DOCSLIB.ORG

Appendix a Sheaves and Abstract Algebraic Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix a Sheaves and Abstract Algebraic Varieties Appendix A Sheaves and Abstract Algebraic Varieties A.1 Sheaves Let X be a topological space. For each open set U eX, consider the set F(U, q of all ((>valued functions on U. This set naturally forms a C-algebra under pointwise addition and multiplication of functions. Definition: A sheafR ofC-valued functions on X assigns to each open set U C X a subalgebra R(U) C F(U, q in a way that is "compatible with both restriction and gluing," that is, • For any open sets Ul C U2 C X and f E R(U2 ), the restriction of f to U1 is in R(U1 ) • • If {U"'}"'EA is an open cover of an open set U ~ X and f E F(U, q is such that flu", E R(U",) for all a, then f E R(U). The functions f E R(U) are called the sections of the sheaf R over the open set U C X. If R is a sheaf of C-valued functions and if f E R(U1 UU2 ), then note that flul and flu2 both have the same restriction to U1 n U2 , namely flu1nu2 • Conversely, if hE R(Ud and 9 E R(U2 ) are such that hl u1nU2 = glu1nU2 , then the map f E F(U1 U U2 , q given by f(x) = {h(X) if x E U1 , g(x) if x E U2 , 146 Appendix A. Sheaves and Abstract Algebraic Varieties is well-defined. Clearly, flu! = hand flu2 = g. So by the second property of sheaves, f E R(U1 U U2 ). We say that hand g glue together to give f. In a similar way, we can define a sheaf of JR.-valued functions, or a sheaf of functions with values in any ring, or even in any set. Examples of sheaves of functions: • Regular functions on a quasi-projective variety V form a sheaf Ov of C-valued functions. The sheaf associates to each open set U C V the C-algebra Ov(U) of regular functions on U. • Continuous JR.-valued functions on a topologiCal space form a sheaf of JR.-valued functions. • COO-functions on a smooth manifold form a sheaf of JR.-valued functions. • HolomorphiC functions on a Riemann surface form a sheaf of C-valued functions. Definition: The sheaf Ov of regular functions on a quasi-projective variety V is called the structure sheaf of the variety. The structure sheaf Ov determines V (up to isomorphism), even if we have only limited information about Ov. For example, in Section 4.3, we proved that for an affine variety, the ring of global sections of Ov is the coordinate ring qV], which in turn recovers the affine variety up to iso­ morphism. In other words, an affine variety is determined by the global sections of its structure sheaf. Only slightly more difficult is the fact that every quasi-projective variety is determined by the rings of sections of the structure sheaf on any affine cover, together with the restriction maps Ov (Ui ) -+ Ov (Ui n Uj ), which recover for us the way these affine pieces are glued together. Later in this appendix we will define an abstract variety, whiCh will be determined by partial information about its structure sheaf in a similar way. This situation is unique to algebraic geometry: A manifold is not determined by such limited information about its sheaf of continuous (or differentiable, or complex holomorphic) functions. For each open set U C X, a sheaf R has a natural restriction Rlu to U. The sections of RI u over an open set U' C U are just the sections R(U') of the original sheaf. Some caution is in order: The ring of sections R(U) and the restriction sheaf Rlu are two different objects; R(U) is a ring, while Rlu is a sheaf (assigning rings to open sets of U). For an open subset V of a quasi-projective variety W, the structure sheaf Ov agrees with the restriCtion of the sheaf Ow to the open set V. A topological space, together with a sheaf of C-valued functions on it, is an example of a ringed space. An understanding of ringed spaces is essential A.I. Sheaves 147 for eventually mastering the definition of a scheme, so we introduce the definition here. Definition: A sheaf of rings R on a topological space X assigns to each open set U C X a ring R(U) in such a way that the following axioms are satisfied • If U1 C U2 , then there is a ring homomorphism R(U2 ) --+ R(U1 ). This map is called "the restriction map from U2 to U1 ," and the image of any element f E R(U2 ) under this map is denoted by flu,. • If U1 C U2 C U3 , then the restriction map R(U3 ) --+ R(Ud is the composition of the restriction maps R(U3 ) --+ R(U2 ) --+ R(U1 ). • If {UoJ"'EA is an open cover of an open set U C X and {g"'}"'EA is a collection of elements g", E R(Uc,) such that for all indices 00, /3, g",lu"nu!3 = g,elua nu!3' then there exists a unique g E R(U) such that glua = g", for all 00. A topological space X, together with a sheaf of rings on X, is called a ringed space. The rings R(U) in the definition above are just abstract rings: They need not be rings of functions on the set U. In particular, the word "restriction" above should not be interpreted literally as the restriction of functions. Our sheaves of C-valued