Lect 2 (summary) In Lect 1 I talked about the Dedekind-Weber paper of 1882 in colloquial or propaganda terms. To be more formal, I should have stated the theory as a definition and theorem. Definition A _Dedekind domain_ is a Noetherian integral domain A that is 1-dimensional and normal (integrally closed in Frac A). You already know what integral domain means. Krull dimension refers to the maximum length of a strictly increasing chain of prime ideals P1 in P2 in .. Since A is an integral domain 0 is prime. Krull dimension 0 means 0 is maximal, which happens iff A is a field. For an integral domain, Krull dimension 1 means that A is not a field, but any prime ideal 0 <> P is maximal. As an integral domain A is in its field of fractions A in Frak A, and normal means that any element x in K that is integral over A is actually in A. Exa. You already know ZZ and k[x]. These are basic results that I will prove later: Theorem. Let A be a Dedekind domain, and K in L a finite extension of its field of fractions. Write B for the integral closure of A in L, giving B in L | | A in K Integral closure means that B consists of all elements of the extension field L that satisfy a monic equation over A. (Exactly the same definition as ring of integers of a number field.) Then B is finite as an A-module, and is again a Dedekind domain. Theorem. Let A be a Dedekind domain and P in Spec A a prime ideal. Then the localisation A_P = S^-1A (where S = A - P) is a discrete valuation ring (DVR). A DVR A_P is the simplest kind of UFD: there is a unique prime element z (up to multiplication by units), and every element f of the field of fractions is f = z^n*(unit of A_P), where the exponent n = v_P(f) is the valuation of f at P (like the order of zeros or poles of a meromorphic function). Theorem. Let A be a Dedekind domain with K = Frak A. Write K^x for the multiplicative group of nonzero elements of K. (I) For every nonzero f in K^x, there are only finitely many P in Spec A for which v_P(f) <> 0. (II) Any nonzero ideal of A is uniquely expressed as a product of prime ideals. More precisely, let I in A be a nonzero ideal, and write v_P(I) for the minimum v_P(I) = min_{f in I} v_P(f). Then I = Product P^v_P(f). ==== Existence of prime ideals. Baby result: a maximal ideal is prime. m in A maximal means that m <> A but any strictly bigger ideal is the whole of A. Thus for x in A - m, m+Ax = A. Let x1, x2 in A-m. Then I can write 1 = a1*x1 + m1 and 1 = a2*x2 + m2 for some m1,m2 in m and a1,a2 in A. Expanding the product 1 = (a1*x1 + m1)*(a2*x2 + m2) gives 1 = a1*a2*x1*x2 + (sum of terms in m) Therefore the ideal m + A*(x1*x2) is strictly bigger than m, so product x1*x2 notin m. This result illustrates: Principle. If an ideal I is _maximal in some restricted class_ or "conditionally maximal", then we can hope that I is prime. This is a main technique for producing prime ideals. ==== Zorn's axiom ("Zorn's lemma"). Let Si be a nonempty partially ordered set. Assume that every totally ordered subset S0 in Si has an upper bound in Si. [_Totally ordered_ means that for all x,y in S0, either x emptyset because I in Si. If {I_j} is a totally ordered subset of Si, write Q = Union_j I_j. Then this is an element of Si: It obviously contains I. It is an ideal (because if x_j in I_j is a finite set of elements, there is a biggest among the I_j, and so they are all in I_j, which is an ideal). Also it is disjoint from S because all the I_j are. So we can use Zorn's lemma to assert that P exists. P is an ideal disjoint from S, but maximal in that class. Copy and paste the above baby argument: Let x1, x2 in A-P. Then P + A*xi is a bigger ideal, so intersects S by the maximal assumption. So I can write s1 = a1*x1 + m1 and s2 = a2*x2 + m2 for some m1,m2 in P and a1,a2 in A. Expanding the product s1*s2 = (a1*x1 + m1)*(a2*x2 + m2) gives s1*s2 = a1*a2*x1*x2 + (sum of terms in P). Therefore the ideal P + A*(x1*x2) is strictly bigger than P, so product x1*x2 notin P, and P is prime. This is all basic stuff from [UCA, Chap. 1], but the same principle applies in many more advanced settings. ==== Ring of fractions S^-1A and modules of fractions S^-1M.