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"A great subject and expert authors!" Nieuw Archief voor Wiskunde,June 2001 "Both Eisenbud and Harris are experienced and compelling educators of modern mathematics. This book is strongly recommended to anyone who would like to know what schemes are all about." Newsletter of the New Zealand Mathematical Society, No. 82, August 2001
E-Book Content
The Geometry of Schemes
David Eisenbud Joe Harris
Springer
. . . the end of all our exploring Will be to arrive where we started And know the place for the first time. – T. S. Eliot, “Little Gidding” (Four Quartets)
Contents
Introduction
I
Basic Definitions I.1 Affine Schemes . . . . . . . . . . . . . . . . . . . I.1.1 Schemes as Sets . . . . . . . . . . . . . . I.1.2 Schemes as Topological Spaces . . . . . . I.1.3 An Interlude on Sheaf Theory . . . . . . References for the Theory of Sheaves . . I.1.4 Schemes as Schemes (Structure Sheaves) I.2 Schemes in General . . . . . . . . . . . . . . . . I.2.1 Subschemes . . . . . . . . . . . . . . . . I.2.2 The Local Ring at a Point . . . . . . . . I.2.3 Morphisms . . . . . . . . . . . . . . . . . I.2.4 The Gluing Construction . . . . . . . . . Projective Space . . . . . . . . . . . . . . I.3 Relative Schemes . . . . . . . . . . . . . . . . . . I.3.1 Fibered Products . . . . . . . . . . . . . I.3.2 The Category of S-Schemes . . . . . . . I.3.3 Global Spec . . . . . . . . . . . . . . . . I.4 The Functor of Points . . . . . . . . . . . . . . .
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II Examples II.1 Reduced Schemes over Algebraically Closed Fields . . . II.1.1 Affine Spaces . . . . . . . . . . . . . . . . . . . . II.1.2 Local Schem