6.2 Spectral Sites  Suppose X is a topological space. If Y is a subset of X, we denote by Y - the closure of Y. A generic point for an irreducible closed subset Z is a point z such that Z = {z}-. A topological space is sober if every closed irreducible set has a unique generic point.  If x, x' are two points of X and x' Î {x}-, then we say that x' is a specialization of x, and x is a generalization of x', written x' £ x. The underlying set of X is a partial ordered set (i.e., a poset) with the partial order £. Any continuous map of topological spaces preserves the partial order. A point x is called a maximal (resp. minimal) point if it is a maximal (resp. minimal) element in the poset X.  A subset Y of a space X is called stable (under specialization) if y Î Y and x £ y implies x Î Y. Any closed set is a stable set. For any subset Y we let Y_ = {x Î X | x £ y for some y Î Y} be the stable set generated by Y. Clearly we have Y_ Í Y -. We say Y is preclosed if Y_ = Y -.  Definition 6.2.1. A continuous map f: Y ® X of topological spaces is called stable (resp. preclosed) if for any closed subset V of Y, f(V) is stable (resp. preclosed).  Remark 6.2.2. A subset Y of X is closed if and only if Y is both stable and preclosed. A continuous map f: Y ® X is a closed map if and only if f is both stable and preclosed.  Remark 6.2.3. (a) Any finite union of preclosed subsets of X is preclosed.  (b) A subset Y of X is preclosed if and only if for any open subspace U of X, Y Ç U is preclosed in U.  (c) A subset Y of X is preclosed if there is an open cover {Ui} of X such that Ui Ç Y is preclosed in Ui for each i.  (d) A continuous map f: Y ® X is preclosed if and only if f sends any preclosed subset of Y to a preclosed subset of X.  Remark 6.2.4. Suppose f: X ® Y is a continuous map of topological spaces. Consider the following conditions on f:  (a) f is a closed map.  (b) For any x Î X and any y' £ f(x), there is a x' £ x such that f(x') = y'.  (c) f is stable.  Then clearly (a) implies (b) and (b) implies (c).  Definition 6.2.5. An affine spectral site is an affine site B having the following properties:  (a) The space of any object is sober.  (b) For any point x of an object, the intersection {x}o of open neighborhoods of x is effective, denoted xo.  (c) Any morphism is preclosed.  A spectral site is an algebraic site C with an affine basis B which is an affine spectral site. (Note that a spectral site also satisfies (a) and (b)).  Example 6.2.6. We show that ASch and GASch are affine spectral sites.  (a) Any irreducible closed subset of an affine scheme Spec A is the closure V(p) of a unique prime ideal p of A, which is the unique generic point of V(p). Thus Spec A is sober.  (b) For any prime ideal p Î Spec A, {p}o is effective because it is the image of the effective morphism Spec Ap ® Spec A induced by the localization map A ® Ap.  (c) Consider a morphism f: Spec A ® Spec B for two rings A and B and V a closed subset of Spec A. We have to prove that f(V) is preclosed. By switching to the reduced closed subschemes of Spec A and Spec B determined by V and the closure of f(V), it suffices to prove that Spec B is a preclosed set under the assumption that Spec A and Spec B are reduced and f is dominated, i.e., the corresponding ring homomorphism B ® A is injective. We claim that Spec B is the stable set generated by f(V). This follows from the following two facts  (1) Any prime ideal of B contains a minimal prime ideal.  (2) Any minimal prime ideal p of B is the pull back of a prime ideal q of A. (Since B is reduced, Bp is a field and Bp Í A Ä Bp, we can take q to be the pull back of any prime ideal of A Ä Bp along the localization map A ® A Ä Bp.)  Example 6.2.7. Sch and GSch are spectral sites.  Proposition 6.2.8. Let f: Y ® X be a quasi-compact morphism in a spectral site. Then   (a) f is preclosed.   (b) f is a closed morphism if and only if f is stable.  Proof. The assertions (b) follows from (a) by (6.2.2). To prove (a) it suffices to prove that the intersection U Ç f(X) for any affine open subset U is a preclosed subset in U (6.2.3.b). Since f is quasi-compact, f-1(U) is a finite union of affine open subsets Vi. Each f(Vi) Í U is preclosed since Vi ® U is a morphism of affine objects (6.2.5c). Thus the finite union U Ç f(X) = È f(Vi) is a preclosed subset of U (6.2.3.a).  An object X in a spectral site C is called a locality if X has exactly one closed point TX and any other point is a generalization of TX. A morphism f: X ® Y of localities is a morphism f from X to Y such that f(TX) = TY.  Example 6.2.9. For any point x of an object X, the effective subobject xo of X is a locality. If f: X ® Y is a morphism, then f induces a morphism of localities xo ® f(x)o.  Note that a locality must be affine because X is covered by affine open subsets and the only open neighborhood of TX is X.  A morphism f: X ® Y of irreducible objects in a spectral site is called birational if f sends the generic point KX of X to the generic point KY of Y, and the induced morphism KX ® KY of spots is an isomorphism. (Note that in a spectral site a generic point x of an irreducible object is effective as we have {x} = {x}o.)       [Next Section][Content][References][Notations][Home]