# 632HW3 - -algebra For any line L in K n 1(i.e a direct...

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Math. 632. Homework 3 1, 2. Hartshorne, Ex. 5.10, 5. 11. 3. Let A = R [ T 0 , . . . , T n ] be the polynomial algebra graded by the condition deg( T i ) = q i > 0 and X = Proj A . Show by an example, that O X (1) is not locally trivial in general and prove that O X ( n ) is locally trivial on the open subset equal to the union of D + ( T i )’s such that q i | n . 4. Let i : Y X be the canonical morphism of a closed subscheme Y of X deﬁned by an Ideal I Y in O X . Show that for any quasi-coherent O X -Module F , the inverse image i * ( F ) is isomorphic to the O Y -Module F / I Y F (make sense of this quotient sheaf and its structure of a quasi-coherent sheaf on Y ). 5. Let φ : R [ T 0 , . . . , T n ] A be a surjective homomorphism of graded R -algebras and I = Ker ( φ ). Show that there exists an isomorophism Φ : X = Proj( A ) Y , where Y is a closed subscheme of P n R with ideal sheaf isomorphic to ˜ I . Show that Φ * ( O P n R (1)) = O X (1) (here we identify Φ with its composition with the natural morphism Y P n R ). 6. Let I be a homogeneous ideal in R [ T 0 , . . . , T n ]. Let K be an R
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Unformatted text preview: -algebra. For any line L in K n +1 (i.e. a direct summand of R n +1 of rank 1) make sense of the equality F ( L ) = 0 , where F ∈ I . Show that there is a natural (make sense of this too) bijection between the set of K-points of Proj( R [ T , . . . , T n ] /I ) and the set of lines L in K n +1 such that F ( L ) = 0 , for any F ∈ I . 7. Show that the canonical projection p : P n R → Spec R deﬁnes a homomorphism of groups p * : Pic(Spec R ) → Pic( P n R ) with quotient isomorphic to Z (prove this ﬁrst in the case when R is a ﬁeld to get the idea). 8. Let S be a scheme and X = A n S → S be the aﬃne space over S . Show that the canonical homomorphism Pic S → Pic X is an isomorphism. 9. Let X = Spec( C [ X, Y, Z ]( XY + Z 2 ). Prove that Pic( X ) is a group of order 2. What is the class group of X ? 1...
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## This note was uploaded on 02/24/2012 for the course MATH 632 taught by Professor Igordolgache during the Fall '08 term at University of Michigan-Dearborn.

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