Math 654
Homework #1
January 15, 2013
Due Thursday, January 24, in class.
Justify all of your work.
A1. Let R be a commutative ring with 1, and let X be the set of all prime ideals of R. For a
subset E ⊆ R, let
Z(E) := {P ∈ X | E ⊆ P }.
Show the following.
(a) If A is the ideal generated by E, then Z(E) = Z(A).
(b) Z(0) = X and Z(1) = ∅.
(c) If Ei , i ∈ I, is a family of subsets fo R, then
!
[
\
Z
Ei =
Z(Ei ).
i∈I
i∈I
(d) For ideals A, B of R, we have Z(A ∩ B) = Z(AB) = Z(A) ∪ Z(B).
(e) Show that we can define a topology on X such that the subsets {Z(E) | E ⊆ R} are the
closed subsets.
Note: The set X is called the prime spectrum of R, and it is often denoted X = Spec(R).
The topology on X is called the Zariski topology.
A2. Continuing with the notation from A1, let X = Spec(Z).
(a) Show that X = {(p) | p is a positive prime number} ∪ {(0)}.
(b) If p is a positive prime, show that {(p)} = {(p)}, where {(p)} denotes the closure of the
set {(p)} in the Zariski topology on X. (So the point (p) ∈ X is a closed point.)
(c) What is {(0)}?
A3. Let R be the category of commutative rings with 1 whose morphisms are ring homomorphisms
φ : R → S such that φ(1) = 1. Let T be the category of topological spaces whose morphisms
are continuous functions. Define a contravariant functor
F :R→T
such that for any ring R in R we have F (R) = Spec(R) under the Zariski topology.
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