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w an exists and is equal to 0 (since each an E U, compact). If not, from I rf= U, it follows that u rf= u W (since K is a field). Since u W is closed, there exists a neighborhood Y of u whose closure Y is disjoint from u W. Now, given any point x E W, and any neighborhood Z of ux, such that Z n Y = 0, by continuity, there exists open neighborhoods X(x) of x, T(x) of u such that T(x)· X(x) S; Z, hence T(x) .

In fact, given any real number p > 0 and x E K, let n be an integer such that p + Ixl < lal- n , if Bp(O) = {z E K Ilzl < p}, then x + Bp(O) ~ a-nU. Ifm is an integer such that 11m < p12, then there exists a point X~l~l () . E K such that x E X~l)m . + Vm ; hence 1 X n• m + Vm ~ X + Bp(O). (fJ) Let Wi, W2 , ... , Wnl ... be a countable fundamental sys- U 52 1. Absolute Values of Fields tern of neighborhoods of K. ;(n) - 0 when x¢: W n . E Wn , If x =I y there exists a neighborhood V of x such that x E V, Y ¢ V; hence there exists Wn such that x E Wn S; V, Y ¢ W n.

So there exists ~ > 0 such that if gl, g2 E K[Xln and 1. Absolute Values of Fields 24 Ilgl - 1211 : :; 6, IIg2 - 1211 : :; 6, then IR(gl, g2) - R(/I ,h) I < r /2. Hence R(gl, g2) i= 0, so gl, g2 are also relatively prime. 0 By considering on K[Xln the topology indicated, we may rephrase (a) by saying that the set of polynomials with distinct roots is an open set. Similarly, the set of pairs of relatively prime polynomials is an open subset of K[Xln x K[Xln' In the next two results we consider roots of polynomials, so it is convenient to assume that K is an algebraically closed field.

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An introduction to the derived category by Theo Bühler


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