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Author Archives: limsup
Commutative Algebra 47
Minkowski Theory: Introduction Suppose is a finite extension and is the integral closure of in K. In algebraic number theory, there is a classical method by Minkowski to compute the Picard group of (note: in texts on algebraic number theory, this … Continue reading
Commutative Algebra 46
Properties of Dedekind Domains Throughout this article, A denotes a Dedekind domain. Proposition 1. Every fractional ideal of A can be written as where each is a maximal ideal. The expression is unique up to permutation of terms. Note In … Continue reading
Commutative Algebra 45
Invertibility is Local In this article, we again let A be an integral domain and K its field of fractions. We continue our discussion of invertible fractional ideals of A. Proposition 1. A fractional ideal M of A is invertible if and … Continue reading
Commutative Algebra 44
Fractional Ideals Throughout this article, let A be an integral domain and K its field of fractions. We do not assume the ring to be noetherian. The objective here is to develop the theory of multiplying and dividing certain classes of non-zero … Continue reading
Posted in Advanced Algebra
Tagged fractional ideals, ideal division, invertible ideals, localization, picard group, principal ideals
5 Comments
Commutative Algebra 43
Catenary Rings Let us look at prime chains in greater detail. Definition. Let be a chain of prime ideals of a ring A. We say the chain is saturated if for any prime ideal of A, ; maximal if it … Continue reading
Posted in Advanced Algebra
Tagged algebras, catenary rings, krull dimension, noether normalization, noetherian
2 Comments
Commutative Algebra 42
Noether Normalization Theorem Throughout this article, k is a field, not necessarily algebraically closed. Definition. Let A be a finitely generated k-algebra which is an integral domain. We say are algebraically independent over k if they are so as elements … Continue reading
Commutative Algebra 41
Basic Definitions The objective of this article is to establish the theory of transcendence bases for field extensions. Readers who are already familiar with this may skip the article. We will focus on field extensions here. Definition. Let be a … Continue reading
Posted in Advanced Algebra
Tagged algebraic extensions, fields, transcendence basis, transcendence degree
2 Comments
Commutative Algebra 40
More on Integrality Lemma 1. Let be an integral extension. If is an ideal and , the resulting injection is an integral extension. Proof Any element of can be written as , . Then x satisfies a monic polynomial relation: . … Continue reading
Posted in Advanced Algebra
Tagged closed maps, fibres, finite extensions, going up, integral extensions, krull dimension, localization
4 Comments
Commutative Algebra 39
Integrality Throughout this article, A is a subring of B; we will also call B a ring extension of A. Definition. An element is said to be integral over A if we can find (where ) such that in B. For example, is integral over since … Continue reading
Posted in Advanced Algebra
Tagged field of fractions, finite extensions, integral closure, integral extensions, normal domains, rings, UFDs
4 Comments
Commutative Algebra 38
Artinian Rings The main result we wish to prove is the following. Theorem. A ring A is artinian if and only if it is noetherian and , where denotes the Krull dimension. Note Recall that means all prime ideals of A … Continue reading
Mathematics and Such 