functions are examples of sheaves of rings on the corresponding spaces. In fact, they are all sheaves of C-algebras, since each ring R(U) is actually a C-algebra. Although an abstract sheaf of rings is not necessarily a sheaf of functions, one should think of every sheaf of rings as very much like a sheaf of functions. The third axiom in the definition of a sheaf of rings-also called the sheaf axiom-guarantees that the elements of R(U) really behave like functions: They are uniquely defined by their values on any open cover of U. It is easy to check that every sheaf of C-valued functions is a sheaf of rings. We could also define a sheaf of Abelian groups, a sheaf of sets, a sheaf of algebras, or even a sheaf of objects in almost any category. Just strike out all occurrences of the word "ring" in the preceding definition and replace it by the word "Abelian group," "set," or "algebra." A topological space may come equipped with several different sheaves of rings or algebras. For example, on cn with its usual Euclidean topology, we have not only the sheaf of continuous functions, but also the sheaf of holomorphic functions. We can also equip the set cn with the Zariski topology, where we have the sheaf of regular functions. Definition: Let Rand S be two sheaves of rings on a space X. A map of sheaves of rings 148 Appendix A. Sheaves and Abstract Algebraic Varieties consists of a ring map R(U) GJ!!) S(U) for each open set U C X such that whenever U1 C U2, the following diagram commutes: G(U2) R(U2) • S(U2) j j R(U1 ) • S(Ud G(Ud where the vertical maps are the restriction maps. If the sheaves of rings are actually sheaves of (C>algebras, the maps are furthermore required to preserve the ([:::-algebra structure, that is, each R(U) GJ!!) S(U) must be I[>linear. It does not make sense to speak of maps of sheaves when the sheaves are on two different topological spaces. However, given a continuous map X ----* Y of topological spaces, there is a way to define a sheaf of rings on Y from any sheaf of rings on X. Definition: Given a sheaf R on a topological space X and a continuous map X ~ Y of topological spaces, the push-forward f* R of R is the sheaf on Y defined as follows. For each open set U C Y, If R is a sheaf of rings on X, then f* R is a sheaf of rings on Y. Definition: A map of ringed spaces (X, Ox) ----* (Y, Oy) is a pair (F, F#) consisting of a continuous map of topological spaces X ~ Y and a map F# of sheaves of rings on Y, Oy ---+ F*Ox. Example: Let V ~ W be a map of quasi-projective algebraic varieties. Then there is a naturally induced map of ringed spaces (V, Ov) ----* (W, Ow) where Ov (respectively Ow) is the sheaf of regular functions on V (respec­ tively W). The map of topological spaces is F, and the map Ow ----* F*Ov A.I. Sheaves 149 of sheaves of rings is defined by the pullback: For each open set U C W, Ow(U) ---+ Ov(F-1 (U)), 9 r------+ F# (g) = 9 0 F. This idea works more generally, as the next example shows. Example: If X ~ Y is any continuous map of topological spaces, there is always a morphism of ringed spaces (X,Fx) ----t (Y,Fy), where Fx is the sheaf of CC-valued functions on X and Fy is the sheaf of CC-valued functions on Y. Indeed, the map of sheaves Fy ----t F*Fx is defined using the pullback, Fy(U) ----t F x (F-1 (U)), 9 r------+ goF. If Fx and Fy instead denote the sheaves of continuous CC-valued functions on X and Y, then (X, F x) ----t (Y, Fy) will be a morphism of these ringed spaces.
Load more

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|, 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.
    [Show full text]
  • Stable Higgs Bundles on Ruled Surfaces 3
    STABLE HIGGS BUNDLES ON RULED SURFACES SNEHAJIT MISRA Abstract. Let π : X = PC(E) −→ C be a ruled surface over an algebraically closed field k of characteristic 0, with a fixed polarization L on X. In this paper, we show that pullback of a (semi)stable Higgs bundle on C under π is a L-(semi)stable Higgs bundle. Conversely, if (V,θ) ∗ is a L-(semi)stable Higgs bundle on X with c1(V ) = π (d) for some divisor d of degree d on C and c2(V ) = 0, then there exists a (semi)stable Higgs bundle (W, ψ) of degree d on C whose pullback under π is isomorphic to (V,θ). As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on X whenever g(C) ≥ 2 and the base field is C. 1. Introduction A Higgs bundle on an algebraic variety X is a pair (V, θ) consisting of a vector bundle V 1 over X together with a Higgs field θ : V −→ V ⊗ ΩX such that θ ∧ θ = 0. Higgs bundle comes with a natural stability condition (see Definition 2.3 for stability), which allows one to study the moduli spaces of stable Higgs bundles on X. Higgs bundles on Riemann surfaces were first introduced by Nigel Hitchin in 1987 and subsequently, Simpson extended this notion on higher dimensional varieties. Since then, these objects have been studied by many authors, but very little is known about stability of Higgs bundles on ruled surfaces.
