DOCSLIB.ORG

Rings of Differential Operators and Étale Homomorphisms G´Isli Másson

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Besides being applicable to rings of higher krull dimension, our method also reveals that the earlier results are a reflection of the (rather natural) fact that differential operators preserve supports of regular functions. We then use our results to prove the existence of minimal reflexive modules for rings of differential operators over domains which satisfy the condition S2. Thisisrelatedto results of Chamarie and Stafford. We also compute the dimension of C(A) (in terms of intersection multiplicities) when A is a completion of an affine plane curve over an algebraically closed field of characteristic zero, and we give a description of H(A) for some algebras A of higher krull dimension, an analogue of a theorem of Brown. Finally we point out that we give a generalization of the formula for applying differ- ential operators to inverses, extending a result of Nakai and Hart. Thesis Supervisor: Michael Artin Title: Professor of Mathematics 2 Acknowledgements First of all I would like to thank my advisor Michael Artin, who always had the right ideas, and who set the record straight on countless matters. I would also like to thank him for the financial support I received as a research assistant the last two years of my studies. The mathematics department at MIT I thank for its support and virtually unlimited resources, and I would in particular like to thank Phyllis Ruby, our graduate administra- tor, for all the letters and other help which made real life considerably simpler. I would like to thank all the friends I made during my time at MIT, especially all my officemates for providing me with good surroundings and nice time on and off work. Special thanks go to Amnon Yekutieli for all the mathematical discussions and for the good time in Texas, to Wolfram Gerdes for the beer and the intercontinental cultural excursions, to Tom Roby for being my primary reference on the english language, and to Jan Pedersen for the coffee and Danish. You all made my time in the Boston area an unforgettable experience. I thank my teachers at the University of Iceland for providing me with a solid under- graduate education, and I also thank them for their hospitality during my stay in Iceland in the spring and summer of 1990. All my friends in Iceland I thank for making it a place where I want to return to, and Hans Jakob Beck and Snorri Bergmann get special thanks for keeping me in the illusion that every place has the same luxurious computer resources as I got spoiled by at MIT. My parents, Sigr´un G´ıslad´ottir and J´ohann M´ar Mar´ıusson, deserve very special thanks for always being supportive of my decisions, and father, this is probably what our early games with a pencil and graph paper finally lead to. Finally my very best thanks go to my wife, my companion and my very best friend, Freyja Hreinsd´ottir, for taking care of me and our son, N¨okkvi, while I was out working. I thank her for the mathematics, and for pointing out all the mistakes and the places where there was room for improvement in my thesis. Last but not least I do thank her for all the patience and willingness to go along even if it meant that she had to sacrifice the most. Freyja, I hope that now I can start giving something in return for all your help. 3 Contents Introduction 6 1 Definition and first properties of rings of differential operators 9 1.1 Basic properties of differential operators .................. 9 1.1.1 Endomorphisms, derivations and the derivation ring ........ 9 1.1.2 Definition of rings of differential operators ............. 11 1.2 Elementary results .............................. 14 1.2.1 A collection of basic lemmas ..................... 14 1.2.2 Modules of differentials ........................ 17 2 Etale´ homomorphisms and differential operators 20 2.1 Basic properties of ´etale homomorphisms .................. 21 2.1.1 Formally ´etale homomorphisms ................... 21 2.1.2 Etale´ homomorphisms, essentially of finite type .......... 24 2.2 Behavior of differential operators under ´etale homomorphisms ...... 26 2.2.1 Differentials and formally ´etale homomorphisms .......... 26 2.2.2 Differential operators and formally ´etale morphisms ........ 30 2.2.3 Finitely presented modules of differentials ............. 36 2.2.4 Etale´ descent for differential operators ............... 43 2.3 Differential operators and algebraic relations ................ 47 2.3.1 Elements satisfying polynomial relations .............. 47 4 2.3.2 Proof of theorem 2.3.2 ........................ 52 2.3.3 Right ideals and ´etale homomorphisms ............... 63 3 Minimal twosided ideals in rings of differential operators 68 3.1 Supports in reduced commutative rings ................... 