A Zariski Topology for Modules
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Lecture 1 Course Introduction, Zariski Topology
18.725 Algebraic Geometry I Lecture 1 Lecture 1: Course Introduction, Zariski topology Some teasers So what is algebraic geometry? In short, geometry of sets given by algebraic equations. Some examples of questions along this line: 1. In 1874, H. Schubert in his book Calculus of enumerative geometry proposed the question that given 4 generic lines in the 3-space, how many lines can intersect all 4 of them. The answer is 2. The proof is as follows. Move the lines to a configuration of the form of two pairs, each consists of two intersecting lines. Then there are two lines, one of them passing the two intersection points, the other being the intersection of the two planes defined by each pair. Now we need to show somehow that the answer stays the same if we are truly in a generic position. This is answered by intersection theory, a big topic in AG. 2. We can generalize this statement. Consider 4 generic polynomials over C in 3 variables of degrees d1; d2; d3; d4, how many lines intersect the zero sets of each polynomial? The answer is 2d1d2d3d4. This is given in general by \Schubert calculus." 4 3. Take C , and 2 generic quadratic polynomials of degree two, how many lines are on the common zero set? The answer is 16. 4. For a generic cubic polynomial in 3 variables, how many lines are on the zero set? There are exactly 27 of them. (This is related to the exceptional Lie group E6.) Another major development of AG in the 20th century was on counting the numbers of solutions for n 2 3 polynomial equations over Fq where q = p . -
1. Introduction Practical Information About the Course. Problem Classes
1. Introduction Practical information about the course. Problem classes - 1 every 2 weeks, maybe a bit more Coursework – two pieces, each worth 5% Deadlines: 16 February, 14 March Problem sheets and coursework will be available on my web page: http://wwwf.imperial.ac.uk/~morr/2016-7/teaching/alg-geom.html My email address: [email protected] Office hours: Mon 2-3, Thur 4-5 in Huxley 681 Bézout’s theorem. Here is an example of a theorem in algebraic geometry and an outline of a geometric method for proving it which illustrates some of the main themes in algebraic geometry. Theorem 1.1 (Bézout). Let C be a plane algebraic curve {(x, y): f(x, y) = 0} where f is a polynomial of degree m. Let D be a plane algebraic curve {(x, y): g(x, y) = 0} where g is a polynomial of degree n. Suppose that C and D have no component in common (if they had a component in common, then their intersection would obviously be infinite). Then C ∩ D consists of mn points, provided that (i) we work over the complex numbers C; (ii) we work in the projective plane, which consists of the ordinary plane together with some points at infinity (this will be formally defined later in the course); (iii) we count intersections with the correct multiplicities (e.g. if the curves are tangent at a point, it counts as more than one intersection). We will not define intersection multiplicities in this course, but the idea is that multiple intersections resemble multiple roots of a polynomial in one variable. -
Invariant Hypersurfaces 3
INVARIANT HYPERSURFACES JASON BELL, RAHIM MOOSA, AND ADAM TOPAZ Abstract. The following theorem, which includes as very special cases re- sults of Jouanolou and Hrushovski on algebraic D-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose φ1,φ2 : Z → X are dominant rational maps from a (possibly nonreduced) irreducible scheme Z of finite-type to an algebraic variety X, with the property that there are infinitely many hyper- surfaces on X whose scheme-theoretic inverse images under φ1 and φ2 agree. Then there is a nonconstant rational function g on X such that gφ1 = gφ2. In the case when Z is also reduced the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou-Hrushovski theorem to generalised algebraic D-varieties and of Cantat’s theorem to self- correspondences. Contents 1. Introduction 2 2. Some differential algebra preliminaries 4 3. The principal algebraic statement 7 4. Proof of the main theorem 12 5. An application to algebraic D-varieties 13 6. Thereducedcaseandanapplicationtorationaldynamics 17 7. Normal varieties equipped with prime divisors 19 8. Positive characteristic 24 References 26 arXiv:1812.08346v2 [math.AG] 5 Jun 2020 Date: June 8, 2020. Keywords: D-variety, dynamical system, foliation, generalised operator, hypersurface, rational map, self-correspondence. 2010 Mathematics Subject Classification. Primary 14E99; Secondary 12H05 and 12H10. Acknowledgements: The authors are grateful for the referee’s careful reading of the original submission, leading to several improvements and corrections. -
