Full paper in PDF:
$%K. Boussaf, N. Maïnetti, and M. Hemdaoui, Tree structure on the set of multiplicative semi-norms
of Krasner algebras H(D), Rev. Mat. Complut. 13 (2000), no. 1, 85–109.%$
Tree Structure on the Set of Multiplicative
Semi-norms of Krasner Algebras H(D)
Kamal BOUSSAF, Nicolas MAÏNETTI,
and Mohamed HEMDAOUI
Laboratoire de Mathématiques Pures
Université Blaise Pascal (Clermont-Ferrand)
Complexe Scientifique des Cézeaux
F 63177 Aubiere Cedex — France | Département de Mathématiques Pures
Université Mohammed I
Oujda — Morocco |
| |
Received: April 12, 1999
Revised: January 21, 2000
Let
be an algebraically closed field, complete for an ultrametric absolute
value, let
be an infinite subset of
and let
be the set of analytic
elements on
(Escassut, 1995). We denote by
the set
of semi-norms y of the
-vector space
which are continuous with
respect to the topology of uniform convergence on
and which satisfy further
whenever
and
. This set is
provided with the topology of simple convergence. By the way of a metric
topology thinner than the simple convergence, we establish the equivalence
between the connectedness of
, the arc-connectedness of
and the infraconnectedness of
. This generalizes a result
of Berkovich given on affinoid algebras (Berkovich, 1990). Next, we study
the filter of neighborhoods of an element of
, and we give
a condition on the field K such that this filter admits a countable basis.
We also prove the local arc-connectedness of
when
is
infraconnected. Finally, we study the metrizability of the topology of simple
convergence on
and we give some conditions to have an
equivalence with the metric topology defined above. The fundamental tool in
this survey consists of circular filters.
1991 Mathematics Subject Classification: 46S10, 11Q25.