Kodaira Vanishing Theorem
The Kodaira vanishing theorem and its applications
Recent posts
Posts sorted by last_modified_at (most recently updated first).
Linear Systems
Complete linear systems, base loci, and ampleness
Divisors
Weil divisors, Cartier divisors, and divisor class groups
Grassmannians
Grassmannians as parameter spaces of linear subspaces
Dimension
Dimension of algebraic varieties
Tangent Spaces and Smoothness
Tangent spaces and smoothness of algebraic varieties
Quasi-projective Varieties
Quasi-projective varieties and regular maps
Rational Maps
Rational maps and birational equivalence
Homology
Definition and Properties of Simplices
Projective Varieties
Projective varieties and homogeneous coordinates
Abelian Groups
Free abelian group, tensor product
Algebraic Structures
Binary operations defined on sets
Direct Products of Groups
Direct product of groups
Field of Fractions
Localization, ring of fraction, prime ideal
Free Products
Free product and universal property
Grothendieck Groups
The Grothendieck group and the definition of integers
Group Actions
Group action
Group Homomorphisms
Definition and properties of group homomorphisms, kernels and images
Semigroups, Monoids, Groups
Definitions of semigroups, monoids, and groups
Isomorphism Theorems
The isomorphism theorems
Modules
Definition of module
Quotient Groups
Normal subgroups and quotient groups
Quotient Rings and Ring Homomorphisms
Quotient ring and ring isomorphism theorems
Restricted Sums
Restricted sum of groups
Definition of Rings
Definition and basic properties of rings
Directed Sets
Directed sets and lattices
Examples of Equivalence Relations
Examples of equivalence relations, saturation of equivalence relations, isomorphism theorems
Filters, Ideals, and Galois Connections
Filter and ideal
Limits
Inverse limit and direct limit
Monotone Functions
Operations on ordered sets and monotone functions
Natural Numbers and Infinite Sets
Definition of natural numbers and properties of infinite sets
Operations on Cardinals
Operations on cardinal numbers
Operations on Functions
Inverse and composition of functions, surjective and injective functions
Order Relations Between Ordinals
Order relations between ordinals and the rigorous definition of cardinals
Properties of Products
Partial products, associativity and distributivity
Axiom of Choice
The axiom of choice and its equivalents
Binary Relations
Definition of Binary Relation