Chapter 2 Affine Algebraic Geometry
Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F. -
Arxiv:1509.06425V4 [Math.AG] 9 Dec 2017 Nipratrl Ntegoercraiaino Conformal of Gen Realization a Geometric the (And in Fact Role This Important Trivial
TRIVIALITY PROPERTIES OF PRINCIPAL BUNDLES ON SINGULAR CURVES PRAKASH BELKALE AND NAJMUDDIN FAKHRUDDIN Abstract. We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of π1 of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve. 1. Introduction It is a consequence of a theorem of Harder [Har67, Satz 3.3] that generically trivial principal G-bundles on a smooth affine curve C over an arbitrary field k are trivial if G is a semisimple and simply connected algebraic group. When k is algebraically closed and G reductive, generic triviality, conjectured by Serre, was proved by Steinberg[Ste65] and Borel–Springer [BS68]. It follows that principal bundles for simply connected semisimple groups over smooth affine curves over algebraically closed fields are trivial. This fact (and a generalization to families of bundles [DS95]) plays an important role in the geometric realization of conformal blocks for smooth curves as global sections of line bundles on moduli-stacks of principal bundles on the curves (see the review [Sor96] and the references therein). An earlier result of Serre [Ser58, Th´eor`eme 1] (also see [Ati57, Theorem 2]) implies that this triviality property is true if G “ SLprq, and C is a possibly singular affine curve over an arbitrary field k. -
Positivity in Algebraic Geometry I
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample. -
MATH 763 Algebraic Geometry Homework 2
MATH 763 Algebraic Geometry Homework 2 Instructor: Dima Arinkin Author: Yifan Wei September 26, 2019 Problem 1 Prove that a subspace of a noetherian topological space is noetherian in the induced topology. Proof Because in the subspace X the closed sets are exactly restrictions from the ambient space, a descending chain of closed sets in X takes the form · · · ⊃ Ci \ X ⊃ · · · , taking closure in the ambient space we get a descending chain which stablizes to, say Cn \ X, and we have for all i Ci \ X ⊂ Cn \ X. And the inequality can clearly be restricted to X, which means the chain in X stablizes. Problem 2 Show that a topological space is noetherian if and only if all of its open subsets are quasi-compact. Proof If the space X is noetherian, then any open subset is noetherian hence quasi-compact. Conversely, if we have a descending chain of closed subsets C1 ⊃ C2 ⊃ · · · , then we have an open covering (indexed by n): \ [ [ X − Ci = X − Ci; n i≤n S − \ − By quasi-compactness we have X Ci is equal to i≤n X Ci for some n, which means the descending chain stablizes to Cn. Problem 3 Let S ⊂ k be an infinite subset (for instance, S = Z ⊂ k = C.) Show that the Zariski closure of the set X = f(s; s2)js 2 Sg ⊂ A2 1 is the parabola Y = f(a; a2)ja 2 kg ⊂ A2: Proof Consider the morphism ϕ : k[x; y] ! k[t] sending x to t and y to t2, its kernel is precisely the vanishing ideal of the parabola Y . -
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 . -
Choomee Kim's Thesis
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE The Zariski Topology on the Prime Spectrum of a Commutative Ring A thesis submitted in partial fulllment of the requirements for the degree of Master of Science in Mathematics by Choomee Kim December 2018 The thesis of Choomee Kim is approved: Jason Lo, Ph.D. Date Katherine Stevenson, Ph.D. Date Jerry Rosen, Ph.D., Chair Date California State University, Northridge ii Dedication I dedicate this to my beloved family. iii Acknowledgments With deep gratitude and appreciation, I would like to rst acknowledge and thank my thesis advisor, Professor Jerry Rosen, for all of the support and guidance he provided during the past two years of my graduate study. I would like to extend my sincere appreciation to Professor Jason Lo and Professor Katherine Stevenson for serving on my thesis committee and taking time to read and advise. I would also like to thank the CSUN faculty and cohort, especially Professor Mary Rosen, Yen Doung, and all of my oce colleagues, for their genuine friendship, encouragement, and emotional support throughout the course of my graduate study. All this has been truly enriching experience, and I would like to give back to our students and colleagues as to become a professor who cares for others around us. Last, but not least, I would like to thank my husband and my daughter, Darlene, for their emotional and moral support, love, and encouragement, and without which this would not have been possible. iv Table of Contents Signature Page.................................... ii Dedication...................................... iii Acknowledgements.................................. iv Table of Contents.................................. v Abstract........................................ vii 1 Preliminaries.................................. -
