Algebraic Groups.Affine Group Schemes

Definition Additive Group Scheme 𝔾_a admitted

Topic Affine Algebraic Groups admitted

Definition Affine Group Scheme over a Base admitted

Theorem Algebra Homomorphisms Form a Group Under Convolution admitted

Definition Algebraic Group over a Base admitted

Theorem Antipode of a Commutative Hopf Algebra Is Multiplicative admitted

Theorem Antipode of a Hopf Algebra Is Anti-Multiplicative admitted

Topic Base Change admitted

Theorem Base Change Preserves Algebraic Group Structure admitted

Theorem Characters Are Group-Like Elements admitted

Definition Comodule over a Coalgebra admitted

Theorem Composition with the Antipode Is an Algebra Hom for Commutative Hopf Algebras admitted

Theorem Convolution Is Pointwise Multiplication admitted

Definition Convolution of Algebra Homomorphisms admitted

Theorem Convolution of Algebra Homomorphisms Is an Algebra Homomorphism admitted

Definition Coordinate Hopf Algebra of an Affine Group Scheme admitted

Topic Coordinate Rings and Hopf Algebras admitted

Theorem Essential Image of Spec Is Affine Group Schemes admitted

Definition Faithful Linear Representation admitted

Lemma Finite-Dimensional Subspace in a Sub-Comodule admitted

Definition Functor of Points of a Group Scheme admitted

Theorem G-Representations Are A-Comodules admitted

Definition General Linear Group Scheme GL_n admitted

Theorem Group Object Structure on Spec A via Yoneda admitted

Theorem Hom Into Spec via Global Sections (Spec ⊣ Γ Adjunction) admitted

Definition Homomorphism of Affine Group Schemes admitted

Lemma Multiplication of a Commutative Bialgebra Is a Coalgebra Hom admitted

Definition Multiplicative Group Scheme 𝔾_m admitted

Theorem Points of GL_n Are Invertible n×n Matrices over R admitted

Theorem Points of 𝔾_a Are the Additive Group of R admitted

Theorem Points of 𝔾_m Are the Units of R admitted

Theorem R-Points of a Group Scheme Form a Group admitted

Example Regular Representation as a Comodule admitted

Definition Roots of Unity Group Scheme μ_n admitted

Definition Spec Functor on Hopf Algebras admitted

Theorem Spec Functor on Hopf Algebras Is Fully Faithful admitted

Definition Special Linear Group Scheme SL_n admitted

Definition Special Orthogonal Group Scheme SO_n admitted

Definition Symplectic Group Scheme Sp_{2n} admitted

Definition The Diagonalizable Group Scheme D(M) admitted

Definition Trivial Group Scheme over a Base admitted

Example Trivial Representation as a Comodule admitted