DOCSLIB.ORG

LECTURE Series

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
LECTURE Series University LECTURE Series Volume 47 Residues and Duality for Projective Algebraic Varieties Ernst Kunz with the assistance of and contributions by David A. Cox and Alicia Dickenstein American Mathematical Society http://dx.doi.org/10.1090/ulect/047 Residues and Duality for Projective Algebraic Varieties University LECTURE Series Volume 47 Residues and Duality for Projective Algebraic Varieties Ernst Kunz with the assistance of and contributions by David A. Cox and Alicia Dickenstein M THE ATI A CA M L 42(4/3 -( N %)3)47 S A O C C I I R E E T !'%7-% Y M A F O 8 U 88 NDED 1 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona NigelD.Higson Eric M. Friedlander (Chair) J. T. Stafford 2000 Mathematics Subject Classification. Primary 14Fxx, 14F10, 14B15; Secondary 32A27, 14M10, 14M25. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-47 Library of Congress Cataloging-in-Publication Data Kunz, Ernst, 1933– Residues and duality for projective algebraic varieties / Ernst Kunz ; with the assistance of and contributions by David A. Cox and Alicia Dickenstein. p. cm. — (University lecture series ; v. 47) Includes bibliographical references and index. ISBN 978-0-8218-4760-2 (alk. paper) 1. Algebraic varieties. 2. Geometry, Projective. 3. Congruences and residues. I. Title. QA564.K86 2009 516.353—dc22 2008038860 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 acknowledgmentofthesourceisgiven. 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. Requests can also be made by e-mail to [email protected]. c 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 131211100908 Contents Preface .............................................................vii GlossaryofNotation ................................................ ix 1. LocalCohomologyFunctors ..........................................1 2. LocalCohomologyofNoetherianAffineSchemes .....................6 3. CechCohomologyˇ ...................................................11 4. KoszulComplexesandLocalCohomology ...........................22 5. ResiduesandLocalCohomologyforPowerSeriesRings .............35 6. TheCohomologyofProjectiveSchemes .............................47 7. DualityandResidueTheoremsforProjectiveSpace .................52 8. Traces,ComplementaryModules,andDifferents .....................65 9. The Sheaf of Regular Differential Forms on an Algebraic Variety . 81 10. ResiduesforAlgebraicVarieties. LocalDuality ......................89 11. DualityandResidueTheoremsforProjectiveVarieties .............100 12. CompleteDuality ..................................................110 13. ApplicationsofResiduesandDuality(AliciaDickenstein) ..........115 14. ToricResidues(DavidA.Cox) .....................................128 Bibliography........................................................151 Index...............................................................155 v Preface The present text is an extended and updated version of my lecture notes Residuen und Dualit¨at auf projektiven algebraischen Variet¨aten (Der Regensburger Trichter 19 (1986)), based on a course I taught in the winter term 1985/86 at the University of Regensburg. I am grateful to David Cox for helping me with the translation and transforming the manuscript into the appropriate LATEX2ε style, to Alicia Dickenstein and David Cox for encouragement and critical comments and for enriching the book by adding two sections, one on applications of algebraic residue theory and the other explaining toric residues and relating them to the earlier text. The main objective of my old lectures, which were strongly influenced by Lip- man’s monograph [71], was to describe local and global duality in the special case of irreducible algebraic varieties over an algebraically closed base field k in terms of dif- ferential forms and their residues. Although the dualizing sheaf of a d-dimensional algebraic variety V is only unique up to isomorphism, there is a canonical choice, the sheaf ωV/k of regular d-forms. This sheaf is an intrinsically defined subsheaf of d the constant sheaf ΩR(V )/k,whereR(V ) is the field of rational functions on V .We construct ωV/k in § 9 after the necessary preparation. Similarly, for a closed point x ∈ V ,thestalk(ωV/k)x is a canonical choice for the dualizing (canonical) module ωOV,x/k studied in local algebra. We have the residue map d −→ Resx : Hx (ωV/k) k defined on the d-th local cohomology of ωV/k. The local cohomology classes can be written as generalized fractions ω f1,...,fd ∈ O where ω ωOV,x/k and f1,...,fd is a system of parameters of V,x.Usingthe residue map, we get the Grothendieck residue symbol ω Resx . f1,...,fd For a projective variety V the residue map at the vertex of the affine cone C(V ) induces a linear operator on global cohomology d : H (V,ωV/k) −→ k V called the integral. The local and global duality theorems are formulated in terms of Resx and V . There is also the residue theorem stating that “the integral is the sum of all of the residues.” Specializing to projective algebraic curves gives the usual residue theorem for curves plus a version of the Serre duality theorem expressed in terms of differentials and their