Commutative Algebra Regular Local Rings June 16, 2020 Throughout
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Associated Graded Rings Derived from Integrally Closed Ideals And
PUBLICATIONS MATHÉMATIQUES ET INFORMATIQUES DE RENNES MELVIN HOCHSTER Associated Graded Rings Derived from Integrally Closed Ideals and the Local Homological Conjectures Publications des séminaires de mathématiques et informatique de Rennes, 1980, fasci- cule S3 « Colloque d’algèbre », , p. 1-27 <http://www.numdam.org/item?id=PSMIR_1980___S3_1_0> © Département de mathématiques et informatique, université de Rennes, 1980, tous droits réservés. L’accès aux archives de la série « Publications mathématiques et informa- tiques de Rennes » implique l’accord avec les conditions générales d’utili- sation (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 contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ASSOCIATED GRADED RINGS DERIVED FROM INTEGRALLY CLOSED IDEALS AND THE LOCAL HOMOLOGICAL CONJECTURES 1 2 by Melvin Hochster 1. Introduction The second and third sections of this paper can be read independently. The second section explores the properties of certain "associated graded rings", graded by the nonnegative rational numbers, and constructed using filtrations of integrally closed ideals. The properties of these rings are then exploited to show that if x^x^.-^x^ is a system of paramters of a local ring R of dimension d, d •> 3 and this system satisfies a certain mild condition (to wit, that R can be mapped -
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. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Arxiv:1810.04493V2 [Math.AG]
MACAULAYFICATION OF NOETHERIAN SCHEMES KĘSTUTIS ČESNAVIČIUS Abstract. To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of “Macaulayfying” a given Noetherian scheme X. For a wide class of X, Kawasaki built the sought Cohen–Macaulay modifications, with a crucial drawback that his blowing ups did not preserve the locus CMpXq Ă X where X is already Cohen–Macaulay. We extend Kawasaki’s methods to show that every quasi-excellent, Noetherian scheme X has a Cohen–Macaulay X with a proper map X Ñ X that is an isomorphism over CMpXq. This completes Faltings’ program, reduces the conjectural resolution of singularities to the Cohen–Macaulay case, and implies thatr every proper, smoothr scheme over a number field has a proper, flat, Cohen–Macaulay model over the ring of integers. 1. Macaulayfication as a weak form of resolution of singularities .................. 1 Acknowledgements ................................... ................................ 5 2. (S2)-ification of coherent modules ................................................. 5 3. Ubiquity of Cohen–Macaulay blowing ups ........................................ 9 4. Macaulayfication in the local case ................................................. 18 5. Macaulayfication in the global case ................................................ 20 References ................................................... ........................... 22 1. Macaulayfication as a weak form of resolution of singularities The resolution of singularities, in Grothendieck’s formulation, predicts the following. Conjecture 1.1. For a quasi-excellent,1 reduced, Noetherian scheme X, there are a regular scheme X and a proper morphism π : X Ñ X that is an isomorphism over the regular locus RegpXq of X. arXiv:1810.04493v2 [math.AG] 3 Sep 2020 r CNRS, UMR 8628, Laboratoirer de Mathématiques d’Orsay, Université Paris-Saclay, 91405 Orsay, France E-mail address: [email protected]. -
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. -
6. Localization
52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i. -
INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 3 Contents 1
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 3 RAVI VAKIL Contents 1. Where we are 1 2. Noetherian rings and the Hilbert basis theorem 2 3. Fundamental definitions: Zariski topology, irreducible, affine variety, dimension, component, etc. 4 (Before class started, I showed that (finite) Chomp is a first-player win, without showing what the winning strategy is.) If you’ve seen a lot of this before, try to solve: “Fun problem” 2. Suppose f1(x1,...,xn)=0,... , fr(x1,...,xn)=0isa system of r polynomial equations in n unknowns, with integral coefficients, and suppose this system has a finite number of complex solutions. Show that each solution is algebraic, i.e. if (x1,...,xn) is a solution, then xi ∈ Q for all i. 1. Where we are We now have seen, in gory detail, the correspondence between radical ideals in n k[x1,...,xn], and algebraic subsets of A (k). Inclusions are reversed; in particular, maximal proper ideals correspond to minimal non-empty subsets, i.e. points. A key part of this correspondence involves the Nullstellensatz. Explicitly, if X and Y are two algebraic sets corresponding to ideals IX and IY (so IX = I(X)andIY =I(Y),p and V (IX )=X,andV(IY)=Y), then I(X ∪ Y )=IX ∩IY,andI(X∩Y)= (IX,IY ). That “root” is necessary. Some of these links required the following theorem, which I promised you I would prove later: Theorem. Each algebraic set in An(k) is cut out by a finite number of equations. This leads us to our next topic: Date: September 16, 1999. -
Arxiv:Math/0005288V1 [Math.QA] 31 May 2000 Ilb Setal H Rjciecodnt Igo H Variety
