8320notes3 - THE NULLSTELLENSATZ; CLOSED POINTS AND...

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Unformatted text preview: THE NULLSTELLENSATZ; CLOSED POINTS AND k-POINTS PETE L. CLARK 1. The Nullstellensatz Our treatment of affine varieties up to this point has been nonstandard in that we have refused to associate to an affine variety a set of points. Recall that the standard definition of an affine variety (or rather, an affine subvariety of A n ) over an algebraically closed field k is as a subset of k n which is defined as the set Z ( S ) of simultaneous zeros of a family S = { P i } R n := k [ t 1 , . . . , t n ] of polynomials. If I ( S ) is the ideal generated by S , one sees immediately that Z ( S ) = Z ( I ( S )). Moreover, by the Hilbert Basis Theorem, every ideal of R n is finitely generated, so even though a priori we allow intersections of possibly infinite sets of polynomials, any such set can also be cut out by a finite set of polynomials. We can also go in the other direction: given an ideal I in R n , we can consider the set V ( I ) of all points of x A n such that f I = f ( x ) = 0. For instance, associated to the ideal ( xy ) C [ x, y ], we get the set of all points for which x = 0 or y = 0, i.e., two lines meeting at the origin. Thus we have two sets: the set of all subsets of A n , and the set of all ideals of R n , and we have two mappings between them, V and I . The following formal properties satisfied by V and I are immediate: (GC1) S 1 S 2 = I ( S 2 ) I ( S 1 ) , I 1 I 2 = V ( I 2 ) V ( I 1 ). That is, both maps are order reversing (or antitone). (GC2) For S A n and I an ideal of R n we have: S V ( I ) I I ( S ) . Indeed, both conditions say: f I, P S, f ( P ) = 0. In general, if we have two partially ordered sets S and T and maps f : S T , g : T S satisfying (GC1) and (GC2) (with replaced by the in the posets), we say that ( S, T, f, g ) form a Galois connection . 1 Notice that the concept of an antitone Galois connection is inherently symmet- rical in the sense that if ( S, T, f, g ) is a Galois connection, so too is ( T, S, g, f ). (An isotone Galois connection breaks this symmetry, and represents the fundamental asymmetry between left and right adjoint functors.) 1 More precisely, an antitone Galois connection. When f and g are both order-preserving, an appropriate modification of (GC2) leads to the notion of an isotone Galois connection. 1 2 PETE L. CLARK Proposition 1. Let ( S, T, f, g ) be a Galois connection on partially ordered sets. a) For s S and t T , f ( g ( f ( s )) = s , g ( f ( g ( t )) = t . b) The operators c S : S S , c T : T T by c S : s 7 f ( g ( s )) , c T : t 7 g ( f ( s )) are both closure operators in the sense of order theory....
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8320notes3 - THE NULLSTELLENSATZ; CLOSED POINTS AND...

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