DOCSLIB.ORG

Derived Algebraic Geometry V: Structured Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Derived Algebraic Geometry V: Structured Spaces February 22, 2011 Contents 1 Structure Sheaves 7 1.1 C-Valued Sheaves . 9 1.2 Geometries . 12 1.3 The Factorization System on StrG(X)................................ 20 1.4 Classifying 1-Topoi . 26 1.5 1-Categories of Structure Sheaves . 30 2 Scheme Theory 35 2.1 Construction of Spectra: Relative Version . 36 2.2 Construction of Spectra: Absolute Version . 43 2.3 G-Schemes . 50 2.4 The Functor of Points . 60 2.5 Algebraic Geometry (Zariski topology) . 69 2.6 Algebraic Geometry (Etale´ topology) . 75 3 Smoothness 82 3.1 Pregeometries . 83 3.2 Transformations and Morita Equivalence . 87 3.3 1-Categories of T-Structures . 93 3.4 Geometric Envelopes . 97 3.5 Smooth Affine Schemes . 105 4 Examples of Pregeometries 109 4.1 Simplicial Commutative Rings . 109 4.2 Derived Algebraic Geometry (Zariski topology) . 115 4.3 Derived Algebraic Geometry (Etale´ topology) . 122 4.4 Derived Differential Topology . 133 1 Introduction: Bezout's Theorem Let C; C0 ⊆ P2 be two smooth algebraic curves of degrees m and n in the complex projective plane P2. If C and C0 meet transversely, then the classical theorem of Bezout (see for example [4]) asserts that C \ C0 has precisely mn points. We may reformulate the above statement using the language of cohomology. The curves C and C0 have fundamental classes [C]; [C0] 2 H2(P2; Z). If C and C0 meet transversely, then we have the formula [C] [ [C0] = [C \ C0]; where the fundamental class [C \ C0] 2 H4(P2; Z) ' Z of the intersection C \ C0 simply counts the number of points where C and C0 meet. Of course, this should not be surprising: the cup product on cohomology classes is defined so as to encode the operation of intersection. However, it would be a mistake to regard the equation [C] [ [C0] = [C \ C0] as obvious, because it is not always true. For example, if the curves C and C0 meet nontransversely (but still in a finite number of points), then we always have a strict inequality [C] [ [C0] > [C \ C0] if the right hand side is again interpreted as counting the number of points in the set-theoretic intersection of C and C0. If we want a formula which is valid for non-transverse intersections, then we must alter the definition of [C \ C0] so that it reflects the appropriate intersection multiplicities. Determination of these intersection multiplicities requires knowledge of the intersection C \ C0 as a scheme, rather than simply as a set. This is one of the classic arguments that nonreduced scheme structures carry useful information: the intersection number [C][[C0] 2 Z, which is defined a priori by perturbing the curves so that they meet transversally, can also be computed directly (without perturbation) if one is willing to contemplate a potentially non-reduced scheme structure on the intersection. In more complicated situations, the appropriate intersection multiplicities cannot always be determined from the scheme-theoretic intersection alone. Suppose that C and C0 are smooth subvarieties of Pn of complementary dimension, having a zero-dimensional intersection. In this case, the appropriate intersection number associated to a point p 2 C \ C0 is not always given by the complex dimension of the local ring 0 0 OC\C ;p = OC;p ⊗OPn;p OC ;p : The reason for this is easy to understand from the point of view of homological algebra. Since the tensor product functor ⊗OPn;p is not exact, it does not have good properties when considered alone. According to Serre's intersection formula, the correct intersection multiplicity is instead the Euler characteristic X i OPn;p (−1) dim Tori (OC;p; OC0;p): This Euler characteristic contains the dimension of the local ring of the scheme-theoretic intersection as its leading term, but also higher-order corrections. We refer the reader to [15] for further discussion of this formula for the intersection multiplicity. If we would like the equation [C] [ [C0] = [C \ C0] to remain valid in the more complicated situations described above, then we will need to interpret the intersection C \ C0 in some way which remembers not 0 only the tensor product OC;p ⊗OPn;p OC ;p, but the higher Tor groups. Moreover, we should not interpret L these Tor-groups separately, but rather should think of the total derived functor O ⊗ O 0 as a kind C;p OPn;p C ;p of generalized ring. These considerations lead us naturally to the subject of derived algebraic geometry. Using an appropriate notion of \generalized ring", we will mimic the constructions of classical scheme theory to obtain a theory of derived schemes in which a version of the formula [C][[C0] = [C \C0] can