Introduction to Commutative Algebra
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The Completion of a Ring with a Valuation 21
proceedings of the american mathematical society Volume 36, Number 1, November 1972 THE COMPLETION OF A RING WITH A VALUATION HELEN E. ADAMS Abstract. This paper proves three main results : the completion of a commutative ring with respect to a Manis valuation is an integral domain; a necessary and sufficient condition is given that the completion be a field; and the completion is a field when the valuation is Harrison and the value group is archimedean ordered. 1. Introduction. In [1] we defined a generalized pseudovaluation on a ring R and found explicitly the completion of R with respect to the topology induced on R by such a pseudovaluation. Using this inverse limit character- ization of the completion of R we show, in §2, that when the codomain of a valuation on R is totally ordered, the completion of R with respect to the valuation has no (nonzero) divisors of zero and the valuation on R can be extended to a valuation on the completion. In §3, R is assumed to have an identity and so, from [1, §2], the completion has an identity: a necessary and sufficient condition is given such that each element of the completion of R has a right inverse. §§2 and 3 are immediately applicable to a Manis valuation 9?on a com- mutative ring R with identity [3], where the valuation <pis surjective and the set (p(R) is a totally ordered group. A Harrison valuation on R is a Manis valuation 9?on R such that the set {x:x e R, <p(x)> y(l)} is a finite Harrison prime [2]. -
Math 296. Homework 3 (Due Jan 28) 1
Math 296. Homework 3 (due Jan 28) 1. Equivalence Classes. Let R be an equivalence relation on a set X. For each x ∈ X, consider the subset xR ⊂ X consisting of all the elements y in X such that xRy. A set of the form xR is called an equivalence class. (1) Show that xR = yR (as subsets of X) if and only if xRy. (2) Show that xR ∩ yR = ∅ or xR = yR. (3) Show that there is a subset Y (called equivalence classes representatives) of X such that X is the disjoint union of subsets of the form yR for y ∈ Y . Is the set Y uniquely determined? (4) For each of the equivalence relations from Problem Set 2, Exercise 5, Parts 3, 5, 6, 7, 8: describe the equivalence classes, find a way to enumerate them by picking a nice representative for each, and find the cardinality of the set of equivalence classes. [I will ask Ruthi to discuss this a bit in the discussion session.] 2. Pliability of Smooth Functions. This problem undertakes a very fundamental construction: to prove that ∞ −1/x2 C -functions are very soft and pliable. Let F : R → R be defined by F (x) = e for x 6= 0 and F (0) = 0. (1) Verify that F is infinitely differentiable at every point (don’t forget that you computed on a 295 problem set that the k-th derivative exists and is zero, for all k ≥ 1). −1/x2 ∞ (2) Let ϕ : R → R be defined by ϕ(x) = 0 for x ≤ 0 and ϕ(x) = e for x > 0. -
The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The
LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction). -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Local Topological Algebraicity of Analytic Function Germs Marcin Bilski, Adam Parusinski, Guillaume Rond
Local topological algebraicity of analytic function germs Marcin Bilski, Adam Parusinski, Guillaume Rond To cite this version: Marcin Bilski, Adam Parusinski, Guillaume Rond. Local topological algebraicity of analytic function germs. Journal of Algebraic Geometry, American Mathematical Society, 2017, 26 (1), pp.177 - 197. 10.1090/jag/667. hal-01255235 HAL Id: hal-01255235 https://hal.archives-ouvertes.fr/hal-01255235 Submitted on 13 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LOCAL TOPOLOGICAL ALGEBRAICITY OF ANALYTIC FUNCTION GERMS MARCIN BILSKI, ADAM PARUSINSKI,´ AND GUILLAUME ROND Abstract. T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or com- plex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety. 1. Introduction and statement of results The problem of approximation of analytic objects (sets or mappings) by algebraic ones has attracted many mathematicians, see e.g. [2] and the bibliography therein. Nevertheless there are very few positive results if one requires that the approximation gives a homeomorphism between the approximated object and the approximating one. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
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. -
Geometric Engineering of (Framed) Bps States
GEOMETRIC ENGINEERING OF (FRAMED) BPS STATES WU-YEN CHUANG1, DUILIU-EMANUEL DIACONESCU2, JAN MANSCHOT3;4, GREGORY W. MOORE5, YAN SOIBELMAN6 Abstract. BPS quivers for N = 2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with \exotic" SU(2)R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determines the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson- Thomas invariants for framed quiver representations. Contents 1. Introduction 2 1.1. A (short) summary for mathematicians 7 1.2. BPS categories and mirror symmetry 9 2. Geometric engineering, exceptional collections, and quivers 11 2.1. Exceptional collections and fractional branes 14 2.2. Orbifold quivers 18 2.3. Field theory limit A 19 2.4. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Introduction to Abstract Algebra (Math 113)
Introduction to Abstract Algebra (Math 113) Alexander Paulin, with edits by David Corwin FOR FALL 2019 MATH 113 002 ONLY Contents 1 Introduction 4 1.1 What is Algebra? . 4 1.2 Sets . 6 1.3 Functions . 9 1.4 Equivalence Relations . 12 2 The Structure of + and × on Z 15 2.1 Basic Observations . 15 2.2 Factorization and the Fundamental Theorem of Arithmetic . 17 2.3 Congruences . 20 3 Groups 23 1 3.1 Basic Definitions . 23 3.1.1 Cayley Tables for Binary Operations and Groups . 28 3.2 Subgroups, Cosets and Lagrange's Theorem . 30 3.3 Generating Sets for Groups . 35 3.4 Permutation Groups and Finite Symmetric Groups . 40 3.4.1 Active vs. Passive Notation for Permutations . 40 3.4.2 The Symmetric Group Sym3 . 43 3.4.3 Symmetric Groups in General . 44 3.5 Group Actions . 52 3.5.1 The Orbit-Stabiliser Theorem . 55 3.5.2 Centralizers and Conjugacy Classes . 59 3.5.3 Sylow's Theorem . 66 3.6 Symmetry of Sets with Extra Structure . 68 3.7 Normal Subgroups and Isomorphism Theorems . 73 3.8 Direct Products and Direct Sums . 83 3.9 Finitely Generated Abelian Groups . 85 3.10 Finite Abelian Groups . 90 3.11 The Classification of Finite Groups (Proofs Omitted) . 95 4 Rings, Ideals, and Homomorphisms 100 2 4.1 Basic Definitions . 100 4.2 Ideals, Quotient Rings and the First Isomorphism Theorem for Rings . 105 4.3 Properties of Elements of Rings . 109 4.4 Polynomial Rings . 112 4.5 Ring Extensions . 115 4.6 Field of Fractions .