3.2 Schemes and their morphisms

Schemes and their morphisms are the building blocks of modern algebraic geometry. They provide a unified framework for studying geometric objects, from classical varieties to more abstract constructions. This powerful concept allows mathematicians to work with diverse structures using a common language.

Morphisms between schemes are essential for understanding relationships between geometric objects. They enable comparisons, classifications, and the study of families of schemes. Properties like separatedness, properness, and flatness play crucial roles in analyzing these morphisms and their geometric implications.

Schemes as locally ringed spaces

Definition and structure

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• A scheme is a locally ringed space that is locally isomorphic to the spectrum of a commutative ring
• The structure sheaf of a scheme is a sheaf of commutative rings that encodes the local algebraic structure
• Schemes provide a unified framework for studying algebraic geometry, encompassing both affine and projective varieties
• The local nature of schemes allows for the study of geometric objects with varying properties at different points

Generalization of algebraic varieties

• Schemes can be thought of as a generalization of algebraic varieties, allowing for more flexibility in their construction and analysis
• Schemes extend the notion of algebraic varieties to include objects defined over arbitrary commutative rings, not just algebraically closed fields
• The structure sheaf of a scheme allows for the encoding of both geometric and algebraic properties in a unified manner
• Schemes provide a natural setting for studying arithmetic geometry, where the base ring is not necessarily an algebraically closed field

Constructing schemes: Affine vs projective

Affine schemes

• Affine schemes are the most basic examples of schemes, constructed from the spectrum of a commutative ring
  • The spectrum of a ring R, denoted Spec(R), is the set of prime ideals of R with a topology defined by the Zariski topology
  • The structure sheaf of an affine scheme is determined by the ring R, with local rings given by localizations at prime ideals
• Examples of affine schemes include the affine line (Spec($\mathbb{A}^1$)), affine plane (Spec($\mathbb{A}^2$)), and affine space (Spec($\mathbb{A}^n$))
• Affine schemes can be glued together to form more general schemes, such as projective schemes or quasi-projective schemes

Projective schemes

• Projective schemes are constructed by gluing together affine schemes, mimicking the construction of projective varieties
  • Projective space $\mathbb{P}^n$ is a fundamental example of a projective scheme, obtained by gluing together n+1 copies of affine space $\mathbb{A}^n$
  • Projective schemes can be defined using homogeneous ideals in polynomial rings, generalizing the notion of projective varieties
• Examples of projective schemes include projective lines ($\mathbb{P}^1$), projective planes ($\mathbb{P}^2$), and projective spaces ($\mathbb{P}^n$)
• Projective schemes are important in the study of algebraic geometry, as they provide a compactification of affine schemes and allow for the study of global properties

Other examples of schemes

• Grassmann schemes parametrize subspaces of a fixed vector space
  • The Grassmannian $Gr(k,n)$ is a scheme that represents the functor of k-dimensional subspaces of an n-dimensional vector space
• Hilbert schemes parametrize closed subschemes of a fixed scheme with given Hilbert polynomial
  • The Hilbert scheme $Hilb^{P}(X)$ represents the functor of closed subschemes of X with Hilbert polynomial P
• Abelian schemes generalize the notion of abelian varieties to the setting of schemes
  • An abelian scheme is a smooth proper group scheme over a base scheme, generalizing the notion of an abelian variety

Morphisms of schemes and their properties

Definition and local description

• A morphism of schemes is a continuous map between the underlying topological spaces that is compatible with the structure sheaves
• Morphisms of schemes can be described locally by ring homomorphisms between the corresponding affine coordinate rings
• The category of schemes, with morphisms as arrows, forms a fundamental object of study in algebraic geometry
• Morphisms allow for the comparison and relation of different schemes, enabling the study of their similarities and differences

Important properties of morphisms

• Separatedness: A morphism is separated if the diagonal morphism is a closed immersion, ensuring that fibers over points are well-behaved
  • Separatedness is a condition that ensures the uniqueness of limits and the Hausdorff property for the underlying topological space
• Properness: A morphism is proper if it is separated, of finite type, and universally closed, generalizing the notion of compact morphisms
  • Proper morphisms behave well under base change and have nice finiteness properties, making them important in algebraic geometry
• Flatness: A morphism is flat if it preserves the structure of fibers, allowing for well-behaved families of schemes
  • Flat morphisms are important in the study of deformations and moduli problems, where the fibers of the morphism represent the objects being classified

Composing and comparing morphisms

• Morphisms can be composed, allowing for the study of relationships between different schemes
• The composition of morphisms is associative and compatible with the structure sheaves, making the category of schemes a well-behaved object
• Fiber products and base change are important constructions in the study of morphisms, allowing for the comparison of schemes over different base schemes
• The study of morphisms and their properties is central to algebraic geometry, as it provides a way to relate and compare different geometric objects

Schemes unifying geometric concepts

Common language for geometric objects

• Schemes provide a common language for studying various geometric objects, including algebraic varieties, complex manifolds, and arithmetic schemes
• The structure sheaf of a scheme encodes both the algebraic and geometric properties of the object, allowing for a unified treatment
• Schemes allow for the study of geometric objects over arbitrary base rings, not just algebraically closed fields, enabling arithmetic and geometric applications

Moduli problems and classification

• The theory of schemes provides powerful tools for studying moduli problems, which aim to classify geometric objects with specified properties
• Moduli spaces, such as the moduli space of curves or the moduli space of vector bundles, can often be realized as schemes or stacks
• The language of schemes allows for a unified approach to the construction and study of moduli spaces, which play a central role in modern algebraic geometry

Applications in various fields

• Number theory: Arithmetic schemes are used to study Diophantine equations and arithmetic properties of varieties
  • The theory of schemes provides a framework for studying the arithmetic of algebraic varieties over number fields or finite fields
• Complex geometry: Schemes provide a framework for studying complex manifolds and their degenerations
  • The GAGA principle (Géométrie Algébrique et Géométrie Analytique) relates the algebraic and analytic properties of complex algebraic varieties
• Representation theory: Schemes arise naturally in the study of representations of algebraic groups and Lie algebras
  • The flag variety of a reductive algebraic group is a projective scheme that parametrizes certain subgroup schemes, playing a key role in representation theory