residues. Basic rules of the residue calculus are formulated and proved, and later generalized to toric residues by David vii viii PREFACE Cox. Because of the growing current interest in performing explicit calculations in algebraic geometry, we hope that our description of duality theory in terms of differential forms and their residues will prove to be useful. The residues ω Resx f1,...,fd can be considered as intersection invariants, and by a suitable choice of the regular d-form ω, a residue can have many geometric interpretations, including intersection multiplicity, angle of intersection, curvature, and the centroid of a zero-dimensional scheme. The residue theorem then gives a global relation for these local invariants. In this way, classical results of algebraic geometry can be reproved and generalized. It is part of the culture to relate current theories to the achievements of former times. This point of view is stressed in the present notes, and it is particularly sat- isfying that some applications of residues and duality reach back to antiquity (the- orems of Apollonius and Pappus). Alicia Dickenstein gives applications of residues and duality to partial differential equations and problems in interpolation and ideal membership. Since the book is introductory in nature, only some aspects of duality the- ory can be covered. Of course the theory has been developed much further in the last decades, by Lipman and his coworkers among others. At appropriate places, the text includes references to articles that appeared after the publication of Hartshorne’s Residues and Duality [38]; see for instance the remarks following Corollary 11.9 and those at the end of § 12. These articles extend the theory of the book considerably in many directions. This leads to a large bibliography, though it is likely that some important relevant work has been missed. For this, I apologize. The students in my course were already familiar with commutative algebra, including K¨ahler differentials, and they knew basic algebraic geometry. Some of them had profited from the exchange program between the University of Regensburg and Brandeis University, where they attended a course taught by David Eisenbud out of Hartshorne’s book [39]. Similar prerequisites are assumed about the reader of the present text. The section by David Cox requires a basic knowledge of toric geometry. I want to thank the students of my lectures who insisted on clearer exposition, especially Reinhold H¨ubl, Martin Kreuzer, Markus N¨ubler and Gerhard Quarg, all of whom also later worked on algebraic residue theory, much to my benefit and the benefit of this book. Thanks are also due to the referees for their suggestions and comments and to Ina Mette for her support of this project. July 2008 Ernst Kunz Fakult¨at f¨ur Mathematik Universit¨at Regensburg D-93040 Regensburg, Germany David A. Cox Department of Mathematics & Computer Science Amherst College Amherst, MA 01002, USA Alicia Dickenstein Departamento de Matem´atica, FCEN Universidad de Buenos Aires Cuidad Universitaria-Pabell´on I (C1428EGA) Buenos Aires, Argentina Glossary of Notation Unless otherwise stated all rings are commutative with 1. A multiplicatively closed set of a ring R always contains 1R. A ring homomorphism R → S maps 1R to 1S. This is also called an algebra S/R. We try to use the standard notation and language of algebraic geometry. N {0, 1, 2,...} N+ {1, 2, 3,...} U(X) set of open subsets of X 1
Load more

Recommended publications
  • Cellular Sheaf Cohomology in Polymake Arxiv:1612.09526V1
    Cellular sheaf cohomology in Polymake Lars Kastner, Kristin Shaw, Anna-Lena Winz July 9, 2018 Abstract This chapter provides a guide to our polymake extension cellularSheaves. We first define cellular sheaves on polyhedral complexes in Euclidean space, as well as cosheaves, and their (co)homologies. As motivation, we summarise some results from toric and tropical geometry linking cellular sheaf cohomologies to cohomologies of algebraic varieties. We then give an overview of the structure of the extension cellularSheaves for polymake. Finally, we illustrate the usage of the extension with examples from toric and tropical geometry. 1 Introduction n Given a polyhedral complex Π in R , a cellular sheaf (of vector spaces) on Π associates to every face of Π a vector space and to every face relation a map of the associated vector spaces (see Definition 2). Just as with usual sheaves, we can compute the cohomology of cellular sheaves (see Definition 5). The advantage is that cellular sheaf cohomology is the cohomology of a chain complex consisting of finite dimensional vector spaces (see Definition 3). Polyhedral complexes often arise in algebraic geometry. Moreover, in some cases, invariants of algebraic varieties can be recovered as cohomology of certain cellular sheaves on polyhedral complexes. Our two major classes of examples come from toric and tropical geometry and are presented in Section 2.2. We refer the reader to [9] for a guide to toric geometry and polytopes. For an introduction to tropical geometry see [4], [20]. The main motivation for this polymake extension was to implement tropical homology, as introduced by Itenberg, Katzarkov, Mikhalkin, and Zharkov in [15].
