Introduction to algebraic geometry and commutative algebra - Patil, Dilip P.; Storch, Uwe. (Singapore : World Scientific, 2010)
"This introductory textbook for a graduate course in pure mathematics provides a gateway into the two difficult fields of algebraic geometry and commutative algebra. Algebraic geometry, supported fundamentally by commutative algebra, is a cornerstone of pure mathematics.
Along the lines developed by Grothendieck, this book delves into the rich interplay between algebraic geometry and commutative algebra. With concise yet clear definitions and synopses a selection is made from the wealth of meterial in the disciplines including the Riemann-Roch theorem for arbitrary projective curves."--pub. desc.
-- Chapter 1. Finitely Generated Algebras -- 1.A. Algebras over a Ring -- 1.B. Factorization in Rings -- 1.C. Noetherian Rings and Modules -- 1.D. Graded Rings and Modules -- 1.E. Integral Extensions -- 1.F. Noether's Normalization Lemma and Its Consequences -- Chapter 2. The K-Spectrum and the Zariski Topology -- 2.A. The K-Spectrum of a K-Algebra -- 2.B. Affine Algebraic Sets -- 2.C. Strong Topology -- Chapter 3. Prime Spectra and Dimension -- 3.A. The Prime Spectrum of a Commutative Ring -- 3.B. Dimension -- Chapter 4. Schemes -- 4.A. Sheaves of Rings -- 4.B. Schemes -- 4.C. Finiteness Conditions on Schemes -- 4.D. Product of Schemes -- 4.E. Affine Morphisms -- Chapter 5. Projective Schemes -- 5.A. Projective Schemes -- 5.B. Main Theorem of Elimination -- 5.C. Mapping Theorem of Chevalley -- Chapter 6. Regular, Normal and Smooth Points -- 6.A. Regular Local Rings -- 6.B. Normal Domains -- 6.C. Normalization of a Scheme -- 6.D. The Module of Kahler Differentials -- 6.E. Quasi-coherent Sheaves and the Sheaf of Kahler Differentials -- Chapter 7. Riemann-Roch Theorem -- 7.A. Coherent Modules on Projective Schemes -- 7.B. Projective Curves -- 7.C. The Projective Line -- 7.D. Riemann-Roch Theorem for General Curves -- 7.E. Genus of a Projective Curve.
Includes bibliographical references (page [199]) and index.
"ILNS 1"--Spine.
Keywords: Geometry, Algebraic; Commutative algebra