By Dan Laksov

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We denote by Fp the sheaf on X such that Fp (Un ) = Fp,n for all n ∈ N and with Um restricions (ρFp )U for all m, n in N with m ≤ n. n Let F be the simple sheaf with stalk Z. (a) For all p ∈ N we have a map sp : F → Fp of sheaves given by (up )Un : F (Un ) → Fp (Un ) which is the identity on Z when n ≤ p and otherwise zero. Show that this defines a map of sheaves F→ such that F (Un ) → ( (b) Show that the map p∈N p∈N Fp Fp )(Un ) is injective for all n ∈ N. Fx → ( p∈N F p )x 52 Sheaves is injective for all x ∈ X.

Nβ }β∈J of positive integers nβ 40 Rings there is a positive integer n and a family {gβ }β∈J of elements gβ of A such that n fn = fβ β gβ . β∈J (3) The open subset D(f ) of Spec(A) is compact. → → → → Proof. (1) We let a = α∈I Afα be the ideal generated by the elements fα for all α ∈ I. We have that D(f ) ⊆ ∪α∈I D(fα ) if and only if every prime ideal p that does not contain f does not contain fα for some α ∈ I. That is, if and only if every prime ideal that contains the elements fα for all α ∈ I also contains f .

Then the bijection is an isomorphism of topological spaces because D(ϕA/a (f )) in Spec(A/a) corresponds to D(f ) in Spec(A) for all f ∈ A. 10) Maps. Let ϕ : A → B be a homomorphism of rings. For every prime ideal q in B we have that ϕ−1 (q) is a prime ideal in A. a ϕ : Spec(B) → Spec(A). 11) Proposition. Let ϕ : A → B be a homomorphism of rings. For each ideal a of A we have that a ϕ−1 (V (a)) = V (ϕ(a)). In particular a ϕ−1 (D(f )) = D(ϕ(f )) for all f ∈ A, and a ϕ is a continuous map of topological spaces.