DOCSLIB.ORG

Algebraic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Algebraic Geometry J.S. Milne Version 5.20 September 14, 2009 These notes are an introduction to the theory of algebraic varieties. In contrast to most such accounts they study abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory. BibTeX information @misc{milneAG, author={Milne, James S.}, title={Algebraic Geometry (v5.20)}, year={2009}, note={Available at www.jmilne.org/math/}, pages={239+vi} } v2.01 (August 24, 1996). First version on the web. v3.01 (June 13, 1998). v4.00 (October 30, 2003). Fixed errors; many minor revisions; added exercises; added two sections/chapters; 206 pages. v5.00 (February 20, 2005). Heavily revised; most numbering changed; 227 pages. v5.10 (March 19, 2008). Minor fixes; TeX style changed, so page numbers changed; 241 pages. v5.20 (September 14, 2009). Minor corrections; revised Chapters 1, 11, 16; 245 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of Lake Sylvan, New Zealand. Copyright c 1996, 1998, 2003, 2005, 2008, 2009 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents iii Notations vi; Prerequisites vi; References vi; Acknowledgements vi Introduction 1 1 Preliminaries 4 Rings and algebras 4; Ideals 4; Noetherian rings 6; Unique factorization 8; Polynomial rings 10; Integrality 11; Direct limits (summary) 14; Rings of fractions 15; Tensor Prod- ucts 18; Categories and functors 21; Algorithms for polynomials 23; Exercises 29 2 Algebraic Sets 30 Definition of an algebraic set 30; The Hilbert basis theorem 31; The Zariski topology 32; The Hilbert Nullstellensatz 33; The correspondence between algebraic sets and ideals 34; Finding the radical of an ideal 37; The Zariski topology on an algebraic set 37; The coor- dinate ring of an algebraic set 38; Irreducible algebraic sets 39; Dimension 41; Exercises 44 3 Affine Algebraic Varieties 45 Ringed spaces 45; The ringed space structure on an algebraic set 47; Morphisms of ringed spaces 50; Affine algebraic varieties 50; The category of affine algebraic varieties 51; Explicit description of morphisms of affine varieties 53; Subvarieties 55; Properties of the regular map defined by specm(˛) 56; Affine space without coordinates 57; Exercises 58 4 Algebraic Varieties 60 Algebraic prevarieties 60; Regular maps 61; Algebraic varieties 62; Maps from varieties to affine varieties 63; Subvarieties 64; Prevarieties obtained by patching 65; Products of vari- eties 66; The separation axiom revisited 70; Fibred products 73; Dimension 74; Birational equivalence 75; Dominant maps 76; Algebraic varieties as a functors 76; Exercises 78 5 Local Study 80 Tangent spaces to plane curves 80; Tangent cones to plane curves 82; The local ring at a m point on a curve 82; Tangent spaces of subvarieties of A 83; The differential of a regular map 85; Etale maps 86; Intrinsic definition of the tangent space 88; Nonsingular points 91; Nonsingularity and regularity 92; Nonsingularity and normality 93; Etale neighbourhoods 94; Smooth maps 96; Dual numbers and derivations 97; Tangent cones 100; Exercises 102 6 Projective Varieties 103 n n n Algebraic subsets of P 103; The Zariski topology on P 106; Closed subsets of A and iii n n P 107; The hyperplane at infinity 107; P is an algebraic variety 108; The homogeneous n coordinate ring of a subvariety of P 110; Regular functions on a projective variety 111; Morphisms from projective varieties 112; Examples of regular maps of projective vari- eties 113; Projective space without coordinates 118; Grassmann varieties 118; Bezout’s theorem 122; Hilbert polynomials (sketch) 123; Exercises 124 7 Complete varieties 125 Definition and basic properties 125; Projective varieties are complete 127; Elimination theory 128; The rigidity theorem 130; Theorems of Chow 131; Nagata’s Embedding The- orem 132; Exercises 132 8 Finite Maps 133 Definition and basic properties 133; Noether Normalization Theorem 137; Zariski’s main theorem 138; The base change of a finite map 140; Proper maps 140; Exercises 141 9 Dimension Theory 143 Affine varieties 143; Projective varieties 150 10 Regular Maps and Their Fibres 152 Constructible sets 152; Orbits of group actions 155; The fibres of morphisms 157; The fibres of finite maps 159; Flat maps 160; Lines on surfaces 161; Stein factorization 167; Exercises 167 11 Algebraic spaces; geometry over an arbitrary field 168 Preliminaries 168; Affine algebraic spaces 172; Affine algebraic varieties. 173; Algebraic spaces; algebraic varieties. 174; Local study 179; Projective varieties. 180; Complete varieties. 181; Normal varieties; Finite maps. 181; Dimension theory 181; Regular maps and their fibres 181; Algebraic groups 181; Exercises 182 12 Divisors and Intersection Theory 183 Divisors 183; Intersection theory. 184; Exercises 189 13 Coherent Sheaves; Invertible Sheaves 190 Coherent sheaves 190; Invertible sheaves. 