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Unformatted text preview: THE NULLSTELLENSATZ; CLOSED POINTS AND kPOINTS 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 orderpreserving, 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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 Fall '10
 Clark
 Algebra, Algebraic geometry, maximal ideals, Nullstellensatz

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