Integral Closure and Normalization
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The Structure Theory of Complete Local Rings
The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings. -
RINGS of MINIMAL MULTIPLICITY 1. Introduction We Shall Discuss Two Theorems of S. S. Abhyankar About Rings of Minimal Multiplici
RINGS OF MINIMAL MULTIPLICITY J. K. VERMA Abstract. In this exposition, we discuss two theorems of S. S. Abhyankar about Cohen-Macaulay local rings of minimal multiplicity and graded rings of minimal multiplicity. 1. Introduction We shall discuss two theorems of S. S. Abhyankar about rings of minimal multiplicity which he proved in his 1967 paper \Local rings of high embedding dimension" [1] which appeared in the American Journal of Mathematics. The first result gives a lower bound on the multiplicity of the maximal ideal in a Cohen-Macaulay local ring and the second result gives a lower bound on the multiplicity of a standard graded domain over an algebraically closed field. The origins of these results lie in projective geometry. Let k be an algebraically closed field and X be a projective variety. We say that it is non-degenerate if it is not contained in a hyperplane. Let I(X) be the ideal of X. It is a homogeneous ideal of the polynomial ring S = k[x0; x1; : : : ; xr]: The homogeneous coordinate ring of X is defined as R = S(X) = S=I(X): Then R is a graded ring and 1 we write it as R = ⊕n=0Rn: Here Rn = Sn=(I(X) \ Sn): The Hilbert function of R is the function HR(n) = dimk Rn: Theorem 1.1 (Hilbert-Serre). There exists a polynomial PR(x) 2 Q[x] so that for all large n; HR(n) = PR(n): The degree of PR(x) is the dimension d of X and we can write x + d x + d − 1 P (x) = e(R) − e (R) + ··· + (−1)de (R): R d 1 d − 1 d Definition 1.2. -
Solutions of Some Exercises
Solutions of Some Exercises 1.3. n ∈ Let Specmax(R) and consider the homomorphism ϕ: R[x] → R/n,f→ f(0) + n. m ∩m n n ∈ The kernel of ϕ is a maximal ideal of R[x], and R = ,so Specrab(R). 2.5. (a)ThatS generates A means that for every element f ∈ A there exist finitely many elements f1,...,fm ∈ S and a polynomial F ∈ K[T1,...,Tm]inm indeterminates such that f = F (f1,...,fm). Let n P1,P2 ∈ K be points with f(P1) = f(P2). Then F (f1(P1),...,fm(P1)) = F (f1(P2),...,fn(P2)), so fi(P1) = fi(P2) for at least one i. This yields part (a). (b) Consider the polynomial ring B := K[x1,...,xn,y1,...,yn]in2n inde- terminates. Polynomials from B define functions Kn × Kn → K.For f ∈ K[x1,...,xn], define Δf := f(x1,...,xn) − f(y1,...,yn) ∈ B. n So for P1,P2 ∈ K we have Δf(P1,P2)=f(P1) − f(P2). Consider the ideal | ∈ ⊆ I := (Δf f A)B B. G. Kemper, A Course in Commutative Algebra, Graduate Texts 217 in Mathematics 256, DOI 10.1007/978-3-642-03545-6, c Springer-Verlag Berlin Heidelberg 2011 218 Solutions of exercises for Chapter 5 By Hilbert’s basis theorem (Corollary 2.13), B is Noetherian, so by Theorem 2.9 there exist f1,...,fm ∈ A such that I =(Δf1,...,Δfm)B . We claim that S := {f1,...,fm} is A-separating. For showing this, take n two points P1 and P2 in K and assume that there exists f ∈ A with f(P1) = f(P2). -
Integral Closures of Ideals and Rings Irena Swanson
Integral closures of ideals and rings Irena Swanson ICTP, Trieste School on Local Rings and Local Study of Algebraic Varieties 31 May–4 June 2010 I assume some background from Atiyah–MacDonald [2] (especially the parts on Noetherian rings, primary decomposition of ideals, ring spectra, Hilbert’s Basis Theorem, completions). In the first lecture I will present the basics of integral closure with very few proofs; the proofs can be found either in Atiyah–MacDonald [2] or in Huneke–Swanson [13]. Much of the rest of the material can be found in Huneke–Swanson [13], but the lectures contain also more recent material. Table of contents: Section 1: Integral closure of rings and ideals 1 Section 2: Integral closure of rings 8 Section 3: Valuation rings, Krull rings, and Rees valuations 13 Section 4: Rees algebras and integral closure 19 Section 5: Computation of integral closure 24 Bibliography 28 1 