Mannheimer Manuskripte 177 gk-mp-9403/3 SOME CONCEPTS OF ...

30 MARTIN SCHLICHENMAIERLemma 2. Let X = Spec(R) and {f i } i∈J a set of elements of R then the union ofthe sets X fi equals X if and only if the ideal generated by the f i equals the whole ringR.Proof. The union of the X fi covers Spec(R) iff no prime ideal of R contains all the f i .But every ideal strictly smaller than the whole ring is dominated by a maximal (andhence prime) ideal. The above can only be the case iff the ideal generated by the f i isthe whole ring. □Lemma 3. The affine scheme X = Spec(R) is a quasicompact space. This says everyopen cover of X has a finite subcover.Proof. Let X = ⋃X jj∈Jbe a cover of X. Because the basis open set X f are a basis ofthe topology, every X j can be given as union of X fi . Altogether, we get a refinement ofthe cover X = ⋃ X fi . By Lemma 2 the ideal generated by these f i is the whole ring.i∈IIn particular, 1 is a finite linear combination of the f i . Taking only these f i which occurwith a non-zero coefficient in the linear combination we get (using Lemma 2 again) thatX fik , k = 1,..,r is a finite subcover of X. Taking for every k just one element X jkcontaining X fik we obtain a finite number of sets which is a subcover from the cover westarted with. □Note that this space is not called a compact space because the Hausdorff conditionthat every distinct two points have disjoint open neighbourhoods is obviously notfulfilled.The following proposition says that the sheaf axioms (1) and (2) from App. A for thebasis open sets are fulfilled.Proposition. Let X f be coverd by {X fi } i∈I .(a) Let g,h ∈ R f = O R (X f ) with g = h as elements in R fi = O R (X fi ) for everyi ∈ I, then g = h also in R f .(b) Let g i ∈ R fi be given for all i ∈ I with g i = g j in R fi f j, then there exist ag ∈ R f with g = g i in R fi .Proof. Because X f = Spec(R f ) is again an affine scheme it is enough to show theproposition for R f = R, where R is an arbitrary ring. Let X = ⋃ X fi .(a) Let g,h ∈ R be such that they map to the same element in R fi . This can onlybe the case if in R we havef n ii · (g − h) = 0, ∀i ∈ I,i∈I