get t not in p (so t.m is nonzero) and a positive integer
k with ts^k in Ann m. Then s annihilates some nonzero ts^i.m in M with i < k.
This improves on a part of what I wrote earlier on associated primes, amidst
more extensie notes stored in several comments just above.
February 2, 2012 at 7:56 am
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Having now checked [Stacks], I see the relevant result is
Lemma 7.60.6, tag 05C3, whose proof is almost the same but perhaps a tiny bit more complicated than mine. The main points are that it is easy (given the key idea of localizing) and captures the “right” context for 6.5.8.
February 2, 2012 at 8:46 am
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Having now paused to refresh memories of this topic, as I should have done before posting in the first place, it is
clear that all that is being said is that (C) holds even without Noetherian assumptions, and this implies 6.5.8.
March 1, 2012 at 1:00 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Supplement to earlier (pre-Ch 7) remarks.
Based on Notes of Feb 25, 2012.
general: Is there ever any noticeable conflict between using
S for a ring and for a multiplicative set?
One tiny divergence from [Stacks] is that there they
hyphenate: quasi-xxx, pseudo-morphism, etc. The only
hyphenated use in these Notes is in the header of 21.9.8.
2.2.20, Name “functor of points” should have been in 2.2.14 (the
etc. “silly” functor), NOT in 2.2.20 – could say “representable”.
Some (not all) cross-refs in 7.3.7, 7.3.8, 7.6.2 are wrong.
How did this early (p21) blooper go unremarked for so long,
with so many people studying the Notes?
p29 At top, two Homs should be Mor. By Sec. 2.6, explain
why/when you prefer to write Hom. Later, much is Hom.
cf. remark after 3.3.C.
p37 The new (Feb 25 2012) Ex 2.5.E turned 2.5.H into a special
case, best omitted except for brief rmks on localization.
2.6.G Ignore my 1-line incorrect note on this in earlier post.
3.3.C In header, avoid small caps font for ALL of “sheaf Hom”.
4.6.K Functions on A <– no, are on Spec(A).
4.6.14 Sorry to be pedantic, but this is false! If Z is empty
n can be 0 or 1. Rarely done, but defn of connected
component should require nonempty to work 100%, so empty
set has 0 connected components. "Should the empty set
really be considered connected?" is a no-win decision,
fortunately of almost no interest or importance.
6.5.2 Mentions a question, then (after 6.5.A), two questions.
Neither were actually asked. What exactly is the first?
March 1, 2012 at 1:03 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Remarks on Ch 7, except beginning (posted before). Based on Feb 25, 2012 Notes
— Peter Johnson
7.3.E Subscript n+1: should have stopped at n.
7.3.F By loc ringed spaces may be easier: A acts directly (through A_p);
for any morphism A –> local ring this holds, for a unique p.
cf. more checks/work to apply results on maps between affine schemes.
7.3.K (also 6.3.E) Analytification: Maybe I now understand properly. C gives
lots of points, easier details. For R: Shafarevich, BAG, V1, Ch 2, 2.2.
6.3.E General def. of analyticity, for certain fields with notion of convergence,
uses power series. Yours seems appropriate just for C.
7.3.7ff After mixup with functor of points in 2.2.14, 2.2.20, need to correct
several (not all) cross-refs in 7.3.7, 7.3.8, 7.6.2.
7.3.L Last part: what’s needed is only a first step towards Yoneda.
7.3.8 There is a close correspondence, but should not say “are precisely”:
that would mean giving “solutions” a too-artificial definition.
7.3.M Name Hom should probably be Mor here.
After 7.3.N: How could examples with affine source possibly exist??
Just after: Best to merge second para here with last part of first para.
7.4.A Fix “-)graded”. “other words” <– bad way to sneak in new def.
Do before Ex, with an informative name: d-weighted (?) morphism.
7.4.F Use 7.4.D, not (maybe once was) the previous Ex. Just after
7.4.G, I don't see what was your plan to use 7.4.F: what you
want follows immediately from 7.4.G, 7.4.D. Term "regrading"
was undefined; OK but also say you keep only some of the terms.
7.5.1 Really a _class of_ .. on dense openS _of X_, but will usually abuse.
7.5.3 last part: "the generic point" B is surjective.
p171 Bad idea (see below): redundant “dominant” in def. of birational.
In para. where “birational” is defined, maybe separate out clearly
which are the ideas relevant for the category of integral schemes.
7.5.4 The present strategy has several less than ideal features beyond a
flawed idea to use dominance, which seems to lie hidden in Step 1,
if I have successfully managed to decipher what is sketched there.
Without going into that, let me suggest my favorite way; standard
sources have others, but I did not even see the exact same result.
(1) X has a open set U_X on which gof is defined and which contains
X’={x: gof(x)=x}, a dense set as it contains a dense open V_X of X.
(2) Define analogs in Y. Note X’ is homoeomorphic to Y´ via f, g.
(3) Let W_X = V_X \cap f^-1(V_Y), say regarding f as f: U_X -> Y.
Then W_X is open in X and a dense open of X’, hence of X.
(4) As maps between W_X and its analog W_Y, f and g are inverses.
