This is a graduate text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebras which are used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth. The book culminates with the theory of curves, including the Riemann-Roch theorem, elliptic curves and the zeta function of a curve over a finite field, and the Riemann hypothesis for elliptic curves.
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A very valuable enhancement to the current textbook literature on algebraic geometry, and a recommendable introduction to arithmetic geometry. -- Mathematics Abstracts
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Inside This Book
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First Sentence:
In these notes, the letter k will be reserved for a fixed algebraically closed field. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
complete nonsingular curve, finite dominant map, nontrivial valuation rings, minimal nonzero prime ideal, integrally closed integral domain, whose coordinate ring, separable morphism, dominant rational map, principal open set, residue class degree, separating transcendence basis, rational divisors, sheaf axiom, ramified points, affine open sets, regular local ring, finite separable extension, affine varieties, affine variety, discrete valuation ring, integral closure, linear disjointness, dominant morphism, ramification index, nonsingular curves Key Phrases - Capitalized Phrases (CAPs): (learn more)
Extension Theorem, Exercises Exercise, Nakayama's Lemma, Going Down, Hensel's Lemma, Noether's Normalization Lemma, Use Proposition New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In these notes, the letter k will be reserved for a fixed algebraically closed field. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
complete nonsingular curve, finite dominant map, nontrivial valuation rings, minimal nonzero prime ideal, integrally closed integral domain, whose coordinate ring, separable morphism, dominant rational map, principal open set, residue class degree, separating transcendence basis, rational divisors, sheaf axiom, ramified points, affine open sets, regular local ring, finite separable extension, affine varieties, affine variety, discrete valuation ring, integral closure, linear disjointness, dominant morphism, ramification index, nonsingular curves Key Phrases - Capitalized Phrases (CAPs): (learn more)
Extension Theorem, Exercises Exercise, Nakayama's Lemma, Going Down, Hensel's Lemma, Noether's Normalization Lemma, Use Proposition New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 17
books:
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3
books
cite this book:
- The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and their Jacobians (Lecture Notes in Mathematics) by David Mumford in Back Matter
- Hilbert Modular Surfaces (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 3 Folge) by Gerard Van Der Geer in Back Matter
- Introduction to Commutative Algebra and Algebraic Geometry by Ernst Kunz in Back Matter
- Introduction To Commutative Algebra by Michael Atiyah in Back Matter
- The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (v. 106) by Joseph H. Silverman in Back Matter
See all 17 books this book cites
- Resolution of Curve and Surface Singularities: in Characteristic Zero (Algebra and Applications) by K. Kiyek in Back Matter
- Lie Groups (Graduate Texts in Mathematics) by Daniel Bump in Front Matter
- Hodge Theory and Complex Algebraic Geometry I: Volume 1 (Cambridge Studies in Advanced Mathematics) by Claire Voisin in Front Matter
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