Integral closure of rings and ideals (How it arises, monomial ideals and algebras) Integral closure of a ring in an overring is a generalization of the notion of the algebraic closure of a field in an overfield: Definition 1.1 Let R be a ring and S an R-algebra containing R. An element x S is ∈ said to be integral over R if there exists an integer n and elements r1,...,rn in R such that n n 1 x + r1x − + + rn 1x + rn =0. ··· − This equation is called an equation of integral dependence of x over R (of degree n). The set of all elements of S that are integral over R is called the integral closure of R in S. -
Finiteness and Homological Conditions in Commutative Group Rings
XXXX, 1–15 © De Gruyter YYYY Finiteness and Homological Conditions in Commutative Group Rings Sarah Glaz and Ryan Schwarz Abstract. This article surveys the known results for several related families of ring properties in the context of commutative group rings. These properties include finiteness conditions, homological conditions, and conditions that connect these two families. We briefly survey the classical results, highlight the recent progress, and point out open problems and possible future directions of investigation in these areas. Keywords. Group rings, Noetherian rings, coherent rings, finite conductor rings, weak global dimension, von Neumann regular rings, semihereditary rings, Prüfer conditions, zero divisors, PP rings, PF rings. AMS classification. 13B99, 13D05, 13E99, 13F05. In memory of James Brewer with respect and affection 1 Introduction Let R be a commutative ring with identity and let G be an abelian group written multi- plicatively. The group ring RG is the free R module on the elements of G with multi- P plication induced by G. An element x in RG has a unique expression: x = g2G xgg, where xg 2 R and all but finitely many xg are zero. With addition, multiplication, and scalar multiplication by elements of R defined analogously to the standard polynomial operations, RG becomes a commutative R algebra. Properties of the group ring RG, particularly in conjunction with questions of de- scent and ascent of these properties between R and RG, have been of interest for at least 70 years. In his book Commutative Semigroup Rings [14], Gilmer traces the be- ginning of a systematic interest in the nature of RG, for general rings R and groups G, to Higman’s article [28] published in 1940. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Commutative Algebra Ii, Spring 2019, A. Kustin, Class Notes
COMMUTATIVE ALGEBRA II, SPRING 2019, A. KUSTIN, CLASS NOTES 1. REGULAR SEQUENCES This section loosely follows sections 16 and 17 of [6]. Definition 1.1. Let R be a ring and M be a non-zero R-module. (a) The element r of R is regular on M if rm = 0 =) m = 0, for m 2 M. (b) The elements r1; : : : ; rs (of R) form a regular sequence on M, if (i) (r1; : : : ; rs)M 6= M, (ii) r1 is regular on M, r2 is regular on M=(r1)M, ::: , and rs is regular on M=(r1; : : : ; rs−1)M. Example 1.2. The elements x1; : : : ; xn in the polynomial ring R = k[x1; : : : ; xn] form a regular sequence on R. Example 1.3. In general, order matters. Let R = k[x; y; z]. The elements x; y(1 − x); z(1 − x) of R form a regular sequence on R. But the elements y(1 − x); z(1 − x); x do not form a regular sequence on R. Lemma 1.4. If M is a finitely generated module over a Noetherian local ring R, then every regular sequence on M is a regular sequence in any order. Proof. It suffices to show that if x1; x2 is a regular sequence on M, then x2; x1 is a regular sequence on M. Assume x1; x2 is a regular sequence on M. We first show that x2 is regular on M. If x2m = 0, then the hypothesis that x1; x2 is a regular sequence on M guarantees that m 2 x1M; thus m = x1m1 for some m1. -
Wheels — on Division by Zero Jesper Carlstr¨Om
