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**Extra info for Algebraic Topology, Aarhus 1978**

**Example text**

This is a purely algebraic matter: we can write B = AI I, and ideals in B are in 1-1 correspondence with ideals in A which contain I. So if A has a unique maximal ideal, so has AI I, and this 0 is preserved by the projection A --+ AI I. We shall see shortly that it is not true that every finitely generated local COO-ring is a quotient of a ring of the form C;o(JRn). Note that inverse limits of local COO-rings need not be local. It is easy commutative algebra to check that directed colimits of local COO -rings are again local.

IIA x B COO(N x N)j(I(x), I(y» COO(N x N)jJ, = OJ. Now choose 0 < e < 1 and u = {(x, Sf) E Ji2 1x > Sf- e} V = {(X,Sf) E Ji2 1x < Sf + e}, and choose characteristic functions u, v: Ji2 - [0,1] for U and V, respectively. Then u + v is invertible in Coo (N x N) j J, but neither u nor v is. For example, if u· h = 1 mod J for some h, then u would have to be invertible on a set of the form A x B with A, B E " which is clearly not the case. 3 Proposition. C;o(Ji n) ®oo C~(IRm} ~ C~,q)(IRn+m}, lor etJery two points p E Ji n , q E IRm.

By hypothesis there exists a map A ~ B into a pointed local COO-ring B such that cp(J) 1= o. e. 10 B ~ Cr(JRR)/J for some ideal J. 9 we can show that if COO(M)/I ~ Cr(JRR)/J evo, JR corresponds to the point p E Z(I) ~ M, then ~ can be factored as / C~(M)/II, So if ~(J) 1= 0, II" 1= o. (¢:) As in (a), this is easy: the map which takes the germ at each point, A -+ is again injective. : Chapter I. Coo -rungs 46 (b) and (c) are proved similarly. [[Xl"" ,Xmlll J with tp(f) :f:. O. •. , XnllITz(I)_ lW.