69 3.1.1 Associated prime ideals ........................ 69 3.1.2 Geometrical unibranches ....................... 76 3.2 Minimal twosided ideals and minimal left submodules ........... 84 3.2.1 Differential operators and supports ................. 84 3.2.2 Essential submodules ......................... 88 3.2.3 Minimal nonzero ideals in rings of differential operators ...... 91 3.2.4 Applications of theorem 3.2.11 .................... 96 4 Reflexive modules and rings of differential operators 101 4.1 Some noncommutative ring theory ...................... 102 4.1.1 Rings of quotients ........................... 102 4.1.2 Reflexive modules ........................... 105 4.2 Rings satisfying the condition S2 ...................... 111 4.2.1 Minimal reflexive modules ...................... 111 5 Algebras related to rings of differential operators 117 5.1 The algebra C(A) ............................... 118 5.1.1 A dimension formula related to affine plane curves ......... 118 5.2 The algebra H(A) ............................... 123 5.2.1 A description when the irreducible components of A are regular . 123 Bibliography 130 5 Introduction In [25] it was proved that if k is an algebraically closed field of characteristic zero, and A is a finitely generated k-algebra, and a domain of krull dimension 1, 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. A similar result was independently obtained in [17], in the slightly more general case that k is (any) field of characteristic zero and A is a finitely generated reduced k-algebra of krull dimension 1. In [25] it was moreover shown that the result above also holds if A is replaced by its localization at a maximal ideal. In both papers, one of the main techniques was to use the fact that if A is as above, and the map Spec Ae → Spec A is set theoretically injective (where Ae denotes the nor- malization of A), then the rings D(A)andD(Ae) are Morita equivalent (a result motivated by an example of [18]). Since Ae is regular in this case, its ring of differential operators is well known, and using the Morita equivalence it is hence possible to derive many of the properties of D(A) from the same properties of D(Ae). There are two reasons why this method fails for rings of higher krull dimension: First of all it may happen that D(A)andD(Ae) are no longer Morita equivalent if A is not of krull dimension 1; and secondly, rings of differential operators over normal domains may be rather unpleasant. A well known example is the ring of differential operators on the 6 cubic cone, originally given in [1]. The goal of this thesis is to generalize the results stated at the beginning of the intro- duction to some rings of higher krull dimension, and to show how ´etale homomorphisms may be used to do that. In particular we show that if A is a commutative, noetherian and reduced k-algebra such that A has an ´etale covering B with irreducible components that are geometrical unibranches, then an analogue of the module J, as above, may be constructed.
Load more

Recommended publications
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Commutative Algebra
    commutative algebra Lectures delivered by Jacob Lurie Notes by Akhil Mathew Fall 2010, Harvard Last updated 12/1/2010 Contents Lecture 1 9/1 x1 Unique factorization 6 x2 Basic definitions 6 x3 Rings of holomorphic functions 7 Lecture 2 9/3 x1 R-modules 9 x2 Ideals 11 Lecture 3 9/8 x1 Localization 13 Lecture 4 9/10 x1 SpecR and the Zariski topology 17 Lecture 5 [Section] 9/12 x1 The ideal class group 21 x2 Dedekind domains 21 Lecture 6 9/13 x1 A basis for the Zariski topology 24 x2 Localization is exact 27 Lecture 7 9/15 x1 Hom and the tensor product 28 x2 Exactness 31 x3 Projective modules 32 Lecture 8 9/17 x1 Right-exactness of the tensor product 33 x2 Flatness 34 Lecture 9 [Section] 9/19 x1 Discrete valuation rings 36 1 Lecture 10 9/20 x1 The adjoint property 41 x2 Tensor products of algebras 42 x3 Integrality 43 Lecture 11 9/22 x1 Integrality, continued 44 x2 Integral closure 46 Lecture 12 9/24 x1 Valuation rings 48 x2 General remarks 50 Lecture 13 [Section] 9/26 x1 Nakayama's lemma 52 x2 Complexes 52 x3 Fitting ideals 54 x4 Examples 55 Lecture 14 9/27 x1 Valuation rings, continued 57 x2 Some useful tools 57 x3 Back to the goal 59 Lecture 15 9/29 x1 Noetherian rings and modules 62 x2 The basis theorem 64 Lecture 16 10/1 x1 More on noetherian rings 66 x2 Associated primes 67 x3 The case of one associated prime 70 Lecture 17 10/4 x1 A loose end 71 x2 Primary modules 71 x3 Primary decomposition 73 Lecture 18 10/6 x1 Unique factorization 76 x2 A ring-theoretic criterion 77 x3 Locally factorial domains 78 x4 The Picard group 78 Lecture 19
    [Show full text]
