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

Format:

Book

Author/Creator:

Vakil, Ravi, author.

Language:

English

Subjects (All):

Geometry, Algebraic.

Physical Description:

1 online resource (689 pages)

Edition:

1st ed.

Place of Publication:

Princeton, New Jersey : Princeton University Press, [2025]

Summary:

An accessible, motivated introduction to one of the most dynamic areas of mathematics 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.

Contents:

Preliminaries. Just enough category theory to be dangerous ; Sheaves

Schemes. Toward affine schemes : the underlying set, and topological space ; The structure sheaf, and the definition of schemes in general ; Some properties of schemes ; Rings are to modules as schemes are to...

Morphisms of schemes. Morphisms of schemes ; Useful classes of morphisms of schemes ; Closed embeddings and related notions ; Fibered products of schemes, and base change ; Separated and proper mophisms, and (finally!) varieties

"Geometric" properties of schemes. Dimension ; Regularity and smoothness

Quasicoherent sheaves on schemes, and their uses. More on qusicoherent and coherent sheaves ; Line bundles, maps to projective space, and divisors ; Maps to projective space, and properties of line bundles ; Projective morphisms, and relative versions of Spec and Proj ; C̆ech cohomology of Quasicoherent sheaves ; Application : curves ; Application : a glimpse of intersection theory ; Differentials ; Blowing up

More cohomoligical tools. Derived functors ; Flatness ; Cohomology and base change theorems ; Depth and Cohen-Macaulayness ; The twenty-seven lines on a cubic surface ; Power series and the theorem on formal function ; Proof of Serre duality.

Notes:

Includes bibliographical references (pages 635-640) and index.

Description based on publisher supplied metadata and other sources.

ISBN:

9780691268682

0-691-26868-1

OCLC:

1531325969