In the course of teaching two graduate courses at UGA in 2008, I found the need to refresh
and extend my knowledge of "basic" commutative algebra. So I started taking notes. In true
epic fashion, although I orginally started with notes on properties of integral extensions (which explains the file name), this section now appears somewhere in the middle of a long set
of notes. After I reached 100 pages, I found it psychologically necessary to post what I already had, although what I have at the moment is clearly a very rough draft. One aspect of this is that I
am continually moving the sections in an attempt to have the material appear in a linear logical order. This sometimes
has the Bourbakistic effect of postponing some important and relatively easy concept until late in the game, e.g. principal
ideal domains. There is also an aspect of wishful thinking to these notes in that some of the more difficult
results are stated as yet without proof.
As a further caveat emptor (and for my own benefit), here is a list of things that need to be completed in the present draft:
Section ?: More systematic discussion on Galois connections (isotone and antitone), with examples from algebra, especially push-pull.
Section 1: OK.
Section 2: finitely presented modules, faithfully flat extensions.
Section 3: Kaplansky's theorem on projective modules.
Section 4: rings of functions
Section 5: locally free modules (possibly should come later)
Section 6: uniqueness in the structure theorem for Artinian rings
Section 7: something on modules?
Section 8: re/write last section on Nullstellensatz as Galois connection; include Noether
Normalization (here or elsewhere)
Section 9: functoriality of spectra, Hochster's theorem
Section 10: proof of going down theorem
Section 11: Gauss' Lemma for GCD-domains, characterization of Noetherian UFDs; Auslander-Buchsbaum
Section 11: algebraic integers are Bezout
Section 12: OK.
Section 14: Kaplansky's example of nonfiniteness of normalization; second normalization theorem
Section 15: Make contact with material on locally free modules; eliminate redundancy; proof of
final lemma
Section 16: maybe omit
Section 17: whole section needs to be written (using e.g. Matsumura)
Section 18: include material on f.g. modules over a Dedekind domain
The reader should also be(a )ware of the fact that the following important topics are not
covered at all in these notes: completions (and filtrations, graded rings,...), the Hilbert
polynomial, depth, regular sequences, the Koszul complex, regular rings, differentials.
These notes, mostly written after I attended the 2003 Arizona Winter School
on model theory and arithmetic, give a sort of introduction to the
model theory of fields (assuming, unfortunately, that you know some
model theory and some arithmetic geometry and have somehow never
managed to combine them!). The point of departure is the search for
"invariants" of elementarily equivalent fields. Among other things, I
point out that a standard conjecture relating period and index in the
Brauer group would play the same role as the Milnor Conjecture did in
defining the transcendence degree of an absolutely finitely generated
field. But I picked a bad time to try an exposition on the model theory
of fields: since this paper has been written, Scanlon has proven the
equivalence of elementary equivalence and isomorphism for finitely
generated fields of characteristic zero, and Poonen and Pop have
presented much stronger definability results than the ones I discuss in
Section 2. Perhaps someday I will update the exposition to include
these exciting results.
Field Theory
There is, I believe, a finite amount of "general field theory". Most graduate level algebra courses concentrate on the structure theory of algebraic field extensions, which is of course
both beautiful and useful. However, many algebraists need to know more than this. Speaking
as an arithmetic algebraic geometer, the structure theory of transcendental extensions --
and especially, the notion of a separable transcendental extension -- inevitably comes up,
as do certain other constructions which don't seem to make it into the standard course:
e.g. linear disjointness, composita, tensor products of fields, a systematic treatment of
norms and traces.
The notes which follow aim to be a "serious" account of all aspects of general field theory.
At the moment they cover about half of the material that they should, unfortunately for the
most part the better known half. Stay tuned.
Part 1: Quadratic Forms Over Fields: Foundations.