  • Arxiv:0904.3561V3 [Math.AC] 7 Jun 2010 Special Algorithms for That Case (Ford, 1987; Cohen, 1993)
    NORMALIZATION OF RINGS GERT-MARTIN GREUEL, SANTIAGO LAPLAGNE, AND FRANK SEELISCH Abstract. We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. A modification, appli- cable in positive characteristic, where actually all computations are over the original ring, is also described. The new algorithm of this paper has been implemented in Singular, for localizations of affine rings with respect to ar- bitrary monomial orderings. Benchmark tests show that it is in general much faster than any other implementation of normalization algorithms known to us. 1. Introduction Computing the normalization of a ring is a major tool in commutative algebra, with applications in algebraic geometry and singularity theory. The first general algorithms were proposed by Stolzenberg (1968) and Seidenberg (1970, 1975). How- ever, the tools involved, such as extensions of the ground field and addition of new indeterminates, make them unsuitable for most practical applications. In recent years several new and more practicable algorithms using Groebner bases have been proposed. The basic approach, continuing the line of the works men- tioned before, is to compute an increasing chain of rings from the original ring to its normalization. This is carried out in the works of Traverso (1986), Vasconcelos (1991, 1998), Brennan and Vasconcelos (2001). To our knowledge none of these algorithms has been implemented and it remains unclear how practical they are. Also de Jong (1998), Decker et al. (1999) follow this path, applying as a new in- gredient a criterion for normality due to Grauert and Remmert (1971). In Decker et al.
    [Show full text]
  • On the Ring of Quotients of a Noetherian Commutative Ring With
    RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA ALBERTO FACCHINI On the ring of quotients of a noetherian commutative ring with respect to the Dickson topology Rendiconti del Seminario Matematico della Università di Padova, tome 62 (1980), p. 233-243 <http://www.numdam.org/item?id=RSMUP_1980__62__233_0> © Rendiconti del Seminario Matematico della Università di Padova, 1980, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On the Ring of Quotients of a Noetherian Commutative Ring with Respect to the Dickson Topology. ALBERTO FACCHINI The aim of this paper is to investigate the structure of the ring of quotients ~3) of a commutative Noetherian ring .R with respect to the Dickson topology Ð. In particular we study under which condi- tions R = Ra) or R’J) is the total ring of fractions of R (§ 2), the structure of Ren when .R is a GCD-domain and when .I~ is local and satisfies con- dition S2 (§ 3), and the endomorphism ring of the R-module (~ 4). 1. Preliminaries. The symbol .R will be used consistently to denote a commutative Noetherian ring with an identity element. Let Ð be the Dickson topology on I~, that is the Gabriel topology on I~ consisting of the ideals I of R such that is an Artinian ring, i.e.
    [Show full text]
  • ON ZERO-DIVISORS of SEMIMODULES and SEMIALGEBRAS 3 That F IB (See Definition 2.1)
    ON ZERO-DIVISORS OF SEMIMODULES AND SEMIALGEBRAS PEYMAN NASEHPOUR Abstract. In Section 1 of the paper, we prove McCoy’s property for the zero-divisors of polynomials in semirings. We also investigate zero-divisors of semimodules and prove that under suitable conditions, the monoid semi- module M[G] has very few zero-divisors if and only if the S-semimodule M does so. The concept of Auslander semimodules are introduced in this section as well. In Section 2, we introduce Ohm-Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. In Sec- tion 3, we investigate the set of zero-divisors of McCoy semialgebras. We also introduce strong Krull primes for semirings and investigate their extension in semialgebras. 0. Introduction The concept of zero-divisors in ring theory was one of the main concepts that Fraenkel introduced in his paper [11] in 1914 [23, p. 59]. Vandiver introduced the concept of zero-divisors in semirings in his 1934 paper [39], where he introduced the algebraic structure of semirings itself as well [13]. The main purpose of the current paper is to focus on zero-divisors in semimodules and semialgebras and continue our study of these elements in our 2016 paper [30]. Before proceeding to explain what we do in this paper, it is important to clarify, from the beginning, what we mean by a semiring in this paper. In this paper, by a semiring, we understand an algebraic structure, consisting of a nonempty set S with two operations of addition and multiplication such that (S, +) is a commutative monoid with the identity element 0, (S, ) is a commutative monoid with identity element 1 = 0, multiplication distributes· over addition, i.e.
