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**About this book :- **
**Foundations of Algebraic Geometry ** written by
** Ravi Vakil **

This book is intended to give a serious and reasonably complete introduction to algebraic geometry, not just for (future) experts in the field. The exposition serves a narrow set of goals, and necessarily takes a particular point of view on the subject.

**Book Detail :- **
** Title: ** Foundations of Algebraic Geometry
** Edition: **
** Author(s): ** Ravi Vakil
** Publisher: **
** Series: **
** Year: ** 2022
** Pages: ** 788
** Type: ** PDF
** Language: ** English
** ISBN: **
** Country: **
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**Book Contents :- **
**Foundations of Algebraic Geometry ** written by
** Ravi Vakil **
cover the following topics.
**Preface **

For the reader, For the expert, Background and conventions, The goals of this book
**PART-I PRELIMINARIES**
** Some category theory **

Motivation, Categories and functors, Universal properties determine an object up to unique isomorphism, Limits and colimits, Adjoints, An introduction to abelian categories, Spectral sequences
**Sheaves**

Motivating example: The sheaf of differentiable functions, Definition of sheaf and presheaf, Morphisms of presheaves and sheaves, Properties determined at the level of stalks, and sheafification, Sheaves of abelian groups, and OX-modules, form abelian categories, The inverse image sheaf, Recovering sheaves from a “sheaf on a base”
**PART-II SCHEMES**
**Toward affine schemes: the underlying set, and topological space **

Toward schemes, The underlying set of affine schemes, Visualizing schemes I: generic points, The underlying topological space of an affine scheme, A base of the Zariski topology on Spec A: Distinguished open sets, Topological (and Noetherian) properties, The function I(·), taking subsets of Spec A to ideals of A
**The structure sheaf, and the definition of schemes in general **

The structure sheaf of an affine scheme, Visualizing schemes II: nilpotents, Definition of schemes, Three examples, Projective schemes, and the Proj construction
** Some properties of schemes**

Topological properties, Reducedness and integrality, Properties of schemes that can be checked “affine-locally”, Normality and factoriality, Where functions are supported: Associated points of schemes 164
**PART-III MORPHISMS **
** Morphisms of schemes**

Introduction, Morphisms of ringed spaces, From locally ringed spaces to morphisms of schemes, Maps of graded rings and maps of projective schemes, Rational maps from reduced schemes, ⋆ Representable functors and group schemes, ⋆⋆ The Grassmannian (initial construction)
**Useful classes of morphisms of schemes**

An example of a reasonable class of morphisms: Open embeddings, Algebraic interlude: Lying Over and Nakayama, A gazillion finiteness conditions on morphisms, Images of morphisms: Chevalley’s theorem and elimination theory
** Closed embeddings and related notions**

Closed embeddings and closed subschemes, More projective geometry, Smallest closed subschemes such that, Effective Cartier divisors, regular sequences and regular embeddings
** Fibered products of schemes, and base change **

They exist, Computing fibered products in practice, Interpretations: Pulling back families, and fibers of morphisms, Properties preserved by base change, ⋆ Properties not preserved by base change, and how to fix them, Products of projective schemes: The Segre embedding, Normalization,
** Separated and proper morphisms, and (finally!) varieties**

Separated morphisms (and quasiseparatedness done properly), Rational maps to separated schemes, Proper morphisms
**Part-IV “GEOMETRIC” properties: Dimension and smoothness **
** Dimension **

Dimension and codimension, Dimension, transcendence degree, and Noether normalization, Codimension one miracles: Krull’s and Hartogs’s Theorems, Dimensions of fibers of morphisms of varieties, ⋆⋆ Proof of Krull’s Principal Ideal and Height Theorems
** Regularity and smoothness**

The Zariski tangent space, Regularity, and smoothness over a field, Examples, Bertini’s Theorem, Discrete valuation rings: Dimension 1 Noetherian regular local rings, Smooth (and etale) morphisms (first definition), ⋆ Valuative criteria for separatedness and properness, ⋆ More sophisticated facts about regular local rings, ⋆ Filtered rings and modules, and the Artin-Rees Lemma
**Part-V QUASICOHERENT SHEAVES**
**Quasicoherent and coherent sheaves **

Vector bundles and locally free sheaves, Quasicoherent sheaves, Characterizing quasicoherence using the distinguished affine base, Quasicoherent sheaves form an abelian category, Module-like constructions, Finite type and coherent sheaves, Pleasant properties of finite type and coherent sheaves, ⋆⋆ Coherent modules over non-Noetherian rings
** Line bundles: Invertible sheaves and divisors **

