Algebraic Geometry

8.

Introduction to Schemes

8.1.

Motivation for Schemes

8.1.1.

Limitations of Classical Algebraic Geometry

8.1.2.

Arithmetic Applications

8.1.3.

Functorial Viewpoint

8.2.

Sheaves and Presheaves

8.2.1.

Presheaves on Topological Spaces

8.2.1.1.

Definition and Examples

8.2.1.2.

Morphisms of Presheaves

8.2.2.

8.2.2.1.

Definition and Sheaf Axioms

8.2.2.2.

Examples of Sheaves

8.2.2.3.

Sheafification Process

8.2.3.

Stalks of Sheaves

8.2.3.1.

Definition and Properties

8.2.3.2.

Local Properties

8.3.

8.3.1.

Definition of Ringed Space

8.3.2.

Morphisms of Ringed Spaces

8.3.3.

Locally Ringed Spaces

8.3.3.1.

Definition and Examples

8.3.3.2.

Local Rings in Stalks

8.4.

8.4.1.

The Spectrum of a Ring

8.4.1.1.

Points as Prime Ideals

8.4.1.2.

Motivation and Examples

8.4.2.

Zariski Topology on Spec(A)

8.4.2.1.

Closed Sets and Basic Opens

8.4.2.2.

Properties of the Topology

8.4.3.

Structure Sheaf on Spec(A)

8.4.3.1.

Definition via Localization

8.4.3.2.

Sections over Open Sets

8.4.4.

Affine Schemes as Locally Ringed Spaces

8.5.

General Schemes

8.5.1.

Definition by Gluing Affine Schemes

8.5.2.

Examples of Schemes

8.5.2.1.

Affine Space over Integers

8.5.2.2.

Projective Space as Scheme

8.5.2.3.

Spec of Fields and Other Rings

8.5.3.

Morphisms of Schemes

8.5.3.1.

Affine Morphisms

8.6.

Properties of Schemes

8.6.1.

Reduced and Integral Schemes

8.6.2.

Noetherian Schemes

8.6.3.

Separated Schemes

8.6.4.

8.6.5.

Dimension of Schemes

8.7.

The Functor of Points

8.7.1.

Yoneda Lemma Perspective

8.7.2.

Schemes as Functors

8.7.3.

Representable Functors

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9. Coherent Sheaves and Cohomology