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The Etale Topology of Schemes
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abelian affine schemes Applying assertion associated assume called canonical category of schemes closed immersion closed subspace coherent commutative completion condition Consider construction contains defined Definition denote dense descent disjoint union Edited effective etale covering etale equivalence relation etale map etale surjective etale topology example exists extension fact fiber field finite presentation finite type functor geometric give given Hence ideal immersion injective integer isomorphism Lemma Let f:X locally locally of finite map of algebraic modules morphism natural noetherian Note object open immersion open subspace particular projective Proof proper Proposition prove quasicoherent sheaf quasicompact quotient R-module reduced representable restriction ring satisfies separated algebraic space sheaf sheaves Spec stable sufficient Suppose Theorem theory unique universally usual variety write