The rising sea : foundations of algebraic geometry / Ravi Vakil.

Format:

Book

Author/Creator:

Vakil, Ravi, author.

Language:

English

Subjects (All):

Geometry, Algebraic.

Physical Description:

xxiii, 662 pages : illustrations ; 26 cm

Place of Publication:

Princeton : Princeton University Press, 2025.

Biography/History:

Ravi Vakil is the Robert Grimmett Professor of Mathematics at Stanford University and president of the American Mathematical Society. He is the author of A Mathematical Mosaic: Patterns and Problem Solving.

Summary:

"Decades ago, Mumford wrote that algebraic geometry 'seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.' The revolution has now fully come to pass and has fundamentally changed how we think about many fields of mathematics. This book provides a thorough foundation in the powerful ideas that now shape the landscape, with an informal yet rigorous exposition that builds intuition for understanding the formidable machinery. It begins with a discussion of categorical thinking and sheaves and then develops the notion of schemes and varieties as examples of 'geometric spaces' before discussing their specific aspects. The book goes on to cover topics such as dimension and smoothness, vector bundles and their natural generalizations, and important cohomological tools and their applications. Important optional topics are included in starred sections."-- Publisher's website.

Contents:

Part I : Preliminaries

1 : Just enough category theory to be dangerous

2 : Sheaves

Part II : Schemes

3 : Toward affine schemes : the underlying set, and topological space

4 : The structure sheaf, and the definition of schemes in general

5 : Some properties of schemes

6 : Rings are to modules as schemes are to...

Part III : Morphisms of schemes

7 : Morphisms of schemes

8 : Useful classes of morphisms of schemes

9 : Closed embeddings and related notions

10 : Fibered products of schemes, and base change

11 : Separated and proper mophisms, and (finally!) varieties

Part IV : "Geometric" properties of schemes

12 : Dimension

13 : Regularity and smoothness

Part V : Quasicoherent sheaves on schemes, and their uses

14 : More on qusicoherent and coherent sheaves

15 : Line bundles, maps to projective space, and divisors

16 : Maps to projective space, and properties of line bundles

17 : Projective morphisms, and relative versions of Spec and Proj

18 : C̆ech cohomology of Quasicoherent sheaves

19 : Application : curves

20 : Application : a glimpse of intersection theory

21 : Differentials

22 : Blowing up

Part VI : More cohomoligical tools

23 : Derived functors

24 : Flatness

25 : Cohomology and base change theorems

26 : Depth and Cohen-Macaulayness

27 : The twenty-seven lines on a cubic surface

28 : Power series and the theorem on formal function

29 : Proof of Serre duality

Notes:

Includes bibliographical references (pages 635-640) and index

Other Format:

Online version: Vakil, Ravi. Rising sea.

ISBN:

9780691268668

0691268665

9780691268675

0691268673

OCLC:

1463662882

Publisher Number:

CIPO000281428