Alphabeta MathSmooth-Projective Serre Duality and Flag-Variety Line Bundles
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Finite Fields and Cyclotomic Extensions
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Dimension Constructible Images and Dimensions of Fibres
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Commutative Algebra
- Algebraic Zariski Main for Quasi-Finite Morphisms
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Homogeneous Resultants and Projective Intersection Length
- Integral Extensions and Going Up
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Differential Geometry
- Distributions Integral Manifolds and the Frobenius Theorem
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Vector Fields Flows and Lie Derivatives
Functional Analysis
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Grothendieck Spectral Sequences and Computations
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Lie Theory
- Cartan Subalgebras and Root Space Decompositions
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Highest Weight Theory for Complex Semisimple Lie Algebras
- Lie Algebra Representations, Enveloping Algebras, and PBW
- Lie Groups, Invariant Fields, and the Exponential Map
- Lie Subgroups, Actions, and Homogeneous Spaces
- Root Systems, Dynkin Diagrams, and the Cartan-Killing Classification
- Semisimple Lie Algebras, Cohomology, and Levi Theory
- Solvable and Nilpotent Lie Algebras
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Applications of the Fundamental Group
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Definition
Dualizing line bundle and trace datum of a smooth projective variety
Lemma
Residue pairing between degree-zero and top cohomology of projective space
Theorem
Serre duality for twisting sheaves on projective space
Lemma
Finite twisted locally free resolutions on projective space
Lemma
Injective modules are flasque and Ext from the structure sheaf is cohomology
Definition
Sheaf Ext of coherent modules
Lemma
Long exact global sheaf Ext sequence in the first variable
Theorem
Serre duality for coherent sheaves on projective space
Lemma
Smooth closed immersion is regular with exact conormal sequence
Lemma
Adjunction for a smooth closed subvariety
Lemma
Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension
Lemma
Local-to-global Ext collapse for a regular immersion
Lemma
Rational-point Koszul residue normalization for a smooth projective embedding
Lemma
Embedding compatibility of smooth-projective Gysin traces
Theorem
Serre duality for locally free sheaves on a smooth projective variety
Definition
Complex semisimple algebraic group, Borel, and flag variety
Lemma
A finite-type affine algebraic group has a faithful rational representation
Lemma
Algebraic root subgroups from root exponentials
Lemma
Rank-one SL2 homomorphism and Weyl representative
Lemma
Borel, opposite unipotent groups and root coordinates
Lemma
The opposite-root big cell is an open chart
Lemma
Bruhat double cosets from rank-one multiplication
Lemma
Minimal parabolic from one negative simple root
Lemma
Fixed point for the specified Borel on a projective variety
Lemma
Rational highest-weight modules from adjoint Plücker vectors
Lemma
Projective orbit constructions for G/B and G/Pₐlpha
Lemma
Zariski sections of Borel and minimal-parabolic orbit maps
Theorem
A semisimple flag variety is smooth and projective
Theorem
Bruhat cells of the flag variety
Definition
The equivariant line bundle associated to a Borel character
Theorem
Borel characters classify equivariant flag line bundles
Lemma
Canonical weight of a flag variety
Theorem
A minimal-parabolic flag projection is a projective-line bundle
Lemma
Flag line-bundle degree on a minimal-parabolic fiber
Lemma
Relative canonical weight for a minimal-parabolic flag projection
Lemma
Local normal form for a line bundle on a projective-line bundle
Theorem
Leray spectral sequence for sheaf cohomology
Lemma
Relative projective-line cohomology and apolarity
Theorem
Relative projective-line cohomology shift
Sources
- The Stacks Project, Duality for Schemes
- Ravi Vakil, Foundations of Algebraic Geometry, Classes 53-54
- The Stacks Project, Cohomology of Schemes
- R. Hartshorne, Algebraic Geometry
- Ravi Vakil, Foundations of Algebraic Geometry Classes 53-54
- The Stacks Project, Sheaves of Modules
- The Stacks Project, Cohomology of Sheaves
- Stacks Project, Duality for Schemes
- J. Schreyer, Sheaves on Schemes (WS 2018/19), Week 9
- The Stacks Project, Algebra
- Ravi Vakil, Foundations of Algebraic Geometry Classes 53–54
- J. S. Milne, Algebraic Groups (2022)
- Brian Conrad, Reductive Group Schemes
- J. S. Milne, Algebraic Groups
- Michel Brion, Lectures on the Geometry of Flag Varieties
- Jacob Lurie, A Proof of the Borel-Weil-Bott Theorem
- Jacob Lurie, A Proof of the Borel–Weil–Bott Theorem
- The Stacks Project, Injectives