12.3 Introduction to schemes and modern algebraic geometry
5 min read•july 30, 2024
Schemes revolutionize algebraic geometry by providing a unified framework for studying geometric spaces. They generalize algebraic varieties, allowing for the examination of geometric objects over arbitrary base rings and enabling the study of both local and global properties.
The theory of schemes introduces powerful tools like cohomology and the functor of points perspective. This modern approach opens up new avenues for research in areas such as arithmetic geometry, complex geometry, and mathematical physics.
Motivation for Schemes
Generalizing Algebraic Varieties
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- Schemes generalize the notion of algebraic varieties and provide a unified framework for studying geometric spaces locally and globally
- Schemes are locally ringed spaces that are locally isomorphic to the spectrum of a commutative ring (affine schemes)
- Affine schemes are the building blocks of schemes
- Defined as the spectrum of a commutative ring R, denoted Spec(R)
- Spec(R) is the set of all prime ideals of R equipped with the Zariski topology
- The structure sheaf on an is determined by the ring of regular functions on open subsets
- Schemes are constructed by gluing together affine schemes along open subsets, similar to the construction of manifolds (gluing together open subsets of Euclidean space)
Morphisms and Categories
- Morphisms between schemes are defined as morphisms of locally ringed spaces
- Continuous maps that respect the structure sheaves
- The category of schemes is a natural generalization of the category of affine varieties
- Provides a rich setting for studying geometric objects and their properties
- Schemes allow for the study of geometric objects over arbitrary base rings, not just fields
- Enables the study of arithmetic and geometric properties of varieties over various base schemes (integers, finite fields)
Local and Global Properties of Schemes
Local Structure and Singularities
- The local structure of a scheme at a point is determined by the at that point
- Obtained by localizing the structure sheaf at the point
- Encodes information about the scheme in a neighborhood of the point
- The dimension of a scheme at a point is defined as the Krull dimension of the local ring at that point
- The dimension of a scheme is the supremum of the dimensions at all points
- Schemes can have singularities, which are points where the local ring is not regular
- The study of singularities is a central topic in algebraic geometry
- Has applications in various areas of mathematics and physics
Global Properties and Cohomology
- Schemes can be reducible or irreducible
- Irreducible if they cannot be written as the union of two proper closed subschemes
- Irreducible schemes are the building blocks of schemes and play a crucial role in the study of their geometry
- The connectedness and connectivity of schemes can be studied using the Zariski topology
- A scheme is connected if it is not the disjoint union of two proper closed subschemes
- Global properties of schemes, such as cohomology and , can be studied using the tools of sheaf theory and homological algebra
- Sheaves and sheaf cohomology are fundamental tools in the study of schemes
- Provide a way to study global properties of geometric objects by considering local data
Schemes vs Algebraic Varieties
Generalizing Algebraic Varieties
- Every algebraic over an algebraically closed field can be naturally associated with a scheme, called the associated scheme
- The associated scheme captures the intrinsic geometry of the variety and is independent of the choice of embedding
- Schemes can have non-reduced structure, which means that the local rings may have nilpotent elements
- Allows for the study of more general geometric objects (infinitesimal deformations, non-reduced fibers of morphisms)
Functor of Points Perspective
- The functor of points perspective provides a way to study schemes by considering their morphisms into other schemes
- Generalizes the classical approach of studying varieties via their points
- Schemes can be defined over arbitrary base rings, not just fields
- Allows for the study of arithmetic and geometric properties of varieties over various base schemes (integers, finite fields)
Key Concepts in Algebraic Geometry
Sheaves and Cohomology
- Sheaves and sheaf cohomology are fundamental tools in the study of schemes
- Provide a way to study global properties of geometric objects by considering local data
- Coherent sheaves are a special class of sheaves that generalize the notion of vector bundles on varieties
- Play a central role in the study of the geometry of schemes
- Derived categories and derived functors provide a powerful framework for studying the homological algebra of sheaves on schemes and the cohomology of algebraic varieties
Divisors, Line Bundles, and Flatness
- Divisors and line bundles are important tools for studying the birational geometry of schemes and the behavior of morphisms between them
- The theory of flatness and base change is crucial for understanding the behavior of schemes under morphisms and the properties of families of schemes
- Flat morphisms preserve important geometric properties and are essential in the study of moduli spaces and deformation theory
Schemes in Algebraic Geometry
Unifying and Generalizing Algebraic Geometry
- Schemes provide a unifying framework for studying various aspects of algebraic geometry
- Includes the theory of algebraic varieties, arithmetic geometry, and complex geometry
- The language of schemes allows for the generalization of many classical results in algebraic geometry
- Examples: Riemann-Roch theorem, Hirzebruch-Riemann-Roch theorem
- Generalized to more general settings using the theory of schemes
Applications and Further Developments
- Schemes have applications in various areas of mathematics
- Number theory, representation theory, mathematical physics
- Provide a common language for studying geometric objects across different fields
- The theory of moduli spaces, which studies families of geometric objects, relies heavily on the language of schemes and the tools of modern algebraic geometry
- The study of schemes has led to the development of new techniques and insights in algebraic geometry
- Theory of stacks and the geometry of derived schemes
- Opened up new avenues for research and applications
Key Terms to Review (17)
Affine Scheme: An affine scheme is a fundamental concept in algebraic geometry that represents a geometric object by using the spectrum of a ring, specifically the spectrum of a commutative ring with unity. It connects algebraic and geometric perspectives by identifying points in the scheme with prime ideals of the ring, enabling the study of varieties in an algebraic framework and establishing a bridge to modern algebraic geometry.