    [Show full text]
  • Sheaves and Homotopy Theory
    SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case.
    [Show full text]
  • The Calabi Complex and Killing Sheaf Cohomology
    The Calabi complex and Killing sheaf cohomology Igor Khavkine Department of Mathematics, University of Trento, and TIFPA-INFN, Trento, I{38123 Povo (TN) Italy [email protected] September 26, 2014 Abstract It has recently been noticed that the degeneracies of the Poisson bra- cket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of dif- ferential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of lo- cally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of general- ized Poincar´eduality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds. Contents 1 Introduction2 2 The Calabi complex4 2.1 Tensor bundles and Young symmetrizers..............5 2.2 Differential operators.........................7 2.3 Formal adjoint complex....................... 11 2.4 Equations of finite type, twisted de Rham complex........ 14 3 Cohomology of locally constant sheaves 16 3.1 Locally constant sheaves....................... 16 3.2 Acyclic resolution by a differential complex............ 18 3.3 Generalized Poincar´eduality...................
    [Show full text]
  • Duality, Part 1: Dual Bases and Dual Maps Notation
    Duality, part 1: Dual Bases and Dual Maps Notation F denotes either R or C. V and W denote vector spaces over F. Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Examples: Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx.
    [Show full text]
  • Introduction to Generalized Sheaf Cohomology
    Introduction to Generalized Sheaf Cohomology Peter Schneider 1 Right derived functors Let M be a category and let S be a class of morphisms in M. The localiza- γ tion M −!Mloc of M with respect to S (see [GZ]) is characterized by the universal property that γ(s) for any s 2 S is an isomorphism and that for any functor F : M!B which transforms the morphisms in S into isomor- phisms in B there is a unique functor F : Mloc !B such that F = F ◦ γ. Now let F : M!B be an arbitrary functor. We then want at least a functor RF : Mloc !B such that RF ◦ γ is as \close" as possible to F . Definition 1.1. A right derivation of F is a functor RF : Mloc −! B together with a natural transformation η : F ! RF ◦ γ such that, for any functor G : Mloc !B, the map ∼ natural transf. (RF; G) −−! natural transf. (F; G ◦ γ) ϵ 7−! (ϵ ∗ γ) ◦ η is bijective. Since the pair (RF; η) if it exists is unique up to unique isomorphism we usually will refer to RF as the right derived functor of F . There seems to be no completely general result about the existence of right derived functors; in various different situations one has different methods to construct them. In the following we will discuss an appropriate variant of the method of \resolutions" which will be applicable in all situations of interest to us. From now on we always assume that the class S satisfies the condition that %& in any commutative diagram −! in M (∗) in which two arrows represent morphisms in S also the third arrow represents a morphism in S.