ISSN: 1401-5617 Wheels | On Division by Zero Jesper Carlstr¨om Research Reports in Mathematics Number 11, 2001 Department of Mathematics Stockholm University Electronic versions of this document are available at http://www.matematik.su.se/reports/2001/11 Date of publication: September 10, 2001 2000 Mathematics Subject Classification: Primary 16Y99, Secondary 13B30, 13P99, 03F65, 08A70. Keywords: fractions, quotients, localization. Postal address: Department of Mathematics Stockholm University S-106 91 Stockholm Sweden Electronic addresses: http://www.matematik.su.se [email protected] Wheels | On Division by Zero Jesper Carlstr¨om Department of Mathematics Stockholm University http://www.matematik.su.se/~jesper/ Filosofie licentiatavhandling Abstract We show how to extend any commutative ring (or semiring) so that di- vision by any element, including 0, is in a sense possible. The resulting structure is what is called a wheel. Wheels are similar to rings, but 0x = 0 does not hold in general; the subset fx j 0x = 0g of any wheel is a com- mutative ring (or semiring) and any commutative ring (or semiring) with identity can be described as such a subset of a wheel. The main goal of this paper is to show that the given axioms for wheels are natural and to clarify how valid identities for wheels relate to valid identities for commutative rings and semirings. Contents 1 Introduction 3 1.1 Why invent the wheel? . 3 1.2 A sketch . 4 2 Involution-monoids 7 2.1 Definitions and examples . 8 2.2 The construction of involution-monoids from commutative monoids 9 2.3 Insertion of the parent monoid . -
Cohen-Macaulay Rings and Schemes
Cohen-Macaulay rings and schemes Caleb Ji Summer 2021 Several of my friends and I were traumatized by Cohen-Macaulay rings in our commuta- tive algebra class. In particular, we did not understand the motivation for the definition, nor what it implied geometrically. The purpose of this paper is to show that the Cohen-Macaulay condition is indeed a fruitful notion in algebraic geometry. First we explain the basic defini- tions from commutative algebra. Then we give various geometric interpretations of Cohen- Macaulay rings. Finally we touch on some other areas where the Cohen-Macaulay condition shows up: Serre duality and the Upper Bound Theorem. Contents 1 Definitions and first examples1 1.1 Preliminary notions..................................1 1.2 Depth and Cohen-Macaulay rings...........................3 2 Geometric properties3 2.1 Complete intersections and smoothness.......................3 2.2 Catenary and equidimensional rings.........................4 2.3 The unmixedness theorem and miracle flatness...................5 3 Other applications5 3.1 Serre duality......................................5 3.2 The Upper Bound Theorem (combinatorics!)....................6 References 7 1 Definitions and first examples We begin by listing some relevant foundational results (without commentary, but with a few hints on proofs) of commutative algebra. Then we define depth and Cohen-Macaulay rings and present some basic properties and examples. Most of this section and the next are based on the exposition in [1]. 1.1 Preliminary notions Full details regarding the following standard facts can be found in most commutative algebra textbooks, e.g. Theorem 1.1 (Nakayama’s lemma). Let (A; m) be a local ring and let M be a finitely generated A-module. -
NOTES on CHARACTERISTIC P COMMUTATIVE ALGEBRA MARCH 3RD, 2017
NOTES ON CHARACTERISTIC p COMMUTATIVE ALGEBRA MARCH 3RD, 2017 KARL SCHWEDE Proposition 0.1. Suppose that R ⊆ S is an inclusion of Noetherian domains such that S ∼= R ⊕ M as R-modules. Then if S is strongly F -regular, so is R. e Proof. Choose 0 6= c 2 R. Since S is strongly F -regular, there exists a φ : F∗ S −! S such e ∼ that φ(F∗ c) = 1. Let ρ : S −! R be such that ρ(1S) = 1R (this exists since S = R ⊕ M). e e φ ρ e Then the composition F∗ R ⊂ F∗ S −! S −! R sends F∗ c to 1 which proves that R is strongly F -regular. Remark 0.2. The above is an open problem in characteristic zero for KLT singularities. Corollary 0.3. A direct summand of a regular ring in characteristic p > 0 is Cohen- Macaulay. The above is obvious if we are taking a finite local inclusion of local rings of the same dimension. It is not so obvious otherwise (indeed, it has perhaps only recently been discovered how to show that direct summands of regular rings in mixed characteristic are Cohen-Macaulay). Proposition 0.4. If R is an F -finite ring such that Rm is strongly F -regular for each maximal m 2 Spec R, then R is strongly F -regular. Proof. Obviously strongly F -regular rings are F -split (take c = 1) and so R is F -split. By e post composing with Frobenius splittings, if we have a map φ : F∗ R −! R which sends e F∗ c 7! 1, then we can replace e by a larger e. -
