Algebraic Varieties.

Summary

Titles

• Full Title: Algebraic Varieties.

Series Statement

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Notes

• Cover; Series Page; Title; Copyright; Contents; Introduction; 1 Algebraic varieties: definition and existence; 1.1 Spaces with functions; 1.2 Varieties; 1.3 The existence of affine varieties; 1.4 The nullstellensatz; 1.5 The rest of the proof of existence of affine varieties / subvarieties; 1.6 An and Pn; 1.7 Determinantal varieties; 2 The preparation lemma and some consequences; 2.1 The lemma; 2.2 The Hilbert basis theorem; 2.3 Irreducible components; 2.4 Affine and finite morphisms; 2.5 Dimension; 2.6 Hypersurfaces and the principal ideal theorem; 3 Products.
• Separated and complete varieties3.1 Products; 3.2 Products of projective varieties; 3.3 Graphs of morphisms and separatedness; 3.4 Algebraic groups; 3.5 Cones and projective varieties; 3.6 A little more dimension theory; 3.7 Complete varieties; 3.8 Chow's lemma; 3.9 The group law on an elliptic curve; 3.10 Blown up An at the origin; 4 Sheaves; 4.1 The definition of presheaves and sheaves; 4.2 The construction of sheaves; 4.3 Abelian sheaves and flabby sheaves; 4.4 Direct limits of sheaves; 5 Sheaves in algebraic geometry; 5.1 Sheaves of rings and modules.
• 5.2 Quasi-coherent sheaves on affine varieties5.3 Coherent sheaves; 5.4 Quasi-coherent sheaves on projective varieties; 5.5 Invertible sheaves; 5.6 Operations on sheaves that change spaces; 5.7 Morphisms to projective space and affine morphisms; 6 Smooth varieties and morphisms; 6.1 The Zariski cotangent space and smoothness; 6.2 Tangent cones; 6.3 The sheaf of differentials; 6.4 Morphisms; 6.5 The construction of affine morphisms and normalization; 6.6 Bertini's theorem; 7 Curves; 7.1 Introduction to curves; 7.2 Valuation criterions; 7.3 The construction of all smooth curves.
• 7.4 Coherent sheaves on smooth curves7.5 Morphisms between smooth complete curves; 7.6 Special morphisms between curves; 7.7 Principal parts and the Cousin problem; 8 Cohomology and the Riemann-Roch theorem; 8.1 The definition of cohomology; 8.2 Cohomology of afflnes; 8.3 Higher direct images; 8.4 Beginning the study of the cohomology of curves; 8.5 The Riemann-Roch theorem; 8.6 First applications of the Riemann-Roch theorem; 8.7 Residues and the trace homomorphism; 9 General cohomology; 9.1 The cohomology of An -- {0} and Pn; 9.2 Čech cohomology and the Künneth formula.
• 9.3 Cohomology of projective varieties9.4 The direct images of flat sheaves; 9.5 Families of cohomology groups; 10 Applications; 10.1 Embedding in projective space; 10.2 Cohomological characterization of affine varieties; 10.3 Computing the genus of a plane curve and Bezout's theorem; 10.4 Elliptic curves; 10.5 Locally free coherent sheaves on P1; 10.6 Regularity in codimension one; 10.7 One dimensional algebraic groups; 10.8 Correspondences; 10.9 The Riemann-Roch Theorem for surfaces; Appendix; A.1 Localization; A.2 Direct limits; A.3 Eigenvectors; Bibliography; Glossary of notation; Index.

Identifiers

• Isbns: 9781107094574; 1107094577; 9781107359956; 1107359953; 9780521426138; 0521426138
• Oclc Number: (OCoLC)850148954

Publication Statement

• Place: Cambridge
• Publisher: Cambridge University Press
• Date: 1993

Physical Description

• Extent: 1 online resource (176 pages)

Table Of Contents

• Cover; Series Page; Title; Copyright; Contents; Introduction; 1 Algebraic varieties: definition and existence; 1.1 Spaces with functions; 1.2 Varieties; 1.3 The existence of affine varieties; 1.4 The nullstellensatz; 1.5 The rest of the proof of existence of affine varieties / subvarieties; 1.6 An and Pn; 1.7 Determinantal varieties; 2 The preparation lemma and some consequences; 2.1 The lemma; 2.2 The Hilbert basis theorem; 2.3 Irreducible components; 2.4 Affine and finite morphisms; 2.5 Dimension; 2.6 Hypersurfaces and the principal ideal theorem; 3 Products.
• Separated and complete varieties3.1 Products; 3.2 Products of projective varieties; 3.3 Graphs of morphisms and separatedness; 3.4 Algebraic groups; 3.5 Cones and projective varieties; 3.6 A little more dimension theory; 3.7 Complete varieties; 3.8 Chow's lemma; 3.9 The group law on an elliptic curve; 3.10 Blown up An at the origin; 4 Sheaves; 4.1 The definition of presheaves and sheaves; 4.2 The construction of sheaves; 4.3 Abelian sheaves and flabby sheaves; 4.4 Direct limits of sheaves; 5 Sheaves in algebraic geometry; 5.1 Sheaves of rings and modules.
• 5.2 Quasi-coherent sheaves on affine varieties5.3 Coherent sheaves; 5.4 Quasi-coherent sheaves on projective varieties; 5.5 Invertible sheaves; 5.6 Operations on sheaves that change spaces; 5.7 Morphisms to projective space and affine morphisms; 6 Smooth varieties and morphisms; 6.1 The Zariski cotangent space and smoothness; 6.2 Tangent cones; 6.3 The sheaf of differentials; 6.4 Morphisms; 6.5 The construction of affine morphisms and normalization; 6.6 Bertini's theorem; 7 Curves; 7.1 Introduction to curves; 7.2 Valuation criterions; 7.3 The construction of all smooth curves.
• 7.4 Coherent sheaves on smooth curves7.5 Morphisms between smooth complete curves; 7.6 Special morphisms between curves; 7.7 Principal parts and the Cousin problem; 8 Cohomology and the Riemann-Roch theorem; 8.1 The definition of cohomology; 8.2 Cohomology of afflnes; 8.3 Higher direct images; 8.4 Beginning the study of the cohomology of curves; 8.5 The Riemann-Roch theorem; 8.6 First applications of the Riemann-Roch theorem; 8.7 Residues and the trace homomorphism; 9 General cohomology; 9.1 The cohomology of An -- {0} and Pn; 9.2 Čech cohomology and the Künneth formula.
• 9.3 Cohomology of projective varieties9.4 The direct images of flat sheaves; 9.5 Families of cohomology groups; 10 Applications; 10.1 Embedding in projective space; 10.2 Cohomological characterization of affine varieties; 10.3 Computing the genus of a plane curve and Bezout's theorem; 10.4 Elliptic curves; 10.5 Locally free coherent sheaves on P1; 10.6 Regularity in codimension one; 10.7 One dimensional algebraic groups; 10.8 Correspondences; 10.9 The Riemann-Roch Theorem for surfaces; Appendix; A.1 Localization; A.2 Direct limits; A.3 Eigenvectors; Bibliography; Glossary of notation; Index.

Summary

• An introduction to the theory of algebraic functions on varieties from a sheaf theoretic standpoint.

Subjects

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