which appear to capture much of the essence of p-adic geometry in a coordinate-free, combinatorial manner. 0. INTRODUCTION. A considerable number of ...
A. W . M. D R E S S A N D W . T E R H A L L E
A COMBINATORIAL
APPROACH
TO p-ADIC GEOMETRY
ABSTRACT. An important tool in p-adic geometry is the process of p-adic completion. It is shown that this process can be performed on the level of valuated matroids, that is, certain structures which appear to capture much of the essence of p-adic geometry in a coordinate-free, combinatorial manner.
0.
INTRODUCTION
A considerable number of well-known technical problems arising in p-adic geometry are caused by the fact that there is just no canonical way to define reduction modulo p for a projective variety, defined over a p-adic field, and t h a t - c o r r e s p o n d i n g l y - t h e r e are many different, equally suitable p-adic (ultra-)metrics definable on such a variety which can be used to deal with specific aspects of p-adic geometry, e.g. to define, construct and analyse padic completions. Consequently, the only systematic way to deal with such a situation and to reveal the intrinsic features of p-adic geometry is to consider simultaneously all possible reductions (or metrics) and to study the laws of transformation according to which they are transformed into one another. Fortunately, an amazingly handy conceptual tool for such studies was introduced recently in [DW]. There the authors defined a valuated matroid of rank m to be a pair M = (E, v) = (E, v)m, consisting of a set E and a map v: E" ~ ( - ~ } u R, satisfying the conditions (VM0) to (VM3), specified at the beginning of Section 1. Two motivations for this concept were offered in [DW-I. One is that it occurs naturally when studying a certain variant of the greedy algorithm. The other one, which is relevant in the present context, is that for any p-adic valuation w: K ~ { - ~ } u ~ of a field K (that is, for a map w:K~{-oo}uR satisfying w ( x ) = - o o ~ x = O , w(xy)=w(x)+w(y), and w(x + y) 1 e2,...,e,e/~sup{v'(e', for all
e'e E'
view of
e2..... e,,)- i=2 ~ p(ei)} =p'(e')
and therefore also
p'(e')~r)(e')= e'~,...,e'eE'SUp{v'(e',
e~ .... , e~,) -- i~=2 "b(e'i)}
0, e~ .... , e~. e E, and choose a base B _q E with (m - 2). v(B) >~ - e . Then
v'(a, b, e'3. . . . . e',,) v(fl ..... fro)" For all %2 . . . . . ak,meE (1 ~< k ~< m) we have v(el . . . . . era) + ~
V(fk, ak,2 . . . . . ak,m)
k=l
~
N.
[]
C O R O L L A R Y 1.18. I f M ' =(E',v')m is a completion o f M = ( E , v ) m , the canonical embedding TM ~ TM', p ~ p', defined in Proposition 1.9 is a bijection. Proof. If q ~ TM,, then qlE is in TM, and if ql, q2 ~ TM, with ql IE = q2 IE, then ql = q2: let e e E ; then for all e > 0 there exist e~ . . . . . e ' ~ E ' , such that
v'(e, e'2. . . . . e ~ - q(e'2) . . . . .
q(e'm)
