INTRODUCTION TO ALGEBRAIC GEOMETRY-page141

INTRODUCTION TO - 137 Since f x m Y,y ⊂ m X,x we can define a homomorphism O Y,y m Y,y → O X,x m X,x and passing to the fields of quotients we

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Unformatted text preview: 137 Since f * x ( m Y,y ) ⊂ m X,x , we can define a homomorphism O Y,y / m Y,y → O X,x / m X,x and passing to the fields of quotients we obtain an extension of fields k ( x ) /k ( y ) . Also, f * x,y induces a linear map m Y,y / m 2 Y,y → m X,x / m 2 X,x , where the target space is considered as a vector space over the subfield k ( y ) of k ( x ) , or equivalently a linear map of k ( x )-spaces ( m Y,y / m 2 Y,y ) ⊗ k ( y ) k ( x ) → m X,x / m 2 X,x The transpose map defines a linear map of the Zariski tangent spaces df zar x : Θ( X ) x → Θ( Y ) y ⊗ k ( y ) k ( x ) . (14.4) It is called the (Zariski) Zariski differential of f at the point x . Let Y be a closed subset of X and f : Y → X be the inclusion map. Let U ⊂ X be an affine open neighborhood of a point x ∈ X and let φ 1 , . . . , φ r be equations defining Y in U . The natural projection O ( X ∩ U ) → O ( Y ∩ U ) = O ( U ∩ X ) / ( φ 1 , . . . , φ r ) defines a surjective homomorphism O X,x → O Y,x...
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This note was uploaded on 01/08/2012 for the course MATH 299 taught by Professor Wei during the Spring '09 term at SUNY Stony Brook.

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