The heart of cohomology

Goro Kato1402050356, 9781402050350

If you have not heard about cohomology, this book may be suited for you. Fundamental notions in cohomology for examples, functors, representable functors, Yoneda embedding, derived functors, spectral sequences, derived categories are explained in elementary fashion. Applications to sheaf cohomology are given. Also cohomological aspects of D-modules and of the computation of zeta functions of the Weierstrass family are provided.

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Table of contents :
Contents……Page 6
Preface……Page 8
1.1 Categories and Functors……Page 13
1.2 Opposite Category……Page 17
1.3 Forgetful Functors……Page 19
1.4 Embeddings……Page 20
1.5 Representable Functors……Page 22
1.6 Abelian Categories……Page 25
1.7 Adjoint Functors……Page 31
1.8 Limits……Page 33
1.9 Dual Notion of Inverse Limit……Page 36
1.10 Presheaves……Page 37
1.11 Notion of Site……Page 39
1.12 Sheaves Over Site……Page 40
1.13 Sieve; Another Notion for a Site……Page 41
1.14 Sheaves of Abelian Groups……Page 45
1.15 The Sheafification Functor……Page 48
2.2 Cohomology……Page 50
2.3 Homotopy……Page 52
2.4 Exactness……Page 53
2.5 Injective Objects……Page 54
2.6 Resolutions……Page 56
2.7 Derived Functors……Page 57
2.8 Properties of Derived Functors……Page 63
2.9 Axioms for Derived Functors……Page 73
2.10 The Derived Functors (Ext[sup(j)])[sub(j≥0)]……Page 74
2.11 Precohomology……Page 75
3.1 Definition of Spectral Sequence……Page 82
3.2 Filtered Complexes……Page 87
3.3 Double Complexes……Page 88
3.4 Cohomology of Sheaves over Topological Space……Page 102
3.5 Higher Derived Functors of lim……Page 119
4.1 Defining Derived Categories……Page 128
4.2 Derived Categorical Derived Functors……Page 133
4.3 Triangles……Page 140
4.4 Triangles for Exact Sequences……Page 143
5.1 Exposition……Page 159
5.2 The Weierstrass Family……Page 161
5.3 Exposition on D-Modules……Page 178
5.4 Cohomological Aspects of D-Modules……Page 181
K……Page 197
S……Page 198
W……Page 199
EPILOGUE (INFORMAL)……Page 200
F……Page 201
T……Page 202
Z……Page 203