192; Invertible sheaves and divisors. 193; Direct images and inverse images of coherent sheaves. 195; Principal bundles 195 14 Differentials (Outline) 196 15 Algebraic Varieties over the Complex Numbers 198 16 Descent Theory 201 Models 201; Fixed fields 201; Descending subspaces of vector spaces 202; Descending subvarieties and morphisms 204; Galois descent of vector spaces 205; Descent data 206; Galois descent of varieties 210; Weil restriction 211; Generic fibres and specialization 212; Rigid descent 212; Weil’s descent theorems 215; Restatement in terms of group actions 216; Faithfully flat descent 218 17 Lefschetz Pencils (Outline) 222 Definition 222 iv 18 Algebraic Schemes and Algebraic Spaces 225 A Solutions to the exercises 226 B Annotated Bibliography 233 Index 236 v Notations We use the standard (Bourbaki) notations: N 0; 1; 2; : : : , Z ring of integers, R D f g D D field of real numbers, C field of complex numbers, Fp Z=pZ field of p elements, p D D D a prime number. Given an equivalence relation, Œ denotes the equivalence class containing . A family of elements of a set A indexed by a second set I , denoted .ai /i I , is a function 2 i ai I A. 7! W ! A field k is said to be separably closed if it has no finite separable extensions of degree > 1. We use ksep and kal to denote separable and algebraic closures of k respectively. All rings will be commutative with 1, and homomorphisms of rings are required to map 1 to 1. For a ring A, A is the group of units in A: A a A there exists a b A such that ab 1 : D f 2 j 2 D g We use Gothic (fraktur) letters for ideals: abcmnpqABCMNPQ abcmnpqABCMNPQ X def YX is defined to be Y , or equals Y by definition; D X YX is a subset of Y (not necessarily proper, i.e., X may equal Y ); X YX and Y are isomorphic; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism). ' Prerequisites The reader is assumed to be familiar with the basic objects of algebra, namely, rings, mod- ules, fields, and so on, and with transcendental extensions of fields (FT, Section 8). References Atiyah and MacDonald 1969: Introduction to Commutative Algebra, Addison-Wesley. Cox et al. 1992: Varieties, and Algorithms, Springer. FT: Milne, J.S., Fields and Galois Theory, v4.20, 2008 (www.jmilne.org/math/). CA: Milne, J.S., Commutative Algebra, v2.10. Hartshorne 1977: Algebraic Geometry, Springer. Mumford 1999: The Red Book of Varieties and Schemes, Springer. Shafarevich 1994: Basic Algebraic Geometry, Springer. For other references, see the annotated bibliography at the end. Acknowledgements I thank the following for providing corrections and comments on earlier versions of these notes: Sandeep Chellapilla, Umesh V. Dubey, Shalom Feigelstock, B.J. Franklin, Sergei Gelfand, Daniel Gerig, Darij Grinberg, Lucio Guerberoff, Guido Helmers, Christian Hirsch, Jasper Loy Jiabao, Lars Kindler, Sean Rostami, David Rufino, Tom Savage, Nguyen Quoc Thang, Soli Vishkautsan, Dennis Bouke Westra, and others. vi Introduction There is almost nothing left to discover in geometry. Descartes, March 26, 1619 Just as the starting point of linear algebra is the study of the solutions of systems of linear equations, n X aij Xj bi ; i 1; : : : ; m; (1) D D j 1 D the starting point for algebraic geometry is the study of the solutions of systems of polyno- mial equations, fi .X1;:::;Xn/ 0; i 1; : : : ; m; fi kŒX1;:::;Xn: D D 2 Note immediately one difference between linear equations and polynomial equations: the- orems for linear equations don’t depend on which field k you are working over,1 but those for polynomial equations depend on whether or not k is algebraically closed and (to a lesser extent) whether k has characteristic zero. A better description of algebraic geometry is that it is the study of polynomial functions and the spaces on which they are defined (algebraic varieties), just as topology is the study of continuous functions and the spaces on which they are defined (topological spaces), differential topology the study of infinitely differentiable functions and the spaces on which they are defined (differentiable manifolds), and so on: algebraic geometry regular (polynomial) functions algebraic varieties topology continuous functions topological spaces differential topology differentiable functions differentiable manifolds complex analysis analytic (power series) functions complex manifolds. The approach adopted in this course makes plain the similarities between these different areas of mathematics. Of course, the polynomial functions form a much less rich class than the others, but by restricting our study to polynomials we are able to do calculus over any field: we simply define d X i X i 1 ai X iai X : dX D 1 For example, suppose that the system (1) has coefficients aij k and that K is a field containing k.
Recommended publications
  • A Zariski Topology for Modules