Some line bundles on projective space, Line bundles and Weil divisors, ⋆ Effective Cartier divisors “=” invertible ideal sheaves
** Quasicoherent sheaves on projective A-schemes **

The quasicoherent sheaf corresponding to a graded module, Invertible sheaves (line bundles) on projective A-schemes, Globally generated and base-point-free line bundles, Quasicoherent sheaves and graded modules
** Pushforwards and pullbacks of quasicoherent sheaves **

Introduction, Pushforwards of quasicoherent sheaves, Pullbacks of quasicoherent sheaves, Line bundles and maps to projective schemes, The Curve-to-Projective Extension Theorem, Ample and very ample line bundles, ⋆ The Grassmannian as a moduli space
** Relative versions of Spec and Proj, and projective morphisms **

Relative Spec of a (quasicoherent) sheaf of algebras, Relative Proj of a sheaf of graded algebras, Projective morphisms, Applications to curves
** Cech cohomology of quasicoherent sheaves **

(Desired) properties of cohomology, Definitions and proofs of key properties, Cohomology of line bundles on projective space, Riemann-Roch, degrees of coherent sheaves, arithmetic genus, and Serre duality, A first glimpse of Serre duality, Hilbert functions, Hilbert polynomials, and genus, ⋆ Serre’s cohomological characterization of ampleness, Higher direct image sheaves, ⋆ Chow’s Lemma and Grothendieck’s Coherence Theorem
** Application: Curvesy**

A criterion for a morphism to be a closed embedding, A series of crucial tools, Curves of genus 0, Classical geometry arising from curves of positive genus 1, Hyperelliptic curves, Curves of genus 2, Curves of genus 3, Curves of genus 4 and 5, Curves of genus 1, Elliptic curves are group varieties, Counterexamples and pathologies using elliptic curves
** ⋆ Application: A glimpse of intersection theory**

Intersecting n line bundles with an n-dimensional variety, Intersection theory on a surface, The Grothendieck group of coherent sheaves, and an algebraic version of homology, ⋆⋆ The Nakai-Moishezon and Kleiman criteria for ampleness
**Differentials **

Motivation and game plan, Definitions and first properties, Smoothness of varieties revisited, Examples, Studying smooth varieties using their cotangent bundles, Unramified morphisms, The Riemann-Hurwitz Formula
**⋆ Blowing up**

Motivating example: blowing up the origin in the plane, Blowing up, by universal property, The blow-up exists, and is projective, Examples and computations
**Part-VI MORE**
** Derived functors**

The Tor functors, Derived functors in general, Derived functors and spectral sequences, Derived functor cohomology of O-modules, Cech cohomology and derived functor cohomology agree
**Flatness **

Introduction, Easier facts, Flatness through Tor, Ideal-theoretic criteria for flatness, Topological aspects of flatness, Local criteria for flatness, Flatness implies constant Euler characteristic
**Smooth, etale, and unramified morphisms revisited **

Some motivation, Different characterizations of smooth and etale morphisms, Generic smoothness and the Kleiman-Bertini Theorem
** Depth and Cohen-Macaulayness**

Depth, Cohen-Macaulay rings and schemes, ⋆⋆ Serre’s R1 + S2 criterion for normality
** Twenty-seven lines**

Introduction, Preliminary facts, Every smooth cubic surface (over k) has 27 lines, Every smooth cubic surface (over k) is a blown up plane
**Cohomology and base change theorems **

Statements and applications, ⋆⋆ Proofs of cohomology and base change theorems, Applying cohomology and base change to moduli problems
**Power series and the Theorem on Formal Functions **

Introduction, Algebraic preliminaries, Defining types of singularities, The Theorem on Formal Functions, Zariski’s Connectedness Lemma and Stein Factorization, Zariski’s Main Theorem, Castelnuovo’s criterion for contracting (−1)-curves, ⋆⋆ Proof of the Theorem on Formal Functions 29.4.2
**⋆ Proof of Serre duality**

Introduction, Ext groups and Ext sheaves for O-modules, Serre duality for projective k-schemes, The adjunction formula for the dualizing sheaf, and ωX = KX
**Bibliography **
**Index **

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- Abstract Algebra
- Calculus
- Differential Equations
- Engineering Mathematics
- Linear Algebra
- Math Magic
- Real Analysis