    [Show full text]
  • Sheaf Theory
    Sheaf Theory Anne Vaugon December 20, 2013 The goals of this talk are • to define a generalization denoted by RΓ(F) of de Rham cohomology; • to explain the notation RΓ(F) (here F is a sheaf and RΓ is a derived functor). 1 Presheaves and sheaves 1.1 Definitions and examples Let X be a topological space. Definition 1.1. A presheaf of k-modules F on X is defined by the following data: • a k-module F(U) for each open set U of X; • a map rUV : F(U) → F(V ) for each pair V ⊂ U of open subsets such that – rWV ◦ rVU = rWU for all open subsets W ⊂ V ⊂ U; – rUU = Id for all open subsets U. Therefore, a presheaf is a functor from the opposite category of open sets to the category of k-modules. If F is a presheaf, F(U) is called the set of sections of U and rVU the restriction from U to V . Definition 1.2. A presheaf F is a sheaf if • for any family (Ui)i∈I of open subsets of X • for any family of elements si ∈ F(Ui) such that rUi∩Uj ,Ui (si) = rUi∩Uj ,Uj (sj) for all i, j ∈ I there exists a unique s ∈ F(U) where U = ∪i∈I Ui such that rUi,U (s) = si for all i ∈ I. This means that we can extend a locally defined section. Definition 1.3. A morphism of presheaves f : F → G is a natural trans- formation between the functors F and G: for each open set U, there exists a morphism f(U): F(U) → G(U) such that the following diagram is commutative for V ⊂ U.
    [Show full text]
  • SHEAVES of MODULES 01AC Contents 1. Introduction 1 2
    SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter.
    [Show full text]
  • Arxiv:1307.5568V2 [Math.AG]
    PARTIAL POSITIVITY: GEOMETRY AND COHOMOLOGY OF q-AMPLE LINE BUNDLES DANIEL GREB AND ALEX KURONYA¨ To Rob Lazarsfeld on the occasion of his 60th birthday Abstract. We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs. Contents 1. Introduction 1 2. Overview of the theory of q-ample line bundles 4 2.1. Vanishing of cohomology groups and partial ampleness 4 2.2. Basic properties of q-ampleness 7 2.3. Sommese’s geometric q-ampleness 15 2.4. Ample subschemes, and a Lefschetz hyperplane theorem for q-ample divisors 17 3. q-Kodaira vanishing for Du Bois divisors and log canonical pairs 19 References 23 1. Introduction Ampleness is one of the central notions of algebraic geometry, possessing the extremely useful feature that it has geometric, numerical, and cohomological characterizations. Here we will concentrate on its cohomological side. The fundamental result in this direction is the theorem of Cartan–Serre–Grothendieck (see [Laz04, Theorem 1.2.6]): for a complete arXiv:1307.5568v2 [math.AG] 23 Jan 2014 projective scheme X, and a line bundle L on X, the following are equivalent to L being ample: ⊗m (1) There exists a positive integer m0 = m0(X, L) such that L is very ample for all m ≥ m0. (2) For every coherent sheaf F on X, there exists a positive integer m1 = m1(X, F, L) ⊗m for which F ⊗ L is globally generated for all m ≥ m1.
    [Show full text]
  • Recent Developments in the Theory of Duality in Locally Convex Vector Spaces
    [ VOLUME 6 I ISSUE 2 I APRIL– JUNE 2019] E ISSN 2348 –1269, PRINT ISSN 2349-5138 RECENT DEVELOPMENTS IN THE THEORY OF DUALITY IN LOCALLY CONVEX VECTOR SPACES CHETNA KUMARI1 & RABISH KUMAR2* 1Research Scholar, University Department of Mathematics, B. R. A. Bihar University, Muzaffarpur 2*Research Scholar, University Department of Mathematics T. M. B. University, Bhagalpur Received: February 19, 2019 Accepted: April 01, 2019 ABSTRACT: : The present paper concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Key Words: LOCALLY CONVEX SPACES We shall be exclusively concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A convex set A in a vector space E is symmetric if —A=A; then 0ЄA if A is not empty. A convex set A is absorbing if for every X≠0 in E), there exists a number α≠0 such that λxЄA for |λ| ≤ α ; this implies that A generates E. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Conversely, if any filter base is given on a vector space E, and consists of convex, symmetric, and absorbing sets, then it defines one and only one topology on E for which x+y and λx are continuous functions of both their arguments.