    A Zariski Topology for Modules

    A Zariski Topology for Modules∗ Jawad Abuhlail† Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals [email protected] October 18, 2018 Abstract Given a duo module M over an associative (not necessarily commutative) ring R, a Zariski topology is defined on the spectrum Specfp(M) of fully prime R-submodules of M. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration. 1 Introduction The interplay between the properties of a given ring and the topological properties of the Zariski topology defined on its prime spectrum has been studied intensively in the literature (e.g. [LY2006, ST2010, ZTW2006]). On the other hand, many papers considered the so called top modules, i.e. modules whose spectrum of prime submodules attains a Zariski topology (e.g. [Lu1984, Lu1999, MMS1997, MMS1998, Zah2006-a, Zha2006-b]). arXiv:1007.3149v1 [math.RA] 19 Jul 2010 On the other hand, different notions of primeness for modules were introduced and in- vestigated in the literature (e.g. [Dau1978, Wis1996, RRRF-AS2002, RRW2005, Wij2006, Abu2006, WW2009]). In this paper, we consider a notions that was not dealt with, from the topological point of view, so far. Given a duo module M over an associative ring R, we introduce and topologize the spectrum of fully prime R-submodules of M. Motivated by results on the Zariski topology on the spectrum of prime ideals of a commutative ring (e.g. [Bou1998, AM1969]), we investigate the interplay between the topological properties of the obtained space and the module under consideration.
  • Lecture 1 Course Introduction, Zariski Topology

    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 .
  • Arxiv:1610.06871V4 [Math.AT] 13 May 2019 Se[A H.7506 N SG E.B.6.2.7]): Rem

    Arxiv:1610.06871V4 [Math.AT] 13 May 2019 Se[A H.7506 N SG E.B.6.2.7]): Rem

    THE MOREL–VOEVODSKY LOCALIZATION THEOREM IN SPECTRAL ALGEBRAIC GEOMETRY ADEEL A. KHAN Abstract. We prove an analogue of the Morel–Voevodsky localization theorem over spectral algebraic spaces. As a corollary we deduce a “derived nilpotent invariance” result which, informally speaking, says that A1-homotopy invariance kills all higher homotopy groups of a connective commutative ring spectrum. 1. Introduction ´et Let R be a connective E∞-ring spectrum, and denote by CAlgR the ∞-category of ´etale E∞-algebras over R. The starting point for this paper is the following fundamental result of J. Lurie, which says that the small ´etale topos of R is equivalent to the small ´etale topos of π0(R) (see [HA, Thm. 7.5.0.6] and [SAG, Rem. B.6.2.7]): Theorem 1.0.1 (Lurie). For any connective E∞-ring R, let π0(R) denote its 0-truncation E ´et ´et (viewed as a discrete ∞-ring). Then restriction along the canonical functor CAlgR → CAlgπ0(R) ´et induces an equivalence from the ∞-category of ´etale sheaves of spaces CAlgπ0(R) → Spc to the ´et ∞-category of ´etale sheaves of spaces CAlgR → Spc. Theorem 1.0.1 can be viewed as a special case of the following result (see [SAG, Prop. 3.1.4.1]): ′ Theorem 1.0.2 (Lurie). Let R → R be a homomorphism of connective E∞-rings that is ´et ´et surjective on π0. Then restriction along the canonical functor CAlgR → CAlgR′ defines a fully ´et faithful embedding of the ∞-category of ´etale sheaves CAlgR′ → Spc into the ∞-category of ´etale ´et sheaves CAlgR → Spc.
  • Fundamental Algebraic Geometry

    Fundamental Algebraic Geometry

    http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
  • Algebraic Stacks (Winter 2020/2021) Lecturer: Georg Oberdieck