(pdf) (22 pages)
Real Analysis II (Math 243, McGill,
2005)
It was great fun teaching a semester of real analysis at McGill. I was
initially surprised at how elementary the syllabus was -- one variable, no
explicit mention of compactness, metric spaces, no Lebesgue integration -- but I
ended up not missing any of that stuff much: it meant that almost every result
was due to Cauchy, Riemann or Weierstrass. The course text was Russell Gordon's
Real Analysis: A First Course, but I ended up writing extensive lecture
notes. One nontraditional feature was that the homework problems are directly
embedded in the notes -- not even necessarily at the end of each section. So the
lecture notes really had to be read, and this worked out rather well. In fact,
even the guy who gave me my only frowny-faced rating on you-know-what infamous
website -- he called me arrogant and sneaky [sneaky, at least, rings false] --
followed by admitting that I made ``good concise course notes.'' Here they are
with some comments.
VII: Power Series and Abel's Theorem (6 pages) (pdf)
Comments: It looks like calculus, but many topics were explored
in greater depth than in any calculus class I have ever taken or taught. The
ratio and root tests are done with lim sup's and lim inf's, and we prove that
the root test is stronger. That the ratio and root tests are ultimately linked
to geometric series is much emphasized, both in terms of limitations -- the
tests are doomed to fail on any series whose terms "decay moderately" -- and
merits: any series shown to be convergent using the ratio test (power series!)
comes with a ready-made error bound on the partial sums. We look at when the
Cauchy product of two convergent series converges to the product of the sums:
as long as it converges at all, it converges to the product of the sums; it
converges if at least one of the two factor series is absolutely convergent;
it need not converge if neither is absolutely convergent. We give a complete
treatment of Riemann's theory of rearrangments, using the term ``conditionally
convergent'' for a series which converges but can be rearranged to diverge,
and showing (in particular) that conditional convergence is the same as
non-absolute convergence. We present the beginnings of the theory of power
series, Abel's theorem and its relation to the Cauchy product.
Regrettably left out: A more explicit treatment of Abel
summation (it appears with little explanation in the middle of some proofs).
Applications of Dirichlet's test to convergence of Dirichlet series; more
discussion of summability versus convergence.
The Riemann-Darboux Integral:
I. Axiomatic Approach to the Integral; Riemann sums (11 pages) (pdf)
II: Darboux Sums; Further Integration Theorems (10 pages) (pdf)
III: Further Topics in Integration: Improper Integrals; Lebesgue's
criterion (7 pages) (pdf)
Comments: Of course to do the Riemann integral in all its glory
is quite a production. Reflecting on this before teaching the course, I found
it strange that the fundamental theorem of calculus is in fact rather easy to
prove: how can this be? I realized that if you write down some simple-looking
axioms that the integral of a function should satisfy, then the fundamental
theorem follows from these axioms, and moreover it is not hard to check that
there is at most one such functional on the space of all continuous functions.
(Just a little later I found an almost identical discussion in Lang's
Analysis. I am happy to follow in his footsteps.) What is harder is
showing that such a functional always exists: I called a particular
description of such a functional an integration process. Now there are
at least two well-known integration processes which construct this functional:
Riemann's approach through convergence, uniformly in the mesh, of arbitrary
Riemann sums, and Darboux's approach via upper and lower sums. (Actually there
is yet a third approach, which will occur to you when you study topology: the
tagged partitions of an interval form a directed set, so one can just require
convergence of the Riemann sums in the sense of nets. This condition,
a priori intermediate in strength between Riemann and Darboux so
ultimately equivalent to both, allows one to define, for instance,
non-Archimedean Riemann integrals.) Darboux's approach, later than Riemann's,
is technically simpler, and is used without comment (i.e., still called the
Riemann integral, which in a sense it is and a sense it isn't) in many modern
treatments. Its only drawback: in many applications of the integral in
numerical analysis and especially in the sciences, one really wants to be able
to compute the integral as a limit of Riemann sums! For instance, in the
applications to volumes, surface areas, forces, etc. one meets in calculus,
the appeal is clearly to the Riemann, rather than the Darboux, integration
process. Even Rudin's Principles of Mathematical Analysis introduces
the Darboux(-Stieltjes, but never mind that) process and then later on applies
it to Riemann sums: no fair!
