• Study Resource
  • Explore
    • Arts & Humanities
    • Business
    • Engineering & Technology
    • Foreign Language
    • History
    • Math
    • Science
    • Social Science

    Top subcategories

    • Advanced Math
    • Algebra
    • Basic Math
    • Calculus
    • Geometry
    • Linear Algebra
    • Pre-Algebra
    • Pre-Calculus
    • Statistics And Probability
    • Trigonometry
    • other →

    Top subcategories

    • Astronomy
    • Astrophysics
    • Biology
    • Chemistry
    • Earth Science
    • Environmental Science
    • Health Science
    • Physics
    • other →

    Top subcategories

    • Anthropology
    • Law
    • Political Science
    • Psychology
    • Sociology
    • other →

    Top subcategories

    • Accounting
    • Economics
    • Finance
    • Management
    • other →

    Top subcategories

    • Aerospace Engineering
    • Bioengineering
    • Chemical Engineering
    • Civil Engineering
    • Computer Science
    • Electrical Engineering
    • Industrial Engineering
    • Mechanical Engineering
    • Web Design
    • other →

    Top subcategories

    • Architecture
    • Communications
    • English
    • Gender Studies
    • Music
    • Performing Arts
    • Philosophy
    • Religious Studies
    • Writing
    • other →

    Top subcategories

    • Ancient History
    • European History
    • US History
    • World History
    • other →

    Top subcategories

    • Croatian
    • Czech
    • Finnish
    • Greek
    • Hindi
    • Japanese
    • Korean
    • Persian
    • Swedish
    • Turkish
    • other →
 
Profile Documents Logout
Upload
Exercise Sheet 3
Exercise Sheet 3

Topology, MM8002/SF2721, Spring 2017. Exercise set 2 Exercise 1
Topology, MM8002/SF2721, Spring 2017. Exercise set 2 Exercise 1

1 Overview 2 Sheaves on Topological Spaces
1 Overview 2 Sheaves on Topological Spaces

PDF
PDF

Sheaf Cohomology 1. Computing by acyclic resolutions
Sheaf Cohomology 1. Computing by acyclic resolutions

PDF
PDF

... A regular space is a topological space in which points and closed sets can be separated by open sets; in other words, given a closed set A and a point x ∈ / A, there are disjoint open sets U and V such that x ∈ U and A ⊆ V . A T3 space is a regular T0 -space. A T3 space is necessarily also T2 , that ...
Universal spaces in birational geometry
Universal spaces in birational geometry

9. Sheaf Cohomology Definition 9.1. Let X be a topological space
9. Sheaf Cohomology Definition 9.1. Let X be a topological space

PDF
PDF

What Is...a Topos?, Volume 51, Number 9
What Is...a Topos?, Volume 51, Number 9

0.1 A lemma of Kempf
0.1 A lemma of Kempf

... for all 0 < i < n and U ∈ A. Suppose α ∈ H n (X, F). Then there is a covering of X by open sets V ∈ A such that the image of α in H n (X, V F) is zero for each V . Proof. We will prove this result by induction on n. First, suppose n > 1, and that the result is valid for n − 1. The base case will be ...
Topology
Topology

Class 3 - Stanford Mathematics
Class 3 - Stanford Mathematics

Pages 31-40 - The Graduate Center, CUNY
Pages 31-40 - The Graduate Center, CUNY

Exercise Sheet 4 - D-MATH
Exercise Sheet 4 - D-MATH

LECTURE NOTES 4: CECH COHOMOLOGY 1
LECTURE NOTES 4: CECH COHOMOLOGY 1

Exercises
Exercises

A BRIEF INTRODUCTION TO SHEAVES References 1. Presheaves
A BRIEF INTRODUCTION TO SHEAVES References 1. Presheaves

... A ringed space (X, O) is a local ringed space if each stalk Ox (x ∈ X) is a local ring with maximal ideal mX ; a morphism (X, OX ) −→ (Y, OY ) of local ringed spaces is one where on each stalk OY,f (x) −→ f∗ OX,x is a homomorphism of local rings (i.e., a ring homomorphism h : R −→ S for which h−1 mS ...
Notes
Notes

... language is supposed to evoke the idea that the set F(U ) lives ‘above’ U in some sense. Sections of F over X are the global sections. An alternate notation, common in algebraic geometry, is to write Γ(U, F) for F(U ). This is often used when U is considered fixed and F is allowed to vary. Example 1 ...
Generalities About Sheaves - Lehrstuhl B für Mathematik
Generalities About Sheaves - Lehrstuhl B für Mathematik

Let X and Y be topological spaces, where the only open
Let X and Y be topological spaces, where the only open

... ...
NOTES ON GROTHENDIECK TOPOLOGIES 1
NOTES ON GROTHENDIECK TOPOLOGIES 1

PDF
PDF

Note - Math
Note - Math

PDF
PDF

< 1 ... 61 62 63 64 65 >

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They exist in several varieties such as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the ""usual"" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and number theory.
  • studyres.com © 2025
  • DMCA
  • Privacy
  • Terms
  • Report