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 is a locally 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
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 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