3.3 Affine and projective schemes

Affine and projective schemes are fundamental objects in algebraic geometry. They provide a unified framework for studying algebraic varieties, generalizing the classical notions of affine and projective varieties to more abstract settings.

Affine schemes are built from commutative rings, while projective schemes come from graded rings. This distinction reflects their different geometric properties and allows for a deeper understanding of algebraic structures in geometry.

Affine Schemes from Rings

Constructing Affine Schemes

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• An affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring
• The spectrum of a commutative ring A, denoted Spec(A), consists of all prime ideals of A with the Zariski topology
  • The Zariski topology on Spec(A) has closed sets V(I) = {p ∈ Spec(A) | I ⊆ p}, where I is an ideal of A
  • The Zariski topology is T0 (Kolmogorov) but not T1 (Fréchet), and compact if and only if A is Noetherian (Noetherian rings have the ascending chain condition on ideals)

Structure Sheaf and Morphisms

• The structure sheaf OX on an affine scheme X = Spec(A) is defined by OX(U) = {s: U → ⨆_{p ∈ U} Ap | s is locally a fraction}, where Ap is the localization of A at the prime ideal p
  • The stalk of the structure sheaf OX at a point p ∈ X is isomorphic to the localization Ap of A at the prime ideal p
  • Localizing a ring at a prime ideal inverts elements outside the prime ideal, allowing the study of local properties
• Morphisms between affine schemes correspond to ring homomorphisms in the opposite direction
  • A morphism f: Spec(B) → Spec(A) of affine schemes induces a ring homomorphism f^#: A → B
  • Affine schemes form a category opposite to the category of commutative rings

Projective Schemes from Graded Rings

Graded Rings and Homogeneous Ideals

• A graded ring A is a ring with a direct sum decomposition A = ⨁_{n ∈ ℤ} An such that An · Am ⊆ An+m for all n, m ∈ ℤ
  • Elements of An are called homogeneous elements of degree n
  • Examples of graded rings include polynomial rings with their natural grading and exterior algebras
• A homogeneous ideal I of a graded ring A is an ideal generated by homogeneous elements, i.e., I = ⨁_{n ∈ ℤ} (I ∩ An)
  • Homogeneous ideals are important for constructing projective schemes
  • The irrelevant ideal A+ = ⨁_{n > 0} An plays a key role in the definition of projective spectrum

Projective Spectrum and Schemes

• The projective spectrum of a graded ring A, denoted Proj(A), is the set of all homogeneous prime ideals of A that do not contain the irrelevant ideal A+, equipped with the Zariski topology
• A projective scheme is a locally ringed space isomorphic to the projective spectrum of a graded ring
  • The structure sheaf OX on a projective scheme X = Proj(A) is defined using the degree zero parts of localizations of A at homogeneous prime ideals
  • Projective schemes are fundamental objects in algebraic geometry, generalizing projective varieties

Affine vs Projective Schemes

Key Differences

• Affine schemes are constructed from arbitrary commutative rings, while projective schemes are constructed from graded rings
• Affine schemes have a unique closed point corresponding to the maximal ideal, while projective schemes do not have a unique closed point
  • The unique closed point in an affine scheme allows for the study of local properties
  • The lack of a unique closed point in projective schemes reflects their global nature

Topological and Geometric Properties

• Affine schemes are quasi-compact (every open cover has a finite subcover), while projective schemes are proper (universally closed and separated)
  • Quasi-compactness is a finiteness condition on the topology of a scheme
  • Properness is a key property in algebraic geometry, related to compactness and separatedness
• Affine schemes can be covered by distinguished open sets D(f) = {p ∈ Spec(A) | f ∉ p}, while projective schemes can be covered by open sets D+(f) = {p ∈ Proj(A) | f ∉ p}, where f is homogeneous
  • Distinguished open sets are important for understanding the local structure of schemes
  • Open sets in projective schemes are defined using homogeneous elements, reflecting the graded structure

Morphisms and Functoriality

• Morphisms between affine schemes correspond to ring homomorphisms, while morphisms between projective schemes correspond to graded ring homomorphisms that preserve the irrelevant ideal
  • Affine schemes form a category opposite to the category of commutative rings
  • Projective schemes form a category with morphisms induced by graded ring homomorphisms
• Affine and projective schemes have different functorial properties
  • The functor Spec from commutative rings to affine schemes is contravariant, while the functor Proj from graded rings to projective schemes is not fully faithful
  • The functorial properties of schemes are essential for studying moduli problems and geometric invariants

Schemes for Algebraic Varieties

Unifying Framework

• Schemes provide a unified framework for studying algebraic varieties over arbitrary fields and their generalizations (algebraic spaces, stacks)
  • Schemes allow for the study of varieties with nilpotent elements and non-reduced structures, important in deformation theory and moduli problems
  • Algebraic spaces and stacks are generalizations of schemes that arise in moduli theory and algebraic geometry

Realizing Varieties as Schemes

• Affine varieties can be realized as affine schemes by taking the spectrum of their coordinate rings
  • The coordinate ring of an affine variety encodes its algebraic structure
  • The spectrum construction translates between the algebraic and geometric perspectives
• Projective varieties can be realized as projective schemes by taking the projective spectrum of their homogeneous coordinate rings
  • Homogeneous coordinate rings capture the graded structure of projective varieties
  • The projective spectrum construction provides a scheme-theoretic perspective on projective varieties

Studying Invariants and Properties

• Schemes can be used to define and study important invariants of varieties (cohomology groups, Hilbert polynomials, intersection numbers)
  • Cohomology groups, such as sheaf cohomology and de Rham cohomology, are fundamental invariants that capture topological and geometric properties
  • Hilbert polynomials encode dimensions of graded pieces of coordinate rings and are used in the study of moduli spaces
  • Intersection numbers measure the complexity of intersections between subvarieties and are central to enumerative geometry
• The language of schemes is essential for modern developments in algebraic geometry (minimal model program, theory of moduli spaces, arithmetic geometry)
  • The minimal model program aims to classify algebraic varieties up to birational equivalence using schemes and related techniques
  • Moduli spaces parametrize geometric objects (curves, surfaces, vector bundles) and are constructed using schemes and their generalizations
  • Arithmetic geometry studies schemes over rings of integers and their connections to number theory and cryptography