8430notes2

# 8430notes2 - 8430 HANDOUT 2 ORDERS PRE/ALMOST/DEDEKIND...

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8430 HANDOUT 2: ORDERS, PRE/ALMOST/DEDEKIND DOMAINS, AND THE PICARD GROUP PETE L. CLARK 1. Dedekind domains To the reader: although this section is concerned with properties of Dedekind domains, it turns out that many of the most important properties of Dedekind domains are characteristic properties, i.e., not only does any Dedekind domain en- joy the property, but conversely any integral domain which enjoys that property is a Dedekind domain. Since the rings of initial interest to us, namely Z [ - n ], are in general non-maximal orders in quadratic fields, in the following section we will be faced with the following task: knowing that certain desirable properties of Dedekind domains cannot possibly hold for non-maximal orders, can we find slightly weaker properties that still hold for these “almost Dedekind domains”? All this is to say that in this section we will be performing the following dance: “Let R be any integral domain; now let R be a Dedekind domain; now let R be any (or maybe Noetherian, etc.) integral domain...” So prepare yourself for it! Convention: As is common in the study of Dedekind rings, we will often use “ideal” to mean “nonzero ideal.” Since it will be immediately apparent whether or not any given assertion pertains to the zero ideal, this ought not to cause confusion. 1.1. Basic properties. Let us collect some basic (not necessarily easy!) properties of Dedekind rings. I do not honestly expect to need to use all of these, but it is comforting to have them in one place if we need them. Theorem 1. (Localization of Dedekind domains) Let R be a Dedekind domain with quotient field K and S R a multiplicatively closed set. a) The localization S - 1 R = { x y K | x R, y S } is a Dedekind domain. b) If p is a prime ideal of R , then R p := ( R - p ) - 1 R is a discrete valuation ring. c) Conversely, a domain whose localization at every (nonzero!) prime ideal is a discrete valuation ring is necessarily a Dedekind domain. In other words, the “local analogue” of a Dedekind domain is a discrete valuation ring (DVR). 1 The intermediate concept of PID is much more ephemeral. The following is a serious theorem: Theorem 2. (Krull-Akizuki) Let R be a one-dimensional Noetherian domain, with quotient field K , and let L/K be a finite field extension. Let S be any intermediate ring R S L . 1 Topics such as localization and DVR’s are covered very casually here. They are not discussed in Cox’s book and hence are not necessary to understand the main theorems in the case of orders in quadratic fields, but they will be used for a theorem at the end of these notes. 1

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2 PETE L. CLARK a) S is (at most) one-dimensional and Noetherian. b) Thus S is a Dedekind domain iff it is integrally closed; this occurs, in particular, if S is the integral closure of R in L .
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