Rings of Differential Operators and Étale Homomorphisms G´Isli Másson
Rings of differential operators and ´etale homomorphisms by G´ısli M´asson B.S., University of Iceland (1985) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1991 c Massachusetts Institute of Technology 1991 Signature of Author .................................................. Department of Mathematics May 2, 1991 Certified by .......................................................... Michael Artin Professor of Mathematics Thesis Supervisor Accepted by .......................................................... Sigurdur Helgason Chairman, Departmental Graduate Committee Department of Mathematics Rings of differential operators and ´etale homomorphisms by G´ısli M´asson Submitted to the Department of Mathematics on May 2, 1991, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Rings of differential operators over rings of krull dimension 1 were studied by Musson, Smith and Stafford, and Muhasky. In particular it was proved that if k is a field of characteristic zero, and A is a finitely generated reduced k-algebra, then (i) D(A), the ring of differential operators on A, has a minimal essential twosided ideal J , (ii) A contains a minimal essential left D(A)-submodule, J,and (iii) the algebras C(A):=A/J and H(A):=D(A)/J are finite dimensional vector spaces over k. In this case the algebras C(A)andH(A) were studied by Brown and Smith. We use ´etale homomorphisms to obtain similar results in a somewhat more general setting. While an example of Bernstein, Gelfand and Gelfand shows that the statements above do not hold for all reduced finitely generated k-algebras of higher krull dimension, we present sufficient conditions on commutative, noetherian and reduced rings so that analogues of the ideal J and module J can be constructed, and for which statements similar to (i) and (ii) hold. -
Modules Over Boolean Like Semiring of Fractions
Modules over Boolean Like Semiring of Fractions By: Ketsela Hailu Demissie Supervisor: Dr.K.Venkateswarlu A thesis submitted to the Department of Mathematics in partial fulfilment to the requirements for the degree Doctor of Philosophy Addis Ababa University, Addis Ababa, Ethiopia November, 2015 Modules over Boolean Like Semiring of Fractions c Ketsela Hailu Demissie [email protected], [email protected] November, 2015 i Declaration I, Ketsela Hailu Demissie, with student number GSR/3396/04, hereby declare that this thesis is my own work and that it has not previously been submitted for assessment or completion of any post graduate qualification to another University or for another qualification. Date: November 12, 2015 Ketsela Hailu Demissie ii Certeficate I hereby certify that I have read this thesis prepared by Ketsela Hailu Demissie under my direction and recommend that it be ac- cepted as fulfilling the dissertation requirement. Date: November 12, 2015 Dr. K. Venkateswarlu, Associate Professor, Supervisor iii Abstract The concept of Boolean like rings is originally due to A.L.Foster, in 1946. Later, in 1982, V. Swaminathan has extensively studied the geom- etry of Boolean like rings. Recently in 2011, Venkateswarlu et al intro- duced the notion of Boolean like semirings by generalizing the concept of Boolean like rings of Foster. K.Venkateswarlu, B.V.N. Murthy, and Y. Yitayew have also made an extensive study of Boolean like semirings. This work is a continued study of the theory of Boolean like semir- ings by introducing and investigating the notions; Boolean like semir- ings of fractions and Modules over Boolean like semiring of fractions.