Description
This textbook, for an undergraduate course in modern algebraic geometry, recognizes that the typical undergraduate curriculum contains a great deal of analysis and, by contrast, little algebra. Because of this imbalance, it seems most natural to present algebraic geometry by highlighting the way it connects algebra and analysis; the average student will probably be more familiar and more comfortable with the analytic component. The book therefore focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis. GAGA provides the unifying theme of the book: we develop enough of the modern machinery of algebraic geometry to be able to give an essentially complete proof, at a level accessible to undergraduates throughout. The book is based on a course which the author has taught, twice, at the Australian National University.
Chapter
3.3 Localization of rings
3.4 The sheaf R on Spec(R)
3.5 A return to the world of simple examples
3.6 Maps of ringed spaces (Spec(S), Š)→(Spec(R), Ř)
3.7 Some immediate consequences
3.8 A reminder of Hilbert’s Nullstellensatz
3.10 Schemes of finite type over C
4.1 Synopsis of the main results
4.2 The subspace Max(X)כ X
4.3 The correspondence between maximal ideals and ϕ : R → C
4.4 The special case of the polynomial ring
4.5 The complex topology on MaxSpec(R)
4.6 The complex topology on schemes
5 The analytification of a scheme
5.1 Synopsis of the main results
5.2 The Hilbert Basis Theorem
5.3 The sheaf of analytic functions on an affine scheme
5.4 A reminder about Fréchet spaces
5.5 The ring of analytic functions as a completion
5.6 Allowing the ring and the generators to vary
5.7 Affine schemes, done without coordinates
5.8 In the world of elementary examples
6 The high road to analytification
6.1 A coordinate-free approach to polydiscs
6.2 The high road to the complex topology
6.3 The high road to the sheaf of analytic functions
7.1 Sheaves of modules on a ringed space
7.3 Localization for modules
7.4 The sheaf of modules more explicitly
7.6 Coherent algebraic sheaves
7.7 Coherent analytic sheaves
7.8 The analytification of coherent algebraic sheaves
7.9 The statement of GAGA
8 Projective space – the statements
8.1 Products of affine schemes
8.3 Affine group schemes acting on affine schemes
8.4 The action of the group of closed points
8.5 Back to the world of the concrete
8.6 Quotients of affine schemes
8.7 Sheaves on the quotient
8.9 What it all means, in a concrete example
9 Projective space – the proofs
9.1 A reminder of symmetric powers
9.3 Finite dimensional representations of C*
9.4 The finite generation of the ring of invariants
9.5 The topological facts about π : X → X/G
9.8 The global statement about coherent sheaves
9.9 The case of the trivial group
10.2 Another visit to the concrete world
10.3 Maps between the sheaves O(m)
10.4 The coherent analytic version
10.6 GAGA in terms of cohomology
10.7 The first half of GAGA
10.9 Skyscraper sheaves on Pn
10.10 The second half of GAGA
Appendix 1 The proofs concerning analytification
A1.3 The faithful flatness
Algebraic and Analytic Geometry
( London Mathematical Society Lecture Note Series )
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