Group schemes
Seven lessons on group schemes and Hopf algebras, Lie algebras and smoothness, field groups, quotients and torsors, diagonalizable groups and tori, abelian varieties and Néron models. Includes complete arguments, worked examples, exercises and solutions.
- Group schemes, actions and Hopf algebras
- Lie algebras and smoothness of group schemes
- Group schemes over a field
- Quotients and torsors
- Diagonalizable groups and groups of multiplicative type
- Abelian varieties
- Néron models
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Prerequisite readings
- Infinitesimal lifting and the invariance of étale morphisms under thickenings
- Quasi-finite morphisms and Chevalley’s theorem
- Regular sequences, depth and Cohen–Macaulay modules
- Smooth morphisms
- Flatness criteria, dimension and the flat locus
- Étale morphisms and their local structure
- Completion
- The bootstrap theorem
- Krull dimension and Noether normalization
- Noetherian and Artinian rings
- Projective dimension and the Auslander–Buchsbaum formula
- Discrete valuation rings, normal rings and Serre's criterion
- Tori, maximal tori and their conjugacy
- Regular elements and centralizers
- Roots and reductive groups of rank one
- Pinnings and the classification of split reductive groups
- Complete reducibility: Casimir elements and Weyl's theorem
- Étale neighbourhoods, henselization and quasi-finite morphisms
- Affine morphisms, relative Spec, and finite morphisms
- Limits of schemes and Noetherian approximation
- Finiteness of morphisms
- Zariski's Main Theorem
- Dimension theory of Noetherian local rings
- Associated primes and primary decomposition
- Resolutions, Tor and Ext
- Coefficient rings and the Cohen structure theorem
- Algebraic spaces
- Integral extensions: lying over, going up and going down
- The Nullstellensatz and Jacobson rings
- Spectra of rings
- Localization, local properties and support
- Tor and flat modules
- Regular local rings
- Root data, Weyl chambers and the Bruhat decomposition
- The Killing form and Cartan's criteria
- Lie algebras: definitions, examples and first constructions
- The diagonal and separated morphisms
- Dimension of fibres
- Graded modules and Hilbert–Samuel functions
- Formally smooth, unramified and étale ring maps
- Spectral spaces and affine realization
- Nilpotent and solvable Lie algebras: Engel's and Lie's theorems
- Kähler differentials
- Proper morphisms and the valuative criterion of properness
- Valuation rings and the valuative criterion of separatedness
- Projective morphisms and Chow’s lemma
- Very ample invertible sheaves, Segre and Veronese embeddings
- Ample invertible sheaves
- Faithful flatness and the local criterion for flatness
- Cohomology of projective space
- The Picard functor and the Picard scheme of a curve
- The structure of the Picard scheme
- Coherent sheaves on projective schemes: Serre's theorems
- Euler characteristics and Hilbert polynomials
- Coherence of higher direct images under proper morphisms
- Base change and the Grothendieck complex
- Semicontinuity and Grauert's theorem
- Ext sheaves and Serre duality on projective space
- Dualizing sheaves and Serre duality for projective schemes
- Smooth algebras over a field and the Jacobian criterion
- Cohomology of affine schemes and Serre's criterion
- Relative divisors and the existence of the Picard scheme
- Čech cohomology
- Cohomology of sheaves on ringed spaces
- Cauchy's theorem for cycles and its consequences
- Faithfully flat descent
- Hahn–Banach, Baire and the basic theorems on Banach spaces
- Fibred categories and descent data
- Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian
- Hilbert and Quot schemes
- Complex analytic spaces and analytification
- Serre's comparison theorems and Chow's theorem
- The theorem on formal functions
- Local tools for bundles and transport
- Theorems A and B on Stein manifolds
- Finiteness on compact complex spaces
- Coherent sheaves and Oka's coherence theorem
- Analytic germs, local parametrization and the Nullstellensatz
- Cartan's coherence theorem and complex spaces
- Laurent series and homogeneous projections
- Holomorphic functions of several variables
- Cauchy's theorem for cycles and its consequences
- Grothendieck's existence theorem
- Coherent sheaves on projective schemes: Serre's theorems
- Fréchet spaces of sections and Schwartz's theorem
- The local ring of holomorphic germs
- Cohomology of affine schemes and Serre's criterion
- Base change and the Grothendieck complex
- Plurisubharmonic functions and Stein manifolds
- Stein domains in complex space
- Cohomology of sheaves on ringed spaces
- Spectra of rings
- Integral extensions: lying over, going up and going down
- Hahn–Banach, Baire and the basic theorems on Banach spaces
- The complex exponential and the circle
- Real analysis on closed intervals
- Dimension theory of Noetherian local rings
- Čech cohomology
- The Dolbeault complex
- Hörmander's L² estimates on pseudoconvex domains
- Localization, local properties and support
- Noetherian and Artinian rings
- Spectra of rings
- Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian
- Constructing the real numbers
- Graded modules and Hilbert–Samuel functions
- Krull dimension and Noether normalization
- Hilbert spaces and compact operators
- Integral extensions: lying over, going up and going down
- The Nullstellensatz and Jacobson rings
- Associated primes and primary decomposition
- Measure and Hilbert space tools for Haar integration
- Sections 1–4, 8–9
- Tor and flat modules
- Formally smooth, unramified and étale ring maps
- Faithful flatness and the local criterion for flatness
- Resolutions, Tor and Ext
- Kähler differentials
- Projective dimension and the Auslander–Buchsbaum formula
- Regular local rings
- Étale morphisms and their local structure
- Zariski's Main Theorem
- Regular sequences, depth and Cohen–Macaulay modules
- Dimension theory of Noetherian local rings
- Smooth algebras over a field and the Jacobian criterion
- Quasi-finite morphisms and Chevalley’s theorem
- Affine morphisms, relative Spec, and finite morphisms
- Limits of schemes and Noetherian approximation
- Finiteness of morphisms
- The diagonal and separated morphisms
- Normalization