    [Show full text]
  • Solutions Sheet 2 Localization and the Hilbert Nullstellensatz
    D-MATH Commutative Algebra HS 2018 Prof. Paul Nelson Solutions Sheet 2 Localization and the Hilbert Nullstellensatz Let A be a commutative ring, k an algebraically closed field. 1. Let A; B be two commutative rings, f : A ! B a ring homomorphism. (a) Show that, for every ideal b ⊂ B, the preimage f −1(b) is an ideal in A (it is called the contraction of the ideal b, and it is denoted by bc). (b) Show that the image f(a) under f of an ideal a ⊂ A need not be an ideal in B. (For any ideal a ⊂ A, the ideal ae of B generated by f(a) is called the extension of the ideal a.) (c) Show that the contraction of a prime ideal is always a prime ideal, while the extension of a prime ideal need not be a prime ideal. (d) Prove the following inclusions: i. a ⊂ aec for every ideal a ⊂ A; ii. b ⊃ bce for every ideal b ⊂ B; iii. ae = aece and bc = bcec for any ideals a ⊂ A, b ⊂ B. (e) Let a1; a2 ⊂ A, b1; b2 ⊂ B be ideals. Prove that: e e e c c c i.( a1 + a2) = a1 + a2,(b1 + b2) ⊃ b1 + b2; e e e c c c ii.( a1 \ a2) ⊂ a1 \ a2,(b1 \ b2) = b1 \ b2; e e e c c c iii.( a1a2) = a1a2,(b1b2) ⊃ b1b2; e e c c iv. r(a1) ⊂ r(a1), r(b1) = r(b1). Solution. (a) Let b ⊂ B be an ideal. Then clearly f −1(b) is an additive subgroup of A because f is in particular an homomorphism between the additive groups (A; +) and (B; +).
    [Show full text]
  • Integral Domains, Modules and Algebraic Integers Section 2 Hilary Term 2014
    Module MA3412: Integral Domains, Modules and Algebraic Integers Section 2 Hilary Term 2014 D. R. Wilkins Copyright c David R. Wilkins 1997{2014 Contents 2 Integral Domains 12 2.1 Factorization in Integral Domains . 12 2.2 Euclidean Domains . 14 2.3 Principal Ideal Domains . 16 2.4 Fermat's Two Squares Theorem . 17 2.5 Maximal Ideals and Prime Ideals . 20 2.6 Unique Factorization Domains . 23 2.7 Prime Ideals of Principal Ideal Domains . 25 2.8 An Integral Domain lacking Unique Factorization . 26 2.9 Rings of Polynomials with Coefficients in a Unique Factoriza- tion Domain . 30 2.10 Polynomial Rings in Several Indeterminates . 36 2.11 Rings of Fractions . 39 2.12 Integrally Closed Domains . 46 2.13 Irreducibility of Polynomials over Fields of Fractions . 47 2.14 Requirements for Unique Factorization Of Ideals . 47 i 2 Integral Domains 2.1 Factorization in Integral Domains An integral domain is a unital commutative ring in which the product of any two non-zero elements is itself a non-zero element. Lemma 2.1 Let x, y and z be elements of an integral domain R. Suppose that x 6= 0R and xy = xz. Then y = z. Proof Suppose that these elements x, y and z satisfy xy = xz. Then x(y − z) = 0R. Now the definition of an integral domain ensures that if a product of elements of an integral domain is zero, then at least one of the factors must be zero. Thus if x 6= 0R and x(y − z) = 0R then y − z = 0R.
    [Show full text]