    [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]
  • Notes on Principal Bundles and Classifying Spaces
    Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG].
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • Group Invariant Solutions Without Transversality 2 in Detail, a General Method for Characterizing the Group Invariant Sections of a Given Bundle
    GROUP INVARIANT SOLUTIONS WITHOUT TRANSVERSALITY Ian M. Anderson Mark E. Fels Charles G. Torre Department of Mathematics Department of Mathematics Department of Physics Utah State University Utah State University Utah State University Logan, Utah 84322 Logan, Utah 84322 Logan, Utah 84322 Abstract. We present a generalization of Lie’s method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations. The char- acterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics, harmonic maps, and general relativity. This work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions. Keywords. Lie symmetry reduction, group invariant solutions, kinematic reduction diagram, dy- namic reduction diagram. arXiv:math-ph/9910015v2 13 Apr 2000 February , Research supported by NSF grants DMS–9403788 and PHY–9732636 1. Introduction. Lie’s method of symmetry reduction for finding the group invariant solutions to partial differential equations is widely recognized as one of the most general and effective methods for obtaining exact solutions of non-linear partial differential equations.
    [Show full text]
  • Composable Geometric Motion Policies Using Multi-Task Pullback Bundle Dynamical Systems
    Composable Geometric Motion Policies using Multi-Task Pullback Bundle Dynamical Systems Andrew Bylard, Riccardo Bonalli, Marco Pavone Abstract— Despite decades of work in fast reactive plan- ning and control, challenges remain in developing reactive motion policies on non-Euclidean manifolds and enforcing constraints while avoiding undesirable potential function local minima. This work presents a principled method for designing and fusing desired robot task behaviors into a stable robot motion policy, leveraging the geometric structure of non- Euclidean manifolds, which are prevalent in robot configuration and task spaces. Our Pullback Bundle Dynamical Systems (PBDS) framework drives desired task behaviors and prioritizes tasks using separate position-dependent and position/velocity- dependent Riemannian metrics, respectively, thus simplifying individual task design and modular composition of tasks. For enforcing constraints, we provide a class of metric-based tasks, eliminating local minima by imposing non-conflicting potential functions only for goal region attraction. We also provide a geometric optimization problem for combining tasks inspired by Riemannian Motion Policies (RMPs) that reduces to a simple least-squares problem, and we show that our approach is geometrically well-defined. We demonstrate the 2 PBDS framework on the sphere S and at 300-500 Hz on a manipulator arm, and we provide task design guidance and an open-source Julia library implementation. Overall, this work Fig. 1: Example tree of PBDS task mappings designed to move a ball along presents a fast, easy-to-use framework for generating motion the surface of a sphere to a goal while avoiding obstacles. Depicted are policies without unwanted potential function local minima on manifolds representing: (black) joint configuration for a 7-DoF robot arm general manifolds.
    [Show full text]
  • 4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves
    4 Sheaves of modules, vector bundles, and (quasi-)coherent sheaves “If you believe a ring can be understood geometrically as functions its spec- trum, then modules help you by providing more functions with which to measure and characterize its spectrum.” – Andrew Critch, from MathOver- flow.net So far we discussed general properties of sheaves, in particular, of rings. Similar as in the module theory in abstract algebra, the notion of sheaves of modules allows us to increase our understanding of a given ringed space (or a scheme), and to provide further techniques to play with functions, or function-like objects. There are particularly important notions, namely, quasi-coherent and coherent sheaves. They are analogous notions of the usual modules (respectively, finitely generated modules) over a given ring. They also generalize the notion of vector bundles. Definition 38. Let (X, ) be a ringed space. A sheaf of -modules, or simply an OX OX -module, is a sheaf on X such that OX F (i) the group (U) is an (U)-module for each open set U X; F OX ✓ (ii) the restriction map (U) (V ) is compatible with the module structure via the F !F ring homomorphism (U) (V ). OX !OX A morphism of -modules is a morphism of sheaves such that the map (U) F!G OX F ! (U) is an (U)-module homomorphism for every open U X. G OX ✓ Example 39. Let (X, ) be a ringed space, , be -modules, and let ' : OX F G OX F!G be a morphism. Then ker ', im ', coker ' are again -modules. If is an - OX F 0 ✓F OX submodule, then the quotient sheaf / is an -module.