Alexander Grothendieck: Alexander Grothendieck was a French mathematician who is considered one of the most influential figures in 20th-century mathematics, particularly in the fields of algebraic geometry and homological algebra. His revolutionary ideas, especially on schemes, fundamentally changed the landscape of modern algebraic geometry, providing a new framework for understanding geometric objects through algebraic methods.
Coherent sheaf: A coherent sheaf is a type of sheaf that behaves nicely with respect to the algebraic structure of a scheme, particularly in the context of the sheaf's ability to be represented by finitely generated modules over rings. This notion is crucial for understanding properties like the support of a sheaf and its relationships with morphisms in algebraic geometry. Coherent sheaves allow us to study varieties and schemes through algebraic means, providing a bridge between geometry and algebra.
David Mumford: David Mumford is a prominent mathematician known for his significant contributions to algebraic geometry, particularly in the areas of rational maps, birational equivalence, and the development of modern algebraic geometry techniques. His work emphasizes the geometric aspects of algebraic structures and has had a lasting impact on the understanding of surfaces, schemes, and dimensions within this mathematical field.
Divisor: A divisor is a formal mathematical object that represents a way to encode the idea of a 'point' or 'subvariety' on an algebraic variety, capturing how functions behave near those points. It provides a way to study the properties of algebraic varieties through their intersections and the associated function field. Understanding divisors is crucial for exploring rational maps, examining genus, and transitioning into modern algebraic geometry concepts like schemes.
Finite morphism: A finite morphism is a type of morphism between schemes where the preimage of any point in the target scheme is a finite set of points in the source scheme. This means that, locally, a finite morphism behaves like a finite number of algebraic maps, resembling a polynomial function with a finite number of solutions. Finite morphisms play a crucial role in understanding how different schemes relate to each other and allow us to study their properties more deeply.
Intersection Theory: Intersection theory is a branch of algebraic geometry that studies the intersection of subvarieties within a given variety. It provides tools to count and analyze the intersection points of geometric objects, often taking into account their dimensions, multiplicities, and other geometric properties. This theory connects algebraic structures with geometric concepts, allowing for a deeper understanding of the relationships between affine and projective varieties, as well as their dimensions and the modern framework of schemes.
Irreducible Scheme: An irreducible scheme is a type of scheme that cannot be represented as the union of two proper closed subschemes. This concept is crucial in understanding the structure of schemes, particularly in modern algebraic geometry, where it signifies that the underlying topological space is irreducible and the corresponding ring of global sections is an integral domain.
Local Ring: A local ring is a type of ring that has a unique maximal ideal, which means it is focused around a single point or a specific 'local' aspect. This structure allows for the study of properties and behaviors of algebraic objects in a neighborhood, making it essential in various areas like algebraic geometry and commutative algebra.
Projective scheme: A projective scheme is a type of geometric object that is defined as a closed subscheme of projective space, which itself is constructed from the spectrum of a graded ring. These schemes allow for the study of properties of algebraic varieties in a more generalized setting, emphasizing their relationships to projective geometry and enabling the application of tools from modern algebraic geometry.
Reduced Scheme: A reduced scheme is a type of scheme in algebraic geometry where the structure sheaf has no nilpotent elements. This means that the local rings of a reduced scheme do not have any elements that, when raised to some power, become zero. Reduced schemes are significant because they correspond to spaces that do not have 'infinitesimally small' points, making them easier to work with when considering geometric properties.
Regular morphism: A regular morphism is a type of morphism between schemes that locally behaves like a regular function on algebraic varieties. In simpler terms, it means that the morphism can be described by polynomials that satisfy certain smoothness conditions, which allows one to connect the geometric intuition with algebraic structures.
Ringed Space: A ringed space is a topological space equipped with a sheaf of rings, which allows one to study algebraic and geometric properties simultaneously. This structure connects local algebraic data with the global topological features of the space, making it essential in the study of schemes and modern algebraic geometry. By associating a ring to each open set, a ringed space helps in defining functions and allows for a systematic way to handle both local and global properties.
Serre's Criterion: Serre's Criterion is a result in algebraic geometry that provides a way to determine whether a given module over a local ring is Cohen-Macaulay. It connects the concepts of depth, regular sequences, and the properties of rings to the geometric notion of varieties being defined by nice conditions, which can be related to the behavior of schemes in modern algebraic geometry.
Sheaf: A sheaf is a mathematical tool used in algebraic geometry that allows us to systematically organize local data attached to open sets of a topological space. It captures how these local pieces fit together to give a global perspective, making it essential for understanding concepts like functions, sections, and cohomology in modern algebraic geometry.
Variety: In algebraic geometry, a variety is a fundamental geometric object that can be defined as the solution set of one or more polynomial equations over a given field. This concept connects to the study of polynomial rings and ideals, where varieties correspond to the zeros of polynomials, highlighting their geometric significance in higher-dimensional spaces. Varieties can also be connected to singularities and the resolution of these points, offering insight into their structure and behavior in algebraic contexts.
Zariski's Main Theorem: Zariski's Main Theorem states that for an irreducible algebraic variety over an algebraically closed field, the points of the variety correspond bijectively to the prime ideals of its coordinate ring. This deep connection between algebra and geometry reveals how the geometric structure of varieties can be understood through their algebraic properties, linking irreducibility, local rings, and regular functions to broader concepts in algebraic geometry.