Of course, dominance now comes for free. By the way, “reduced”
could be omitted if you weren’t so set on throwing away information
by taking function values not in germs but in quotients k(p). By
Sec. 7.6, “function” shifts meaning, now referring to a section.
7.5.6 In proof, should quote 6.3.3(c), which shows B will always be f.g.
Before 7.5.7 “In particular,” <– thus this belongs immediately after 7.5.6.
7.5.J Make it clearer that at (a) a solution sketch begins.
7.6.A Confusing: nothing special about C here (?); won't rings A do the same?
In addition, false for your kind of functions, e.g. in C[\eps]/(\eps^2),
assuming you still maintain the function vs section distinction.
Worse would be to avoid such problems by restricting to reduced cases.
7.6.B Since Ex was moved to Sec 7.6.1, now "will be .. in 7.6.1" <– "was just"
7.6.C At end, "extends" is misleading; can apply results via a covariant functor
on Sch^op, having nothing to do with schemes. Is there anything new best
TREATED here? Won't clear reminders about material in Sec 2.3 suffice?
Is there much point in doing a small variation much later, in Sec 10.1?
Is it OK to call so many scattered but related results Yoneda's Lemma?
Around here, did not quite state what is the main (?) point: representing
the global section functor. Don't wait for the more general 7.6.D(a).
7.6.D(a) "replaced by any ring" <– Better expressed: works similarly for A-schemes.
7.6.3 Read for entertainment? The going (verifying everything) is a little heavy, but much would be lost by just accepting things without examining details.
In (i) associativity: writing XxXxX is lax, probably deliberately so. But
since the topic is in fact associativity, it seems a little inappropriate.
Could even question XxX. Enough to say "unique up to unique isom, so all OK". Could also mention that ZxX (with maps) can be chosen to be X. Actually, I found the identity `element' hardest to deal with. Without it, all works smoothly given enough extra axioms analogous to identities like g.g^-1.h = h.
7.6.L(b) Should define GL_n (its first appearance). Easier to grasp via a functorial approach, as in 7.6.J.
7.6.N The clarification about G_n belongs earlier.
p182 line 4: [Permutations of a given basis (ordered) suffice to give a cover.]
line 5: "harder" <– still fairly easy.
7.7.B The imposing is only for i,j from 1 to k.
March 1, 2012 at 1:27 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
The posting software truncated something just above, a part between
arrows (), leaving nonsense at 7.5.3, so again it is:
7.5.3 last part: “the generic point” B is surjective.
March 1, 2012 at 1:35 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Software is STILL truncating. Will remove arrows/carets, to try again.
Strangely, all other parts, some with arrows both ways posted just fine.
What should be at 7.5.3 (could insert it in relevant place above):
7.5.3 last part: “the generic point” :– IF X is integral, an assumption
(before/in Ex 7.5.A) that no longer precedes what’s here. Rearrange.
Can rephrase to avoid assuming that Y has a unique generic point.
*** “A little thought”: FALSE unless X is irreducible. This error
*** seems to seriously compromise your idea for proving 7.5.4 (Step 1).
p170-1 Dominance result: are assuming here that \phi: A \to B is surjective.
March 11, 2012 at 1:00 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Just a few pre-Ch. 8 comments. Based on Notes of Mar 05, 2012.
In my posts of March 1, ignore remarks about the functor of
points (one ref. to p21, other to 7.3.7ff) – in haste it was I
who got things the wrong way round. Sorry about “blooper”.
p11 “familiar with .. localization” <– but is done in 2.2.3.
3.1 near end: Should give explicitly your def. of cover, where
the union _equals_ — not, as in many other sources, contains.
This distinction will frequently matter in later arguments.
7.3.N (A) <– Put "EXERCISE" on previous line; (A) <– (a); also CF.
Not sure if it was a good idea for something so central to
be just an exercise, with not even a hint. The newly added
comment in 7.3.9 could well give the impression that we won't
need to glue — after all, P^n has been seamless since 5.5.7.
March 11, 2012 at 1:58 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Just in case anyone wants to read my comments, I could provide
readable (better line breaks) and possibly updated files by email.
Things posted here can get a little distorted: putting accents, say
on etale, produces strange symbols, and parts with carets can
get truncated. One example is in the first para. of my Dec 29 post,
which I’ll try repeating here with an extra “:” in the arrows.
7.2.1 … 3.6.B … Looks better with arrows going down and left, like
F(U) Y: morphisms are compositions with pushforward;
V determines U and the situation seems less rich.] …..
March 11, 2012 at 2:02 pm
[Peter Johnson collated many of his comments and emailed them to me on March 21. I'll work through his email rather than his blog postings. - R.]
Sorry, that also failed, and can’t go back to edit. Next try:
7.2.1 … 3.6.B … Looks better with arrows going down and left, like
F(U) \leftarrow G(V). Want to regard phi there as in Mor(F,G), not
Mor(G,F). Can’t have it both ways! [cf. case with phi in same
direction as X \to Y: morphisms are compositions with pushforward;
V determines U and the situation seems less rich.] ….
March 1, 2013 at 10:52 am
A random note: I have a copy of the 27 lines on a cubic surface that I ordered from Oliver Labs at shapeways. Brian Conrad also sent me this amazing link of the (literal) construction of the 27 lines.