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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 material in the disciplines including the Riemann-Roch theorem for arbitrary projective curves. Table of Contents Series Preface Preface Chapter 1: Finitely Generated Algebra Chapter 2: The K-Algebra and the Zariski Topology Chapter 3: Prime Spectra and Dimension Chapter 4: Schemes Chapter 5: Projective Schemes Chapter 6: Regular, Normal and Smooth Points Chapter 7: Riemann:Roch Theorem References List of Symbols Index Biography of Authors