Happily, Russell Gordon is an integration
theorist, so his text is unusually sympathetic to these issues: reading it
carefully one learns about both processes. I decided to be quite heavy-handed
about this: first I mentioned the axiomatic Riemann integral, then the Riemann
process, then in order to prove some of the stickier theorems on integration
of possibly discontinuous functions I switched to the Darboux integral, then I
explained that the two were equivalent. (Except that the proof of the
equivalence does not appear in the notes!! This will be fixed.) None of this
was easy, but actually the repetition of some of the properties of the
integral with a second (and easier) definition seemed helpful for the
students.
In stating the fundamental theorem, I was careful to
emphasize that the Riemann integrability of the derivative is a nonvacuous
hypothesis. I mentioned that there is a best integration theory which
integrates all derivatives. The Lebesgue integral is not such a theory: the
fact that f is Lebesgue-integrable implies |f| is Lebesgue integrable means
that Lebesgue integration is not suitably for highly oscillatory functions.
Rather an innocuous-looking generalization of the tagged partition leads to
the Kurzweil-Henstock integral. Amazingly, it is both simpler and more
powerful than the Lebesgue integral, albeit much less general. I have seen
passionate letters (e.g. click
here) written by integration theorists urging that the Kurzweil-Henstock
integral replace the Riemann integral in all courses starting with calculus.
Their argument seems to be that almost no calculus student completely
understands what a tagged partition is, so their understanding of a tagging
which is d-fine with respect to a gauge function
d will be about the same. I don't quite buy it at the
calculus level but it might be interesting to teach this integral in an
undergraduate real analysis course. Indeed, segueing into improper integrals,
I found the ad hoc nature of their definitions to be a bit annoying, to
the point that I gave a more complicated definition of an improperly
integrable function on the whole real line (rather than the ``just pick a
point a to break it up; what's that? yes, any point will do'' approach). What
if an unbounded function has a complicated set of discontinuities: do we need
to know what the set is in order to define the improper integral? (Apparently
the Kurzweil-Henstock integral handles improper integrals automatically and
thus takes care of this!) I included some discussion of the relationship to
infinite series, noting in particular that every infinite series can be viewed
as the improper integral of a step function, and noted that the integral test
gives good asymptotics for divergent series and error bounds for convergent
series. I ended with Lebesgue's criterion for Riemann integrability in terms
of the set of discontinuities having measure zero (of course one does not need
a full-blown theory of Lebesgue measure to define measure zero), without
proof.
Regrettably left out: A better discussion of regulated
functions. A proof of Lebesgue's criterion. If you know of a short,
self-contained (no measure theory!) one, please let me know.
Sequences and Series of Functions:
I: Pointwise and Uniform Convergence (10 pages) (pdf)
II: Power Series and Taylor Series (7 pages) (pdf)
III: Rigorous Treatment of Elementary Functions (7 pages) (pdf)
IV: The (Stone-)Weierstrass Approximation Theorem (6 pages) (pdf)
Comments: I began with a systematic discussion of pointwise
convergence, how many desirable properties of the functions in a pointwise
convergent sequence need not be inherited by the limit function, how this can
be traced to the fact that limiting operations cannot, in general be
interchanged, and that we are not about to take this lying down. This business
about the interchange of limit operations points at one of the distinctive
charms of real analysis: for many natural questions the answer is both
yes and no (``....oohhh; short answer: yes with an if; long answer: no with a
but...'' -- Reverend Lovejoy) so we get the fun of both constructing
counterexamples and proving the positive results we will later use. (On the
other hand, in complex analysis the answer is always ``yes'', and this
certainly has its charms as well.) Then we introduce uniform convergence which
fixes (almost) everything.
It took the students quite a bit of time to
understand the difference between pointwise and uniform convergence; until the
end of the course, rather strong ones dropped by to my office hours asking for
clarification just on the difference between the definitions. I tried to
explain that uniform convergence was the more geometrically natural definition
and that a pointwise convergent sequence of functions doesn't have to ``look
more and more like'' the limit function in any reasonable sense. Maybe it
would have been better to allow the backwards E's and upside-down A's onto
center stage and show that their dancing past one another was visibly
responsible for both the the difference in the definitions and the different
order of limiting operations. The contemporary mathematician's post-formalist
distaste for a proliferation of logical symbols comes at the expense of a
certain amount of clarity. (How many times have you been to a seminar or
colloquium talk in which the quantifiers were missing, ambiguous or incorrect?