Exercises for Helsinki Algebraic Geometry Mini-Course
Exercises for Helsinki Algebraic Geometry Mini-Course Professor Karen E Smith 2010 Toukokuun 17. 18. ja 19. Reading: Most of the material of the lectures, can be found in the book Johdettelua Algebralliseen Geometriaan by Kahanp¨a¨aet al, or in An Invitation to algebraic geometry, by Smith et al, which is the same book in english. Two other good (much more comprehensive) books for beginners are Shafarevich's Basic Algebraic Geometry, I (original language: Russian) and Harris's Algebraic Geom- etry: An first course. Students who wish to get credit for the course can complete (or attempt) the following problems and send a pdf file (handwritten and scanned is fine) to [email protected]. If you are a student at any Finnish University, I will read these and assign a grade any time; for deadlines regarding whether or not credits can be assigned, please ask your professor(s) at your own university; of course, I am happy to contact them regarding the amount of work you completed for the course. Problem 1: The Zariski Toplogy. n n 1. Show that an arbitrary intersection of subvarieties of C is a subvariety of C . n n 2. Show that a finite union of subvarieties of C is a subvariety of C . n 3. Show that C can be given the structure of a topological space whose closed sets n n are the subvarieties of C . This is called the Zariski Topology on C . n 4. Show that the Zariski topology on C is strictly coarser than the standard Eu- n clidean topology on C . -
Lecture 7 - Zariski Topology and Regular Elements Prof
18.745 Introduction to Lie Algebras September 30, 2010 Lecture 7 - Zariski Topology and Regular Elements Prof. Victor Kac Scribe: Daniel Ketover Definition 7.1. A topological space is a set X with a collection of subsets F (the closed sets) satisfying the following axioms: 1) X 2 F and ; 2 F 2) The union of a finite collection of closed sets is closed 3) The intersection of arbitrary collection of closed sets is closed 4) (weak separation axiom) For any two points x; y 2 X there exists an S 2 F such that x 2 S but y 62 S A subset O ⊂ X which is the complement in X of a set in F is called open. The axioms for a topological space can also be phrased in terms of open sets. Definition 7.2. Let X = Fn where F is a field. We define the Zariski topology on X as follows. A set in X is closed if and only if if is the set of common zeroes of a collection (possibly infinite) of n polynomials P훼(x) on F Exercise 7.1. Prove that the Zariski topology is indeed a topology. Proof. We check the axioms. For 1), let p(x) = 1, so that V(p) = ;, so ; 2 F . If p(x) = 0, then n n V(p) = F , so F 2 F . For 2), first take two closed sets V(p훼) with 훼 2 I and V(q훽) with 훽 2 J , and consider the family of polynomials f훼훽 = fp훼q훽 j 훼 2 I; 훽 2 Jg. -
[Li HUISHI] an Introduction to Commutative Algebra(Bookfi.Org)
An Introduction to Commutative Algebra From the Viewpoint of Normalization This page intentionally left blank From the Viewpoint of Normalization I Jiaying University, China NEW JERSEY * LONDON * SINSAPORE - EElJlNG * SHANGHA' * HONG KONG * TAIPEI CHENNAI Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library AN INTRODUCTION TO COMMUTATIVE ALGEBRA From the Viewpoint of Normalization Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof. may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rasewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-951-2 Printed in Singapore by World Scientific Printers (S)Pte Ltd For Pinpin and Chao This page intentionally left blank Preface Why normalization? Over the years I had been bothered by selecting material for teaching several one-semester courses. When I taught senior undergraduate stu- dents a first course in (algebraic) number theory, or when I taught first-year graduate students an introduction to algebraic geometry, students strongly felt the lack of some preliminaries on commutative algebra; while I taught first-year graduate students a course in commutative algebra by quoting some nontrivial examples from number theory and algebraic geometry, of- ten times I found my students having difficulty understanding.