By Dan Laksov

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If s and t are sections of F (U ) such that sx = tx for all x ∈ U , then, for each x ∈ U , U there exists a neighbourhood Ux of x contained in U such that ρU Ux (s) = ρUx (t). Since the sets Ux for all x ∈ U cover U it follows from property (F1) for sheaves that s = t. Hence property (i) holds. The images by πU of the sections of F (U ) satisfy property (ii) of assertion (2) since for s ∈ F (U ) we can take Ux = U and s(x) = s for all x ∈ U . Conversely let (sx )x∈U ∈ x∈U Fx satisfy the condition of (ii), We shall show that (sx )x∈U is in the image of πU .

Let K be a field and let A = K n be the product of the field K with itself n times. (1) Describe Spec(A). (2) What is the dimension of Spec(A)? 7. Let A = Zn be the cartesian product of the ring Z with itself n times. (1) Describe Spec(A). (2) What is the dimension of Spec(A)? 8. A topological space X is connected if there is no non-empty subset of X different from X. Let A be a ring and let X = Spec(A). Show that the following assertions are equivalent: (1) The space X = Spec(A) is not connected.

F2) For every collection {sα }α∈I of sections sα ∈ F (Uα ) that satisfy the condition U β α ρU V (sα ) = ρV (sβ ) for all α, β in I, and all V belonging to B with V ⊆ Uα ∩Uβ , there is a section s in F (U ) restricting to Uα for all α ∈ I, that is, ρU Uα (s) = sα for all α ∈ I. A presheaf, or sheaf, that is defined on all the open subsets of X is called a presheaf respectively a sheaf on X. 2) Remark. If follows from property (F1) that the section s of property (F2) i unique. Moreover from the equality ∅ = ∅ ∪ ∅ is follows from property (F2) for sheaves that when not all the F (U ) with U belonging to B are empty we have that F (∅) consists of exactly one element.

Categories: Elementary