be shown to hold with (essentially) no hypotheses on C and C0. Here, we must interpret the intersection C \ C0 in the sense of derived schemes, 2 and we must take great care to give the proper definition for the fundamental classes (the so-called virtual fundamental classes of [1]). What sort of objects should our generalized rings be? To answer this question, we begin by considering the simplest case of Bezout's theorem, in which C and C0 are lines in the projective plane P2. In this case, we know that [C] [ [C0] is the cohomology class of a point, and that C intersects C0 transversely in one point so long as C and C0 are distinct. However, when the equality C = C0 holds, the scheme-theoretic intersection C \ C0 does not even have the correct dimension. Let us now try to formulate a theory of intersections which will handle the degenerate situation where C = C0. To simplify the discussion, we will consider only lines in the affine plane A2 ⊆ P2, with coordinate ring C[x; y]. Two distinct lines in A2 may be given (without loss of generality) by the equations x = 0 and y = 0. The scheme-theoretic intersection of these two lines is the spectrum of the ring C[x; y]=(x; y) ' C, obtained from C[x; y] by enforcing the defining equations of both lines. This ring has Krull dimension 0 because C[x; y] has Krull dimension 2 and we have imposed 2 independent conditions. Now suppose that instead of C and C0 being two distinct lines, they are actually two identical lines, both of which are defined by the equation x = 0. In this case, the affine ring of the scheme theoretic intersection is given by C[x; y]=(x; x) ' C[y]. This ring has Krull dimension 1, rather than the expected dimension 0, because the two equations are not independent: setting x equal to zero twice has the same effect as setting x equal to zero once. To obtain the theory we are looking for, we need a notion of generalized ring which remembers not only whether or not x is equal to 0, but how many different ways x is equal to 0. One way to obtain such a formalism is by categorifying the notion of a commutative ring. That is, in place of ordinary commutative rings, we consider categories equipped with addition and multiplication operations (which are encoded by functors, rather than ordinary functions). For purposes of the present discussion, let us call such an object a categorical ring. We will not give a precise axiomatization of this notion; this turns out to be somewhat complicated (see [7], for example). Example 0.0.1. Let Z≥0 denote the semiring of nonnegative integers. We note that Z≥0 arises in nature through the process of decategorification. The nonnegative integers were originally introduced in order to count: in other words, in order to measure the size of finite sets. To make this statement more precise, let us denote by Fin the category whose objects are finite sets, and whose morphisms are isomorphisms of finite sets. Then we can identify Z≥0 with the set of isomorphism classes of objects in Fin. The addition and multiplication operations on Z≥0 are induced by functors Fin × Fin ! Fin, given by the formation of disjoint union and Cartesian product. Moreover, all of the axioms for a commutative semiring have analogues that hold at the categorical level: for example, the distributive law xy+xz = x(y+z) translates into the existence of a canonical isomorphism a a (X × Y ) (X × Z) ' X × (Y Z) for every triple of objects X; Y; Z 2 Fin. (In order to complete the analogy with the above discussion, we should \complete" the category Fin by formally adjoining inverses, to obtain a categorical ring rather than a categorical semiring, but we will ignore this point for the time being.) To simplify the discussion, we will consider only categorical rings which are groupoids: that is, every morphism in the underlying category is an isomorphism. If C is a categorical ring, then the collection of isomorphism classes of objects in C forms an ordinary commutative ring, which we will denote by π0 C. Every commutative ring R arises in this way: for example, we may take C to be a category whose objects are the elements of R and which contains only identity maps for morphisms. The categorical rings which arise in this way are very special: their objects have no nontrivial automorphisms. For a given commutative ring R, there are usually many other ways to realize an isomorphism of R with the collection of isomorphism classes of objects in a categorical ring C. Although C is not uniquely determined by R, there is often a natural choice for C which is dictated by the manner in which R is constructed.
Recommended publications
  • EXERCISES 1 Exercise 1.1. Let X = (|X|,O X) Be a Scheme and I Is A