    Algebraic Stacks (Winter 2020/2021) Lecturer: Georg Oberdieck

    Algebraic Stacks (Winter 2020/2021) Lecturer: Georg Oberdieck Date: Wednesday 4-6, Friday 12-2. Oral Exam: February 19 and March 26, 2021. Exam requirement: You have to hand in solutions to at least half of all exercise problems to be admitted to the exam. Summary: The course is an introduction to the theory of algebraic stacks. We will discuss applications to the construction of moduli spaces. Prerequisites: Algebraic Geometry I and II (e.g. on the level of Hartshorne's book Chapter I and II plus some background on flat/etale morphisms). Some category theory (such as Vakil's Notes on Algebraic Geometry, Chapter 1). References: (1) Olsson, Algebraic Spaces and Stacks (2) Stacks Project (3) Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (available online) (4) Halpern-Leistner, The Moduli space, https://themoduli.space (5) Guide to the literature by Jarod Alper (6) Behrend, Introduction to algebraic stacks, lecture notes. The main reference of the coarse is Olsson and will be used throughout, in particular for many of the exercises. However, we will diverge in some details from Olsson's presentation, in particular on quasi-coherent sheaves on algebraic stacks where we follow the stacks project. The notes of Vistoli are also highly recommended for the first part which covers aspects of Grothendieck topologies, fibered categories and descent theory. In the second part we will use the notes of Halpern-Leistner. Exercises and student presentations: Friday's session will be used sometimes for informal discussion of the material covered in the problem sets. From time to time there will also be student presentations, see: http://www.math.uni-bonn.de/~georgo/stacks/presentations.pdf Schedule1 Oct 28: Motivation mostly.
  • Algebra & Number Theory Vol. 10 (2016)

    Algebra & Number Theory Vol. 10 (2016)

    Algebra & Number Theory Volume 10 2016 No. 4 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Anand Pillay University of Notre Dame, USA Brian D. Conrad Stanford University, USA Victor Reiner University of Minnesota, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Sergey Fomin University of Michigan, USA Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Vasudevan Srinivas Tata Inst. of Fund. Research, India Andrew Granville Université de Montréal, Canada J. Toby Stafford University of Michigan, USA Joseph Gubeladze San Francisco State University, USA Ravi Vakil Stanford University, USA Roger Heath-Brown Oxford University, UK Michel van den Bergh Hasselt University, Belgium Craig Huneke University of Virginia, USA Marie-France Vignéras Université Paris VII, France Kiran S. Kedlaya Univ. of California, San Diego, USA Kei-Ichi Watanabe Nihon University, Japan János Kollár Princeton University, USA Efim Zelmanov University of California, San Diego, USA Yuri Manin Northwestern University, USA Shou-Wu Zhang Princeton University, USA Philippe Michel École Polytechnique Fédérale de Lausanne PRODUCTION [email protected] Silvio Levy, Scientific Editor See inside back cover or msp.org/ant for submission instructions.
  • Arxiv:1509.06425V4 [Math.AG] 9 Dec 2017 Nipratrl Ntegoercraiaino Conformal of Gen Realization a Geometric the (And in Fact Role This Important Trivial

    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

    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

    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

    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 .
  • Moving Codimension-One Subvarieties Over Finite Fields

    Moving Codimension-One Subvarieties Over Finite Fields

    Moving codimension-one subvarieties over finite fields Burt Totaro In topology, the normal bundle of a submanifold determines a neighborhood of the submanifold up to isomorphism. In particular, the normal bundle of a codimension-one submanifold is trivial if and only if the submanifold can be moved in a family of disjoint submanifolds. In algebraic geometry, however, there are higher-order obstructions to moving a given subvariety. In this paper, we develop an obstruction theory, in the spirit of homotopy theory, which gives some control over when a codimension-one subvariety moves in a family of disjoint subvarieties. Even if a subvariety does not move in a family, some positive multiple of it may. We find a pattern linking the infinitely many obstructions to moving higher and higher multiples of a given subvariety. As an application, we find the first examples of line bundles L on smooth projective varieties over finite fields which are nef (L has nonnegative degree on every curve) but not semi-ample (no positive power of L is spanned by its global sections). This answers questions by Keel and Mumford. Determining which line bundles are spanned by their global sections, or more generally are semi-ample, is a fundamental issue in algebraic geometry. If a line bundle L is semi-ample, then the powers of L determine a morphism from the given variety onto some projective variety. One of the main problems of the minimal model program, the abundance conjecture, predicts that a variety with nef canonical bundle should have semi-ample canonical bundle [15, Conjecture 3.12].
  • Choomee Kim's Thesis

    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..................................