    [Show full text]
  • 4 Fibered Categories (Aaron Mazel-Gee) Contents
    4 Fibered categories (Aaron Mazel-Gee) Contents 4 Fibered categories (Aaron Mazel-Gee) 1 4.0 Introduction . .1 4.1 Definitions and basic facts . .1 4.2 The 2-Yoneda lemma . .2 4.3 Categories fibered in groupoids . .3 4.3.1 ... coming from co/groupoid objects . .3 4.3.2 ... and 2-categorical fiber product thereof . .4 4.0 Introduction In the same way that a sheaf is a special sort of functor, a stack will be a special sort of sheaf of groupoids (or a special special sort of groupoid-valued functor). It ends up being advantageous to think of the groupoid associated to an object X as living \above" X, in large part because this perspective makes it much easier to study the relationships between the groupoids associated to different objects. For this reason, we use the language of fibered categories. We note here that throughout this exposition we will often say equal (as opposed to isomorphic), and we really will mean it. 4.1 Definitions and basic facts φ Definition 1. Let C be a category. A category over C is a category F with a functor p : F!C. A morphism ξ ! η p(φ) in F is called cartesian if for any other ζ 2 F with a morphism ζ ! η and a factorization p(ζ) !h p(ξ) ! p(η) of φ p( ) in C, there is a unique morphism ζ !λ η giving a factorization ζ !λ η ! η of such that p(λ) = h. Pictorially, ζ - η w - w w 9 w w ! w w λ φ w w w w - w w w w ξ w w w w w w p(ζ) w p(η) w w - w h w ) w (φ w p w - p(ξ): In this case, we call ξ a pullback of η along p(φ).
    [Show full text]
  • Chapter 3 Connections
    Chapter 3 Connections Contents 3.1 Parallel transport . 69 3.2 Fiber bundles . 72 3.3 Vector bundles . 75 3.3.1 Three definitions . 75 3.3.2 Christoffel symbols . 80 3.3.3 Connection 1-forms . 82 3.3.4 Linearization of a section at a zero . 84 3.4 Principal bundles . 88 3.4.1 Definition . 88 3.4.2 Global connection 1-forms . 89 3.4.3 Frame bundles and linear connections . 91 3.1 The idea of parallel transport A connection is essentially a way of identifying the points in nearby fibers of a bundle. One can see the need for such a notion by considering the following question: Given a vector bundle π : E ! M, a section s : M ! E and a vector X 2 TxM, what is meant by the directional derivative ds(x)X? If we regard a section merely as a map between the manifolds M and E, then one answer to the question is provided by the tangent map T s : T M ! T E. But this ignores most of the structure that makes a vector 69 70 CHAPTER 3. CONNECTIONS bundle interesting. We prefer to think of sections as \vector valued" maps on M, which can be added and multiplied by scalars, and we'd like to think of the directional derivative ds(x)X as something which respects this linear structure. From this perspective the answer is ambiguous, unless E happens to be the trivial bundle M × Fm ! M. In that case, it makes sense to think of the section s simply as a map M ! Fm, and the directional derivative is then d s(γ(t)) − s(γ(0)) ds(x)X = s(γ(t)) = lim (3.1) dt t!0 t t=0 for any smooth path with γ_ (0) = X, thus defining a linear map m ds(x) : TxM ! Ex = F : If E ! M is a nontrivial bundle, (3.1) doesn't immediately make sense because s(γ(t)) and s(γ(0)) may be in different fibers and cannot be added.
    [Show full text]
  • Cohomology of Local Systems on XΓ Cailan Li October 1St, 2019
    Cohomology of Local Systems on XΓ Cailan Li October 1st, 2019 1 Local Systems Definition 1.1. Let X be a topological space and let S be a set (usually with additional structure, ring module, etc). The constant sheaf SX is defined to be SX (U) = ff : U ! S j f is continuous and S has the discrete topologyg Remark. Equivalently, SX is the sheaf whose sections are locally constant functions f : U ! S and also is equivalent to the sheafification of the constant presheaf which assigns A to every open set. Remark. When U is connected, SX (U) = S. Definition 1.2. Let A be a ring. Then an A−local system on a topological space X is a sheaf L 2 mod(AX ) s.t. there exists a covering of X by fUig s.t. LjUi = Mi where Mi is the constant sheaf associated to the R−module Mi. In other words, a local system is the same thing as a locally constant sheaf. Remark. If X is connected, then all the Mi are the same. Example 1. AX is an A−local system. Example 2. Let D be an open connected subset of C. Then the sheaf F of solutions to LODE, namely n (n) (n−1) o F (U) = f : U ! C j f + a1(z)f + ::: + an(z) = 0 where ai(z) are holomorphic forms a C−local system. Existence and uniqueness of solutions of ODE on simply connected regions means that by choosing a disc D(z) around each point z 2 D, we see that the (k) initial conditions f = yk give an isomorphism ∼ n F jD(z) = C Example 3.
    [Show full text]