    EXERCISES 1 Exercise 1.1. Let X = (|X|,O X) Be a Scheme and I Is A

    EXERCISES 1 Exercise 1.1. Let X = (|X|, OX ) be a scheme and I is a quasi-coherent OX -module. Show that the ringed space X[I] := (|X|, OX [I]) is also a scheme. Exercise 1.2. Let C be a category and h : C → Func(C◦, Set) the Yoneda embedding. Show ∼ that for any arrows X → Y and Z → Y in C, there is a natural isomorphism hX×Z Y → hX ×hY hZ . Exercise 1.3. Let A → R be a ring homomorphism. Verify that under the identification 2 of DerA(R, ΩR/A) with the sections of the diagonal map R ⊗A R/J → R given in lecture, 1 the universal derivation d : R → ΩR/A corresponds to the section given by sending x ∈ R to 1 ⊗ x. Exercise 1.4. Let AffZ be the category of affine schemes. The Yoneda functor h gives rise to a related functor hAff : Sch → Func(Aff◦ , Set) Z Z which only considers the functor of points for affine schemes. Is this functor still fully faithful? Cultural note: since Aff◦ is equivalent to the category of rings ( -algebras!), we can also Z Z view h as defining a (covariant) functor on the category of rings. When X is an affine scheme Aff of the form Z[{xi}]/({fj}], the value of hX on A is just the set of solutions of the fj with coordinates in A. Exercise 1.5. Let R be a ring and H : ModR → Set a functor which commutes with finite products. Verify the claim in lecture that H(I) has an R-module structure.
  • Elements of -Category Theory Joint with Dominic Verity

    Elements of -Category Theory Joint with Dominic Verity

    Emily Riehl Johns Hopkins University Elements of ∞-Category Theory joint with Dominic Verity MATRIX seminar ∞-categories in the wild A recent phenomenon in certain areas of mathematics is the use of ∞-categories to state and prove theorems: • 푛-jets correspond to 푛-excisive functors in the Goodwillie tangent structure on the ∞-category of differentiable ∞-categories — Bauer–Burke–Ching, “Tangent ∞-categories and Goodwillie calculus” • 푆1-equivariant quasicoherent sheaves on the loop space of a smooth scheme correspond to sheaves with a flat connection as an equivalence of ∞-categories — Ben-Zvi–Nadler, “Loop spaces and connections” • the factorization homology of an 푛-cobordism with coefficients in an 푛-disk algebra is linearly dual to the factorization homology valued in the formal moduli functor as a natural equivalence between functors between ∞-categories — Ayala–Francis, “Poincaré/Koszul duality” Here “∞-category” is a nickname for (∞, 1)-category, a special case of an (∞, 푛)-category, a weak infinite dimensional category in which all morphisms above dimension 푛 are invertible (for fixed 0 ≤ 푛 ≤ ∞). What are ∞-categories and what are they for? It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category. —Barry Mazur, “When is one thing equal to some other thing?” An ∞-category frames a template with nouns, verbs, adjectives, adverbs, pronouns, prepositions, conjunctions, interjections,… which has: • objects • and 1-morphisms between them • • • • composition witnessed by invertible 2-morphisms 푓 푔 훼≃ ℎ∘푔∘푓 • • • • 푔∘푓 푓 ℎ∘푔 훾≃ witnessed by • associativity • ≃ 푔∘푓 훼≃ 훽 ℎ invertible 3-morphisms 푔 • with these witnesses coherent up to invertible morphisms all the way up.
  • Hirschdx.Pdf

    Hirschdx.Pdf

    130 5. Degrees, Intersection Numbers, and the Euler Characteristic. 2. Intersection Numbers and the Euler Characteristic M c: Jr+1 Exercises 8. Let be a compad lHlimensionaJ submanifold without boundary. Two points x, y E R"+' - M are separated by M if and only if lkl(x,Y}.MI * O. lSee 1. A complex polynomial of degree n defines a map of the Riemann sphere to itself Exercise 7.) of degree n. What is the degree of the map defined by a rational function p(z)!q(z)? 9. The Hopfinvariant ofa map f:5' ~ 5' is defined to be the linking number Hlf) ~ 2. (a) Let M, N, P be compact connected oriented n·manifolds without boundaries Lk(g-l(a),g-l(b)) (see Exercise 7) where 9 is a c~ map homotopic 10 f and a, b are and M .!, N 1. P continuous maps. Then deg(fg) ~ (deg g)(deg f). The same holds distinct regular values of g. The linking number is computed in mod 2 if M, N, P are not oriented. (b) The degree of a homeomorphism or homotopy equivalence is ± 1. f(c) * a, b. *3. Let IDl. be the category whose objects are compact connected n-manifolds and whose (a) H(f) is a well-defined homotopy invariant off which vanishes iff is nuD homo- topic. morphisms are homotopy classes (f] of maps f:M ~ N. For an object M let 7t"(M) (b) If g:5' ~ 5' has degree p then H(fg) ~ pH(f). be the set of homotopy classes M ~ 5".
  • Diagrammatics in Categorification and Compositionality

