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Sheaf Theory

 
지은이 : Yong Seung Cho
출판사 : 경문사
판수 : 1판
페이지수 : 224
ISBN : 978-89-6105-497-3
예상출고일 : 입금확인후 2일 이내
주문수량 :
도서가격 : 15,000원
적립금 : 450 Point
   

 
Sheaf theory has had profound effects on several mathematical subjects,
especially, topology, differential and algebraic geometry. It gives a
tool for dealing with problems by piecing together solutions of local
problems in a coherent manner to get global solutions. For example, several
geometric structures of a manifold, scheme on the space and general
cohomology theory can be expressed in terms of sheaves. The sheaf
cohomology measures the lack of exactness of the global solutions.
In this book we introduce the foundations of sheaf theory to define
the cohomologies of topological, differential and complex manifolds. In
Chapter 1 we introduce Riemann surfaces and motivation of sheaf
cohomology. In Chapter 2 we introduce direct limits, and in Chapter 3
presheaf, sheaf and examples. In Chapter 4 we introduce ringed spaces,
geometric spaces and module over ringed spaces. In Chapter 5 we deal
with �ech cohomology and Grothendieck cohomology. In Chapter 6 applications
to Riemann surfaces. In Chapter 7 complex manifolds, in Chapter
8 the de Rham theory, and in Chapter 9 Hodge decomposition theorems
on compact oriented manifolds and K�hler manifolds. In Appendix we
introduce characteristic classes, low dimensional manifolds, and categories.
The final manuscript and galley proof are read by my students Ahram
Lim and Semin Yoo. I am sincerely grateful to them.

Chapter 1 Riemann Surface
1.1 Meromorphic Function / 3
1.2 Motivation / 7
1.3 Obstruction to the Construction of Function / 7


Chapter 2 Direct limit
2.1 Direct Limit of Set / 13
2.2 Direct Limit of Abelian Group / 16


Chapter 3 Sheaf
3.1 Sheaf / 21
3.2 Godement Sheaf and Cartan Sheaf / 27
3.3 Induced Sheaf / 33


Chapter 4 Geometric Space
4.1 Ringed Space / 41
4.2 Prime Spectrum of Ring / 51
4.3 Geometric Space / 58
4.4 Module over Ringed Space / 64
4.5 Locally Free Module / 73


Chapter 5 Sheaf Cohomology
5.1 �ech Cohomology / 82
5.2 Grothendieck Cohomology / 88


Chapter 6 Sheaf on Riemann Surface
6.1 Riemann-Roch Theorem / 101
6.2 Divisor Class Group / 102
6.3 Jacobian / 103
6.4 Differential / 106
6.5 Laurent Theorem / 109
6.6 Weierstrass Theorem / 110


Chapter 7 Complex Manifold
7.1 Compact Complex Manifold / 116
7.2 Stein Manifold / 119


Chapter 8 The de Rham Theorem
8.1 Exterior Product of Vector Space / 127
8.2 Differential Form / 129
8.3 The de Rham Theorem for Differential Manifold / 131
8.4 The de Rham Theorem for Stein Manifold / 138
8.5 Lefschetz Theorem / 140


Chapter 9 Hodge Theorem
9.1 Laplace Operator / 145
9.2 Hodge Decomposition Theorem on Compact Oriented Manifold / 148
9.3 K�hler Manifold / 156
9.4 Hodge Decomposition Theorem on Compact K�hler Manifold / 164


Appendix A: Characteristic Class
1. Vector Bundle / 171
2. Characteristic Class for Line Bundle / 173
3. Chern Class / 174
4. Gysin Homomorphism / 179
5. Euler Class / 184
6. Pontrjagin Class / 188


Appendix B: Low-Dimensional Manifold
1. One-Manifold / 191
2. Two-Manifold / 191
3. Three-Manifold / 193
4. Four-Manifold / 195


Appendix C: Category
1. Category / 200
2. Additive Category / 202
3. Abelian Category / 205


REFERENCES / 207


INDEX / 209

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