Deep down, the speaker knows what he really means, but I sometimes don't and
in my waning youth I am getting less shy about persistently asking that this
be made clear.) I decided to use a different notation for uniform convergence:
an arrow with a ``u'' on top, and this was well received. In several places I
found that Rudin's book contained stronger results than Gordon's and I
faithfully copied several of his proofs. I mention Borel's theorem that any
formal power series is the Taylor series of a smooth function.
The
course listing (last modified in 19??) sternly stated that the course would
close with a rigorous treatment of the elementary functions. This seemed
anti-climactic to me, but I was wrong: it's a very nice -- and not overly
simple -- application of a lot of the course material to verify that these
crazy functions called sine, cosine and exp indeed have the properties the
calculus books tell us they do. For the most part I again copied from Rudin's
book. But I wanted to end with a bang so I discussed the Stone-Weierstrass
theorem in the last (optional) set of notes. I can't say I am especially
pleased with the treatment: the strategy to reduce to the case of |x| is well
and good, but the proof of this, taken from an exercise in Rudin, is
completely opaque to me: what on earth is going on? I noticed that Noam
Elkies' webpage sketches a much nicer proof based on the ``identity'' |x| =
sqrt(x^2) and the binomial expansion (which I did not cover!); when I get the
chance I'll rewrite it according to his hints.
Regrettably left
out: So many things. A more systematic treatment of analytic functions. A
proof of Borel's theorem. Lebesgue's theorem on differentiability of monotone
functions. A nowhere-differentiable continuous function. The fact that a
pointwise limit of continuous functions has a dense set of points of
continuity. (I hadn't heard of this result until I started flipping through
analysis books in preparation for the course. I distinctly remembered that
Rudin had an example exhibiting the characteristic function of the rationals
as a pointwise limit of continuous functions. Of course I was wrong.) Some
discussion of Fourier series would have been nice. In fact if I had the time I
would have faithfully lectured from the entire "Sequences and series of
functions'' chapter of Rudin, which is surely the book's high point. (Probably
the book could have ended here and we would still like it as much; the
remaining topics are equally well covered in other texts.) Somehow it seems
disappointing to give the Stone-Weierstrass theorem and none of its many
applications. One very nice application is a proof of Weyl's criterion for
uniform distribution on the unit interval and the consequence that for
irrational z, the fractional parts of nz are uniformly distributed. Wouldn't this be a beautiful
way to end a course?
There is also the possibility of presenting
results which motivate the transition to a more explicitly topological
approach, e.g., the Baire Category theorem, the Arzela-Ascoli theorem, a
characterization of the subsets of R which are loci of discontinuities of some
function.
Final thoughts: The course was packed full,
contentwise: I wouldn't want to cover much more material in a single semester.
However, one might imagine getting to a bit more by having covered at least some
of the material on series in the first semester; indeed, this seems more
traditional. I did have time to say once or twice, ``No, not good enough; you
need to go back over that material and I'll test you again later," but one
should have the luxury of doing this in any undergraduate course. Also the
homework was quite an integral part of the course, and the students found it
very challenging: I myself led a weekly problem session, in which the vast
majority of the solutions were presented by me. It was telling that all in all
it was the top half of the class who showed up most consistently for office
hours, including some of the very best students. I should also mention that the
first semester was taught (not by me) commonly to a group of 60 students who
then split up into two sections for the second semester: mine was the
non-honors section (although I had a couple of ``ringers.'') If you
attended both sections you would certainly be able to tell which was which --
the honors section covered things like metric spaces, and had trickier problems
-- but I think that, although my course was easier, I was offering almost as
much ``content.'' I say this in part as a warning (and partly to myself, at
that): I worked the students long and hard; several of them said that the
homework took at least 10 hours a week. In general, I found that Canadian
students responded quite well to large work loads (which is not to say that this
was the norm in all of their classes; sometimes the course syllabi struck me as
old-fashioned and unambitious compared to what I was used to from American
universities). I was frankly rather horrified at how little most of the students
knew at the beginning of the course (e.g. I assumed that the first batch of
convergence tests would be familiar from calculus, but this really seemed not to
be the case) and I can honestly say that almost everyone improved steadily and
significantly throughout. The final exam I gave was (truly) hard, and though
they looked traumatized as they walked out most of them did quite well, in some
cases turning in their best performance. All in all this was an outcome I will
be trying to replicate the next time I teach such a course.