    Diagrammatics in Categorification and Compositionality

    Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT.
  • The Hecke Bicategory

    The Hecke Bicategory

    Axioms 2012, 1, 291-323; doi:10.3390/axioms1030291 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Communication The Hecke Bicategory Alexander E. Hoffnung Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122, USA; E-Mail: [email protected]; Tel.: +215-204-7841; Fax: +215-204-6433 Received: 19 July 2012; in revised form: 4 September 2012 / Accepted: 5 September 2012 / Published: 9 October 2012 Abstract: We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Keywords: Hecke algebras; categorification; groupoidification; Yang–Baxter equations; Zamalodchikov tetrahedron equations; spans; enriched bicategories; buildings; incidence geometries 1. Introduction Categorification is, in part, the attempt to shed new light on familiar mathematical notions by replacing a set-theoretic interpretation with a category-theoretic analogue. Loosely speaking, categorification replaces sets, or more generally n-categories, with categories, or more generally (n + 1)-categories, and functions with functors. By replacing interesting equations by isomorphisms, or more generally equivalences, this process often brings to light a new layer of structure previously hidden from view.
  • New Ideas in Algebraic Topology (K-Theory and Its Applications)

    New Ideas in Algebraic Topology (K-Theory and Its Applications)

    NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems.
  • On the Homology of Configuration Spaces

    On the Homology of Configuration Spaces

    TopologyVol. 28, No. I. pp. I II-123, 1989 Lmo-9383 89 53 al+ 00 Pm&d I” Great Bntam fy 1989 Pcrgamon Press plc ON THE HOMOLOGY OF CONFIGURATION SPACES C.-F. B~DIGHEIMER,~ F. COHEN$ and L. TAYLOR: (Receiued 27 July 1987) 1. INTRODUCTION 1.1 BY THE k-th configuration space of a manifold M we understand the space C”(M) of subsets of M with catdinality k. If c’(M) denotes the space of k-tuples of distinct points in M, i.e. Ck(M) = {(z,, . , zk)eMklzi # zj for i #j>, then C’(M) is the orbit space of C’(M) under the permutation action of the symmetric group Ik, c’(M) -+ c’(M)/E, = Ck(M). Configuration spaces appear in various contexts such as algebraic geometry, knot theory, differential topology or homotopy theory. Although intensively studied their homology is unknown except for special cases, see for example [ 1, 2, 7, 8, 9, 12, 13, 14, 18, 261 where different terminology and notation is used. In this article we study the Betti numbers of Ck(M) for homology with coefficients in a field IF. For IF = IF, the rank of H,(C’(M); IF,) is determined by the IF,-Betti numbers of M. the dimension of M, and k. Similar results were obtained by Ldffler-Milgram [ 171 for closed manifolds. For [F = ff, or a field of characteristic zero the corresponding result holds in the case of odd-dimensional manifolds; it is no longer true for even-dimensional manifolds, not even for surfaces, see [S], [6], or 5.5 here.
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology

    Floer Homology, Gauge Theory, and Low-Dimensional Topology

    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
  • SHEAVES of MODULES 01AC Contents 1. Introduction 1 2

    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.
  • Why Category Theory?

    Why Category Theory?

