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.