Homework Assignment #7 - UCB Mathematicsogus/Math_256B-2017/Homework/homew… · Homework...
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Homework Assignment #7: Due April 7 Let θ: R → A be a homomorphism of commutative rings. Let ω A/R := Hom R (A, R), viewed as an A-module via A, and let t A/R : ω A/R → R be the homomorphism sending h to h(1). 1. Show that the pair (t A/R ,ω A/R ) represents the functor taking an A- module M to the R-module Hom R (θ * (M ),R). 2. If A = R[X ]/(f ), where f is a monic polynomial, show that ω A/R is a free A-module of rank one. 3. If A = R[x, y]/(x 2 , xy, y 2 ), show that ω A/R is not free as an A-module. 4. If θ: R → A is finite and flat, then for each a ∈ A, let tr A/R (a) be the trace of the endomorphism of A defined by multiplication by a. Then tr A/R is an R-linear map A → R and hence corresponds to an R-linear map Tr A/R : A → ω A/R . This map is an isomorphism if and only if A/R is smooth (hence ´ etale). Prove this fact by hand if A = R[X ]/(f ) where f is monic of degree 2. 5. Hartshorne III, 7.2 6. Hartshorne III, 7.3 1
Transcript of Homework Assignment #7 - UCB Mathematicsogus/Math_256B-2017/Homework/homew… · Homework...
Homework Assignment #7:
Due April 7
Let θ:R → A be a homomorphism of commutative rings. Let ωA/R :=HomR(A,R), viewed as an A-module via A, and let tA/R:ωA/R → R be thehomomorphism sending h to h(1).
1. Show that the pair (tA/R, ωA/R) represents the functor taking an A-module M to the R-module HomR(θ∗(M), R).
2. If A = R[X]/(f), where f is a monic polynomial, show that ωA/R is afree A-module of rank one.
3. If A = R[x, y]/(x2, xy, y2), show that ωA/R is not free as an A-module.
4. If θ:R → A is finite and flat, then for each a ∈ A, let trA/R(a) bethe trace of the endomorphism of A defined by multiplication by a.Then trA/R is an R-linear map A → R and hence corresponds toan R-linear map TrA/R:A → ωA/R. This map is an isomorphism ifand only if A/R is smooth (hence etale). Prove this fact by hand ifA = R[X]/(f) where f is monic of degree 2.
5. Hartshorne III, 7.2
6. Hartshorne III, 7.3
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