    The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories Why category theory? Daniel Tubbenhauer - 02.04.2012 Georg-August-Universit¨at G¨ottingen Gr G ACT Cat C Hom(-,O) GRP Free - / [-,-] Forget (-) ι (-) Δ SET n=1 -x- Forget Free K(-,n) CxC n>1 AB π (-) n TOP Hom(-,-) * Drop (-) Forget * Hn Free op TOP Join AB * K-VS V - Hom(-,V) The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories A generalisation of well-known notions Closed Total Associative Unit Inverses Group Yes Yes Yes Yes Yes Monoid Yes Yes Yes Yes No Semigroup Yes Yes Yes No No Magma Yes Yes No No No Groupoid Yes No Yes Yes Yes Category No No Yes Yes No Semicategory No No Yes No No Precategory No No No No No The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories Leonard Euler and convex polyhedron Euler's polyhedron theorem Polyhedron theorem (1736) 3 Let P ⊂ R be a convex polyhedron with V vertices, E edges and F faces. Then: χ = V − E + F = 2: Here χ denotes the Euler Leonard Euler (15.04.1707-18.09.1783) characteristic. The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories Leonard Euler and convex polyhedron Euler's polyhedron theorem Polyhedron theorem (1736) 3 Polyhedron E K F χ Let P ⊂ R be a convex polyhedron with V vertices, E Tetrahedron 4 6 4 2 edges and F faces.
  • Compact Ringed Spaces

    Compact Ringed Spaces

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 52, 41 l-436 (1978) Compact Ringed Spaces CHRISTOPHER J. MULVEY * Department of Mathematics, Columbia University, New York, New York 10027 and Mathematics Division, University of Sussex, Falmer, Brighton, BNl9QH, England Communicated by Saunders MacLane Received April 16, 1976 The properties of complete regularity, paracompactness, and compactness of a topological space X all may be described in terms of the ring R(X) of continuous real functions on X. In this paper we are concerned with extending these concepts from topological to ringed spaces, replacing the ring of continuous real functions on the space by the ring of sections of the ringed space. Our interest in the problem arose from attempting to construct the tangent module 7M of a smooth manifold IM directly from the sheaf QM of smooth real functions. For each open set U of M, the module rM( U)is the s2,( U)-module of derivations of In,(U). Yet, since taking derivations is not functorial, the evident restriction maps are not canonically forthcoming. For paracompact manifolds the construction needed was known to Boardman [5]. In general, it can be carried out using the complete regularity of the ringed space (M, 62,) [18, 321. Although spaces satisfying these conditions appear naturally in analytical contexts, there emerges a fundamental connection with sectional representations of rings [19, 21, 231. In turn, this has provided a technique for using intuitionistic mathematics in applying these representations [22]. Extending these properties to ringed spaces, we prove a compactness theorem for completely regular ringed spaces generalizing the Gelfand-Kolmogoroff criterion concerning maximal ideals in the ring R(X) of continuous real functions on a completely regular space X.
  • Introduction to Algebraic Geometry

    Introduction to Algebraic Geometry

    Introduction to Algebraic Geometry Jilong Tong December 6, 2012 2 Contents 1 Algebraic sets and morphisms 11 1.1 Affine algebraic sets . 11 1.1.1 Some definitions . 11 1.1.2 Hilbert's Nullstellensatz . 12 1.1.3 Zariski topology on an affine algebraic set . 14 1.1.4 Coordinate ring of an affine algebraic set . 16 1.2 Projective algebraic sets . 19 1.2.1 Definitions . 19 1.2.2 Homogeneous Nullstellensatz . 21 1.2.3 Homogeneous coordinate ring . 22 1.2.4 Exercise: plane curves . 22 1.3 Morphisms of algebraic sets . 24 1.3.1 Affine case . 24 1.3.2 Quasi-projective case . 26 2 The Language of schemes 29 2.1 Sheaves and locally ringed spaces . 29 2.1.1 Sheaves on a topological spaces . 29 2.1.2 Ringed space . 34 2.2 Schemes . 36 2.2.1 Definition of schemes . 36 2.2.2 Morphisms of schemes . 40 2.2.3 Projective schemes . 43 2.3 First properties of schemes and morphisms of schemes . 49 2.3.1 Topological properties . 49 2.3.2 Noetherian schemes . 50 2.3.3 Reduced and integral schemes . 51 2.3.4 Finiteness conditions . 53 2.4 Dimension . 54 2.4.1 Dimension of a topological space . 54 2.4.2 Dimension of schemes and rings . 55 2.4.3 The noetherian case . 57 2.4.4 Dimension of schemes over a field . 61 2.5 Fiber products and base change . 62 2.5.1 Sum of schemes . 62 2.5.2 Fiber products of schemes .