Ebook: Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Diferential Geometry

The main topics of this book are cohomology, sheaves, and sheaf cohomology. Why? Mostly
because for more than thirty years the senior author has been trying to learn algebraic
geometry. To his dismay, he realized that since 1960, under the in
uence and vision of A.
Grothendieck and his collaborators, in particular Serre, the foundations of algebraic geometry
were built on sheaves and cohomology. But the invasion of these theories was not restricted
to algebraic geometry. Cohomology was already a pillar of algebraic topology but sheaves
and sheaf cohomology sneaked in too.
For a novice the situation seems hopeless. Even before one begins to discuss curves or
surfaces, one has to spend years learning
(1) Some algebraic topology (especially homology and cohomology).
(2) Some basic homological algebra (chain complexes, cochain complexes, exact sequences,
chain maps, etc.). Some commutative algebra (injective and projective modules, injective
and projective resolutions).
(3) Some sheaf theory.
This book represents the result of an unnished journey in attempting to accomplish
the above. What we discovered on the way is that algebraic topology is a captivating and
beautiful subject. We also believe that it is hard to appreciate sophisticated concepts such as
sheaf cohomology without prior exposure to fundamentals of algebraic topology, simplicial
homology, singular homology, and CW complexes, in particular.

UPLOAD BY ZAMPIVA