Mannheimer Manuskripte 254 math/0005288 SINGULAR PROJECTIVE VARIETIES AND QUANTIZATION MARTIN SCHLICHENMAIER Abstract. By the quantization condition compact quantizable K¨ahler mani- folds can be embedded into projective space. In this way they become projec- tive varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular vari- eties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, sin- gularities, and quotients appearing in geometric invariant theory are recalled. Contents Introduction 1 1. From quantizable compact K¨ahler manifolds to projective varieties 3 2. Projective varieties 6 2.1. The definition of a projective variety 6 2.2. Embeddings into Projective Space 9 2.3. The projective coordinate ring 12 3. Singularities 14 4. Quotients 17 4.1. Quotients in algebraic geometry 17 4.2. The relation with the symplectic quotient 19 References 20 Introduction arXiv:math/0005288v1 [math.QA] 31 May 2000 Compact K¨ahler manifolds which are quantizable, i.e. which admit a holomor- phic line bundle with curvature form equal to the K¨ahler form (a so called quantum line bundle) are projective algebraic manifolds. This means that with the help of the global holomorphic sections of a suitable tensor power of the quantum line bundle they can be embedded into a projective space of certain dimension. -
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. -
Supplementary Notes on Commutative Algebra1
Supplementary Notes on Commutative Algebra1 April 11, 2017 1These notes are meant as a supplement to the Lectures on Commutative Algebra, that are avail- able online at http://www.math.iitb.ac.in/∼srg/Lecnotes/AfsPuneLecNotes.pdf and are primarily meant for the students of MA 526 at IIT Bombay in Spring 2017. The contents in these notes are mainly drawn from parts of Chapters 2 and 3 of Combinatorial Topology and Algebra, RMS Lecture Notes Series No. 18, Ramanujan Math. Soc., 2012. Contents 1 Graded Rings and Modules 3 1.1 Graded Rings . 3 1.2 Graded Modules . 5 1.3 Primary Decomposition in Graded Modules . 7 2 Hilbert Functions 10 2.1 Hilbert Function of a graded module . 10 2.2 Hilbert-Samuel polynomial of a local ring . 12 3 Dimension Theory 14 3.1 The dimension of a local ring . 14 3.2 The Dimension of an Affine Algebra . 18 3.3 The Dimension of a Graded Ring . 20 References 23 2 Chapter 1 Graded Rings and Modules This chapter gives an exposition of some rudimentary aspects of graded rings and mod- ules. It may be noted that exercises are embedded within the text and not listed separately at the end of the chapter. 1.1 Graded Rings In this section, we shall study some basic facts about graded rings. The study extends to graded modules, which are discussed in the next section. The notions and results in the first two sections will allow us to discuss, in Section 1.3, some special properties of associated primes and primary decomposition in the graded situation. -
Rings of Quotients and Localization a Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Ar
RINGS OF QUOTIENTS AND LOCALIZATION A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREME N TS FOR THE DEGREE OF MASTER OF ARTS IN MATHEMATICS IN THE GRADUATE SCHOOL OF THE TEXAS WOMAN'S UNIVERSITY COLLEGE OF ARTS AND SCIENCE S BY LILLIAN WANDA WRIGHT, B. A. DENTON, TEXAS MAY, l974 TABLE OF CONTENTS INTRODUCTION . J Chapter I. PRIME IDEALS AND MULTIPLICATIVE SETS 4 II. RINGS OF QUOTIENTS . e III. CLASSICAL RINGS OF QUOTIENTS •• 25 IV. PROPERTIES PRESERVED UNDER LOCALIZATION 30 BIBLIOGRAPHY • • • • • • • • • • • • • • • • ••••••• 36 iii INTRODUCTION The concept of a ring of quotients was apparently first introduced in 192 7 by a German m a thematician Heinrich Grell i n his paper "Bezeihungen zwischen !deale verschievener Ringe" [ 7 ] . I n his work Grelt observed that it is possible to associate a ring of quotients with the set S of non -zero divisors in a r ing. The elements of this ring of quotients a r e fractions whose denominators b elong to Sand whose numerators belong to the commutative ring. Grell's ring of quotients is now called the classical ring of quotients. 1 GrelL 1 s concept of a ring of quotients remained virtually unchanged until 1944 when the Frenchman Claude C hevalley presented his paper, "On the notion oi the Ring of Quotients of a Prime Ideal" [ 5 J. C hevalley extended Gre ll's notion to the case wher e Sis the compte- ment of a primt: ideal. (Note that the set of all non - zer o divisors and the set- theoretic c om?lement of a prime ideal are both instances of .multiplicative sets -- sets tha t are closed under multiplicatio n.) 1According to V. -
Emmy Noether, Greatest Woman Mathematician Clark Kimberling
Emmy Noether, Greatest Woman Mathematician Clark Kimberling Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). With more than 100,000 members, NCTM is the largest organization dedicated to the improvement of mathematics education and to the needs of teachers of mathematics. Founded in 1920 as a not-for-profit professional and educational association, NCTM has opened doors to vast sources of publications, products, and services to help teachers do a better job in the classroom. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright March 1982 by the National Council of Teachers of Mathematics. All rights reserved. mmy Noether was born over one hundred years ago in the German university town of Erlangen, where her father, Max Noether, was a professor of Emathematics. At that time it was very unusual for a woman to seek a university education. In fact, a leading historian of the day wrote that talk of “surrendering our universities to the invasion of women . is a shameful display of moral weakness.”1 At the University of Erlangen, the Academic Senate in 1898 declared that the admission of women students would “overthrow all academic order.”2 In spite of all this, Emmy Noether was able to attend lectures at Erlangen in 1900 and to matriculate there officially in 1904.