Continuous valuations
Continuous valuations
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持续估值
DOI：
10.1007/bf02571668
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发表时间：
1993
影响因子：
0.8
通讯作者：
R. Huber
中科院分区：
文献类型：
--
作者：
Mathematische Zeitschrift;Springer;R. Huber
本文研究了一类拓扑环A的连续赋值的等价类构成的拓扑空间ContA，定义如下：设v：A-+Fw{0}是A的赋值，其中F是由im(V)\{0}生成的有序乘法群。在FW[0}上，我们引入了这样的拓扑：U_~F~{0}是开的当且仅当0~Q~U或{xeFlx<；7}C_U对于某个yF。如果映射v：A-~fw{0}关于A上的环拓扑和刚刚定义的fw{0}上的拓扑是连续的，我们称v为连续的。两个连续赋值v：A~Fw{0}和w：A~Au{0}称为等价的，如果存在序么半群的同构f：r~{0i~A~{0}，使得w=fov.则Cont A是A的连续赋值的所有等价类的集合，配以由集合{veCont A[v(A)<；v(B)+O}(a，be A)]生成的拓扑。我们对拓扑空间Cont A的研究受到以下结果的启发。设A是完备的非阿基米德赋值域[2，4，13]上的Tate代数。我们可以联系到A的一个拓扑空间X A，它是唯一确定的，直到同胚。也就是说，设~是伴随A的刚性解析簇Sp A的Grothendieck拓扑[4，III.2.1]。由此可以看出，JA的Topos SHV(JA)是空间的，即存在一个清醒的拓扑空间XA，使得SHV(.~)等价于XA的Topos SHV(XA)。由[6，IV.4.2.4]XA唯一地确定到同胚。本文证明了XA与拓扑子空间Spa(A，A~={v e Cont A i v(A)<；1)是同胚的，对Cont A的每个a e A~}(A~表示A的幂有界元的集合)。我们甚至将证明SHV(，~A)正则等价于Shy(Spa(A，A~))。我们已经看到，对于Tate代数A，拓扑空间ContA非常自然地出现在刚性解析几何中，人们可以要求ContA在更一般的拓扑环A上的应用。在本文中，我们将自己限制在一类拓扑环上，我称之为f-ady环：一个拓扑环是f-addy环当…
0 Introduction In this paper we study, for a certain type of topological rings A, the topological space Cont A of all equivalence classes of continuous valuations of A. The space ContA is defined as follows. Let v: A-+Fw{0} be a valuation of A, where F is an ordered multiplicative group generated by im(v)\{0}. On Fw [0} we introduce the topology such that U _~ F ~ {0} is open iff 0 q~ U or {xeFlx < 7} c_ U for some yeF. We call v continuous if the mapping v: A-~Fw{0} is continuous with respect to the ring topology on A and the topology on Fw {0} just defined. Two continuous valuations v: A ~Fw {0} and w: A ~ A u {0} are called equivalent if there exists an isomorphism f: r~{0I~A~{0} of ordered monoids such that w=fov. Then Cont A is the set of all equivalence classes of continuous valuations of A equipped with the topology generated by the sets {veCont A [ v(a) < v(b) + O} (a, be A). Our study of the topological spaces Cont A is motivated by the following result. Let A be a Tate algebra over a complete, non-archimedean, valued field [2, 4, 13]. One can associate with A a topological space X A which is uniquely determined up to homeomorphism. Namely, let ,~ be the Grothendieck topolo-gy of the rigid analytic variety Sp A associated with A [4, III.2.1]. Then it is easily seen that the topos Shv(JA) of JA is spatial, i.e. there exists a sober topological space X A such that Shv(.~) is equivalent to the topos Shv(XA) of XA. By [6, IV.4.2.4] XA is uniquely determined up to homeomorphism. In this paper we will show that XA is homeomorphic to the topological subspace Spa(A, A ~ = { v e Cont A I v(a) < 1 for every a e A ~ } of Cont A (A ~ denotes the set of power bounded elements of A). We will even show that Shv(,~A) is canonically equivalent to Shy (Spa (A, A~ Having seen that, for Tate algebras A, the topological space Cont A occurs very naturally in rigid analytic geometry, one can ask for applications of Cont A for more general topological rings A. In this paper we restrict ourselves to a class of topological rings which I call f-adic rings: A topological ring is f-adic if …