1.2 Basic algebraic structures and geometric concepts

Algebraic geometry blends algebra and geometry, using rings, fields, and ideals to study geometric shapes. It's like using math tools to build and analyze intricate structures, giving us a new way to see shapes and spaces.

In this chapter, we'll explore how algebraic concepts connect to geometric objects. We'll see how polynomial equations define curves and surfaces, and how algebraic tools help us understand their properties and relationships.

Algebraic Structures in Geometry

Rings and Fields

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• Rings are algebraic structures with addition and multiplication operations that satisfy certain axioms, such as associativity, commutativity, and distributivity
  • Examples include the integers, polynomial rings, and coordinate rings of algebraic varieties
  • The axioms ensure that arithmetic operations behave consistently and predictably within the structure
• Fields are rings in which every nonzero element has a multiplicative inverse
  • Examples include the rational numbers, real numbers, and complex numbers
  • Fields are used as the base field for defining algebraic varieties, providing a foundation for geometric constructions

Ideals and Modules

• Ideals are subsets of rings that are closed under addition and absorption (multiplication by ring elements)
  • They play a crucial role in defining algebraic varieties and studying their properties
  • Principal ideals are generated by a single element, while prime and maximal ideals have special properties related to factorization and quotient rings
• Modules are generalizations of vector spaces, where the scalars come from a ring instead of a field
  • They are used to study the properties of algebraic varieties and their sheaves
  • Important examples include free modules, which are direct sums of copies of the ring, and torsion modules, which contain elements annihilated by non-zero ring elements

Geometric Concepts and Their Algebraic Counterparts

Varieties and Morphisms

• Affine varieties are geometric objects defined by the vanishing of a set of polynomials in an affine space
  • They correspond to ideals in polynomial rings, with the ideal consisting of all polynomials that vanish on the variety
  • Examples include curves, surfaces, and hypersurfaces defined by polynomial equations
• Projective varieties are geometric objects defined by the vanishing of a set of homogeneous polynomials in a projective space
  • They are the projective counterparts of affine varieties and are used to study the global properties of algebraic varieties
  • Projective varieties are defined by homogeneous ideals and have a well-defined dimension and degree
• Morphisms between algebraic varieties are maps that preserve the algebraic structure
  • They correspond to ring homomorphisms between the coordinate rings of the varieties
  • Examples include regular maps, which are defined by polynomials, and rational maps, which are defined by ratios of polynomials

Sheaves and Cohomology

• Sheaves are geometric objects that assign algebraic data (e.g., rings or modules) to open sets of a topological space, such as an algebraic variety
  • They are used to study the local properties of algebraic varieties and to define cohomology theories
  • Important examples include the structure sheaf, which assigns the local ring to each point, and the sheaf of regular functions, which assigns the ring of regular functions to each open set
• Cohomology theories, such as sheaf cohomology and de Rham cohomology, are algebraic tools for studying the global properties of algebraic varieties
  • They assign algebraic invariants (e.g., vector spaces or modules) to algebraic varieties that capture their topological and geometric features, such as their connectivity and the existence of certain types of differential forms
  • Cohomology groups measure the obstructions to solving certain algebraic or geometric problems, such as finding global sections of a sheaf or constructing meromorphic functions with prescribed poles and zeros

Connecting Algebra and Geometry

Nullstellensatz and Coordinate Rings

• The Nullstellensatz is a fundamental theorem that establishes a correspondence between ideals in polynomial rings and algebraic subsets of affine space
  • It states that every maximal ideal in a polynomial ring over an algebraically closed field is the ideal of a point in the corresponding affine space
  • The Nullstellensatz allows us to study algebraic varieties using the tools of commutative algebra
• The coordinate ring of an affine variety is the quotient of a polynomial ring by the ideal defining the variety
  • It encodes the algebraic properties of the variety, such as its dimension and singularities
  • The coordinate ring is a finitely generated k-algebra, where k is the base field, and its prime ideals correspond to subvarieties of the original variety

Function Fields and Zariski Topology

• The function field of an algebraic variety is the field of rational functions on the variety
  • It is the field of fractions of the coordinate ring and encodes the birational geometry of the variety
  • Two varieties are birationally equivalent if their function fields are isomorphic, which means they have the same rational curves and rational maps
• The Zariski topology is a topology on algebraic varieties defined by taking the closed sets to be the algebraic subsets (i.e., subsets defined by polynomial equations)
  • It is used to study the global properties of algebraic varieties, such as their irreducible components and dimension
  • The Zariski topology is coarser than the classical topology, meaning it has fewer open sets, but it is better suited for studying algebraic properties

Solving Geometric Problems with Algebra

Gröbner Bases and Resolutions

• Gröbner bases are special generating sets of ideals in polynomial rings that have useful computational properties
  • They are used to solve systems of polynomial equations, compute the dimension and degree of algebraic varieties, and study their singularities
  • Gröbner bases can be computed using algorithms such as Buchberger's algorithm and Faugère's F5 algorithm
• Resolutions of singularities are methods for transforming singular algebraic varieties into smooth ones by a sequence of blowups
  • They are used to study the local properties of singularities and to compute invariants such as the genus and the canonical class
  • Examples of resolution techniques include embedded resolution, which resolves singularities of subvarieties, and the minimal model program, which aims to find the simplest birational model of a variety

Intersection Theory and Cohomology

• Intersection theory is the study of how algebraic varieties intersect each other
  • It involves computing intersection numbers, which are numerical invariants that measure the multiplicity of the intersection
  • Intersection theory is used to study the geometry of algebraic curves and surfaces and to compute their invariants, such as the degree and the genus
• Cohomology theories, such as sheaf cohomology and de Rham cohomology, are algebraic tools for studying the global properties of algebraic varieties
  • They assign algebraic invariants (e.g., vector spaces or modules) to algebraic varieties that capture their topological and geometric features, such as their connectivity and the existence of certain types of differential forms
  • Examples of cohomology theories include Čech cohomology, which is defined using open covers, and étale cohomology, which is defined using algebraic analogues of topological coverings