    [Show full text]
  • Geometry of Algebraic Curves
    Geometry of Algebraic Curves Fall 2011 Course taught by Joe Harris Notes by Atanas Atanasov One Oxford Street, Cambridge, MA 02138 E-mail address: [email protected] Contents Lecture 1. September 2, 2011 6 Lecture 2. September 7, 2011 10 2.1. Riemann surfaces associated to a polynomial 10 2.2. The degree of KX and Riemann-Hurwitz 13 2.3. Maps into projective space 15 2.4. An amusing fact 16 Lecture 3. September 9, 2011 17 3.1. Embedding Riemann surfaces in projective space 17 3.2. Geometric Riemann-Roch 17 3.3. Adjunction 18 Lecture 4. September 12, 2011 21 4.1. A change of viewpoint 21 4.2. The Brill-Noether problem 21 Lecture 5. September 16, 2011 25 5.1. Remark on a homework problem 25 5.2. Abel's Theorem 25 5.3. Examples and applications 27 Lecture 6. September 21, 2011 30 6.1. The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33 Lecture 7. September 23, 2011 35 7.1. More on divisors 35 7.2. Riemann-Roch, finally 36 7.3. Fun applications 37 7.4. Sheaf cohomology 37 Lecture 8. September 28, 2011 40 8.1. Examples of low genus 40 8.2. Hyperelliptic curves 40 8.3. Low genus examples 42 Lecture 9. September 30, 2011 44 9.1. Automorphisms of genus 0 an 1 curves 44 9.2.
    [Show full text]
  • Math 632: Algebraic Geometry Ii Cohomology on Algebraic Varieties
    MATH 632: ALGEBRAIC GEOMETRY II COHOMOLOGY ON ALGEBRAIC VARIETIES LECTURES BY PROF. MIRCEA MUSTA¸TA;˘ NOTES BY ALEKSANDER HORAWA These are notes from Math 632: Algebraic geometry II taught by Professor Mircea Musta¸t˘a in Winter 2018, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from May 24, 2018. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at [email protected]. The problem sets, homeworks, and official notes can be found on the course website: http://www-personal.umich.edu/~mmustata/632-2018.html This course is a continuation of Math 631: Algebraic Geometry I. We will assume the material of that course and use the results without specific references. For notes from the classes (similar to these), see: http://www-personal.umich.edu/~ahorawa/math_631.pdf and for the official lecture notes, see: http://www-personal.umich.edu/~mmustata/ag-1213-2017.pdf The focus of the previous part of the course was on algebraic varieties and it will continue this course. Algebraic varieties are closer to geometric intuition than schemes and understanding them well should make learning schemes later easy. The focus will be placed on sheaves, technical tools such as cohomology, and their applications. Date: May 24, 2018. 1 2 MIRCEA MUSTA¸TA˘ Contents 1. Sheaves3 1.1. Quasicoherent and coherent sheaves on algebraic varieties3 1.2. Locally free sheaves8 1.3.
    [Show full text]
  • SHEAVES, TOPOSES, LANGUAGES Acceleration Based on A
    Chapter 7 Logic of behavior: Sheaves, toposes, and internal languages 7.1 How can we prove our machine is safe? Imagine you are trying to design a system of interacting components. You wouldn’t be doing this if you didn’t have a goal in mind: you want the system to do something, to behave in a certain way. In other words, you want to restrict its possibilities to a smaller set: you want the car to remain on the road, you want the temperature to remain in a particular range, you want the bridge to be safe for trucks to pass. Out of all the possibilities, your system should only permit some. Since your system is made of components that interact in specified ways, the possible behavior of the whole—in any environment—is determined by the possible behaviors of each of its components in their local environments, together with the precise way in which they interact.1 In this chapter, we will discuss a logic wherein one can describe general types of behavior that occur over time, and prove properties of a larger-scale system from the properties and interaction patterns of its components. For example, suppose we want an autonomous vehicle to maintain a distance of some safe R from other objects. To do so, several components must interact: a 2 sensor that approximates the real distance by an internal variable S0, a controller that uses S0 to decide what action A to take, and a motor that moves the vehicle with an 1 The well-known concept of emergence is not about possibilities, it is about prediction.
    [Show full text]