Semigroups
Introduction to Semigroups and Monoids.
(pdf) (11 pages)
Set Theory
All the set theory I have ever needed to know. (40 pages total) Most
of the time, most of us don't need to know more about set theory than
the distinction between finite, countably infinite, and uncountable
sets. But once in a while it's nice to know a little bit more: e.g. the
least uncountable ordinal comes up in topology, or your colleague asks
you for a counterexample to the variant of Zorn's Lemma with "chain"
replaced by "countable chain." But it is hard to find a treatment of
set theory that goes a little beyond Halmos' Naive Set Theory or Kaplansky's Set Theory and Metric Spaces
(both excellent texts) but that isn't off-puttingly foundational and/or
axiomatic (i.e., that treats set theory as mathematics, rather than a
confusing amalgamation of mathematics and meta-mathematics). For
instance, the standard axioms only allow the elements of sets to
themselves be sets (so that, e.g., mathematicians do not form a set)
and forbid a set from containing itself, although neither of these
options seem logically contradictory. But assuming that we are only
interested in sets up to equivalence (i.e., bijection), it doesn't
matter -- a trick of von Neumann allows us to put any set containing
"individuals" and perhaps containing itself into bijection with a
"pure" set that does not have either of these properties. I wish
someone had told me this a long time ago! So, you guessed it, I am
writing up my own notes.
Chapter 1: Finite, countable and uncountable sets.
(pdf) (11 pages)
Chapter 2: Order and Arithmetic of Cardinalities.
(pdf) (8 pages)
Chapter 3: Ordinalities and their arithmetic; von Neumann's ordinals and cardinals.
(pdf) (18 pages)
Chapter 4: Cardinality Questions.
(pdf) (3 pages)
Shimura Curves
Notes from my 2005 ISM course: (106 pages total)
These are all the notes I typed up for my 2005 ISM course on Shimura
varieties. In the lectures, I presented more material on Hilbert and
Siegel modular varieties, adelic double coset constructions, and strong
approximation than has survived in the lecture notes. Most of the
omitted material is of a rather standard sort -- it appears in many
places -- which is not to say that it shouldn't appear here as well.
The reader will notice that the notes are significantly more polished
at the beginning and the end than in the middle. I am quite pleased
with the very last lecture, which seems to put some of the pieces of
the theory together in a new way. I would like to see more detail on
arithmetic groups and lots more detail on quaternion orders and trace
formulas. Inevitably for notes of this length, the most important
results -- like the existence of rational and integral canonical models
-- get stated and kicked around a bit but not proved. To remedy this
will require significantly more work.
Lecture 10: Integral structures, genera and class numbers.
(pdf) (16 pages)
(I do not have any good explanation for the bizarre numbering. In
actuality there were many more than 12 lectures, and there was nothing
exceptional about the lecture I gave on linear algebraic groups, except
that when I defined unipotent groups one of the attendees had the guts
and honesty to ask, "What is the point of all this?" The point is that
you need to know about a whole lot of different things to understand
the definition of a Shimura variety!)
Uniform Distribution
Some (unpolished) notes on uniform distribution.
(pdf) (20 pages)
These are notes on the basics of uniform distribution of sequences, taken on occasion
while the very nice book on this topic by Kuipers and Niederreiter. The notes
are incomplete, not including a full-blown treatment of uniform distribution in a compact (or locally compact) group.