## Introduction to Algebraic Geometry and Algebraic GroupsIntroduction to Algebraic Geometry and Algebraic Groups |

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accordingly affine open subschemes affine scheme assertion assigns automorphism bialgebra bijection canonical isomorphism canonical map canonical projection closed embedding closed subscheme closed subset Comm commutative components composition Corollary defined definition denotes diagram direct limits element equivalent etale exact sequence faithfully flat field finitely presented follows functor gé G(R geometric space hence homomorphism identified inclusion morphism inverse limit irreducible isomorphism k-algebra k-functor k-G-module k-group-functor k-group-scheme k-module k-monoid k-monoid-functor Kdim kernel lemma Let f Let G Lie algebra Lie G Lie H linear representation local ring monoid monomorphism morphism f morphism of schemes noetherian notation open subfunctor open subset phism prime ideal Proof prop Proposition quasicoherent quasicoherent sheaf quasicompact quasiseparated resp ring satisfies Section sheaf of modules sheaves Spec structure subfunctor subgroup of G subspace Suppose surjective theorem write