Harmless introduction (4 lectures or so)
set the scene, fill in some items from Chunyi's 2020 course
Zariski topology of Spec A
maximal in some class => prime
the formal substitute of NSS for a general ring A:
A -> S^-1A
OX = local functions P -> element of local ring A_P
category of A-modules -> category of S^-1A modules
characterisation of DVR
short summary of general valuations from [AM]
I hope to make each lecture into a coherent narrative
making one or two points.
What is commutative algebra?
In school and intro univ. algebra you studied _division with
remainder_ for integers and polynomials in k[x].
Integers: given a,b positive integers, you can write
a = b*q + r with 0 <= r < b.
If you have a cakes to give to b kids, hand them out equally b
at a time until the remainder r < b is not enough to go around.
Polynomials: given A, B in k[x]
A = an x^n + a_{n-1}x^{n-1} + .. a0
B = bm x^m + .. b0
with deg A = n, deg B = m (deg means top term <> 0). If m <= n we
can subtract a multiple of g to cancel the top term in f:
A - an/bm*x^{n-m}*B of deg <= n-1.
Just continue decreasing deg A until deg (A - (mult. q*B)) < m.
(We will see later that removing the leading term is also an
important idea in constructing a Groebner basis.)
In either case we have a notion of size of A, and can successively
reduce it by subtraction to < size B, (and the logic has an initial
case size B = 0).
The point I want to make is that the objects we are talking
about are quite different: integers versus polynomial functions
or abstract polynomials. Nevertheless, the methods of argument
are exactly the same.
Please think through the argument used to show:
ZZ and k[x] have division with remainder, so are PID: every
ideal I is generated by a single element, I = (f)). And
PID is a UFD: every element factorises as a product of a unit
times a product of prime powers, uniquely up to units and order of
the factors. This gives the usual properties of GCD and LCM,
including the important a*f + b*g = h property of h = GCD(f,g).
Magma -- runs in online calculator http://magma.maths.usyd.edu.au/calc
A := 49; B := 175; XGCD(A,B);
h, a, b := XGCD(A,B);
printf("The HCF of %o, %o is %o, and %o*%o + %o*%o equals %o.\n"),
A,B,h,a,A,b,B,h;
a*A + b*B;
KK := PolynomialRing(Rationals());
A := x^3 + 7*x + 2;
B := x^5 - 2*x^4 + 3;
// XGCD(A,B);
h, a, b := XGCD(A,B);
printf("The HCF of %o, %o is %o.\nAnd
a * (%o) + b * (%o) = %o, where\na = %o, and\nb = %o.\n"),
A,B,h,A,B,h,a,b;
a*A + b*B;
// end of Magma
What is commutative algebra?
The 1882 paper [DW] extended this analogy in elementary algebra
to a theory that encompasses both the ring of integers of a
number field and the ring of functions on an algebraic curve.
Their paper is a landmark in the development of modern algebra,
and marks the starting point of commutative algebra.
[DW] Richard Dedekind and Heinrich Weber, Theorie der
algebraischen Funktionen einer Veraenderlichen, J. reine angew.
Math. 92 (1882), 181--290
I explain this briefly (don't worry about the details -- I will
return to the full arguments later).
==
Ring of integers of an algebraic number field
An _algebraic number field_ is a finite extension field QQ in K.
Corresponding to the ring of integers ZZ in QQ, the field K also
has a subring O_K of integers, the subset of K of integral
elements (details later). In any fairly complicated case, the
division with remainder that we used for ZZ does not work for
O_K, and it is _not_ a UFD.
==
Integral closure of an algebraic function field
You know the polynomial ring k[x] over a field k (say k = CC to
be definite). Its field of fractions k(x) consists of rational
functions f(x)/g(x) with f,g polynomials and g <> 0. An
_algebraic function field_ in one variable is a finite
extension field k(x) in K (where x is transcendental).
Corresponding to the polynomial ring k[t] in k(t), the same
definition as the number field case gives the integral closure
A of k[x] in K: A is the subset of elements of K that are
_integral_ over k[x] (satisfy a monic equation with
coefficients in k[x] -- no denominators allowed, and leading
coefficient 1). This integral closure A = k[C] is the
coordinate ring of a nonsingular affine algebraic curve C over
k. (I am not saying that this is obvious.) In any fairly
complicated case, this A does not have division with remainder,
and is _not_ a UFD.
==
Dedekind and Weber's synthesis
The preceding paragraphs set up the ring of integers O_K of a
number field K, and the coordinate ring k[C] of a nonsingular
affine curve C. These objects are major protagonists of
algebraic number theory and algebraic geometry, and are clearly
very different in nature. However, Dedekind and Weber [DW] say
that these two rings can be studied using the same algebraic
apparatus. As I said, they are usually not UFDs.
The good news: if A is a ring of either type (a Dedekind
domain), the ideals of A have _unique factorisation into prime
ideals_.
The key method of argument is _localisation_ (partial ring of
fractions). If P is a prime ideal of A, the localisation of A
at P is A_P = S^-1A where S = multiplicative set S = A - P.
(I will go through this in detail later.) In arithmetic,
A_P in K is the algebraic numbers that have an expression f/g
with g notin P. For a point P of and algebraic curve C, A_P
consists of the rational functions in k(C) that have an
expression f/g with denominator g not vanishing at P in C.
For either kind of ring A_P is a discrete valuation ring (DVR).
Although when the ring A is not a UFD, its localisation A_P is
the simplest possible UFD: it has a single prime element z (up
to units), and every nonzero element h in K has the
factorisation
h = z^n*(unit), where n = v_P(h) is the valuation of h at P.
Valuations then determine everything about A in K and the
ideals of A: an element h in K is in A if and only if it has
valuation >= 0 at every P. Moroever, every ideal I in A also
has a valuation at P (namely, min v_P(i) taken over i in I).
For any given nonzero ideal I of A, there are just finitely
many primes P such that v_P(I) > 0, and I equals the product of
P^v_P(I).
==
Modern abstract algebra
Notice the breakthrough aspect of Dedekind and Weber: modern
algebra has axioms and abstract arguments, and you often
work with objects in a symbolic way. In this case, without
reference to what the elements of the ring actually are.
==
Background reading:
I will treat localisation S^-1A in the next week or so, or see
any textbook on commutative algebra.
For Dedekind domain and unique factorisation into prime ideals,
see Matsumura p.82 or Atiyah and Macdonald, p.95.
For integral closure and DVR, see [UCA], Chap. 8, or later in
this course.