Locally Semialgebraic Spaces
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1173
Hans Delfs Manfred Knebusch
Locally Semialgebraic Spaces
Springer-Verlag Berlin Heidelberg New York Tokyo
Authors Hans Delfs Manfred Knebusch Fakultat fur Mathematik, Universitat Regensburg Universitatsstr. 31,8400 Regensburg Federal Republic of Germany
Mathematics Subject Classification (1980): 14G30, 54E99, 55005, 57R05 ISBN 3-540-16060-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16060-4 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1985 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-54321 0
To Christl and Gisela
Preface
The primary occupation of real alaebraic geometry, or better "semialgebraic geometry", is to study the set of solutions of a finite system of polynomial inequalities in a finite number of variables over the field ~
of real numbers. One wants to do this in a conceptual way, not always
mentioning the polynomial data, similarly as in algebraic geometry, say over C, where one most often avoids working explicitelywith the systems of polynomial equalities (and non-equalities f'" 0) involved.
But a semialgebraic geometry which deserves its name should be able to work - at least - over an arbitrary real closed field R instead of the field~.
Such fields are useful and even unavoidable in semialgebraic
geometry for much the same reason as algebraically closed fields of characteristic zero - at least - are unavoidable in algebraic geometry over
~,
as soon as one tries to avoid transcendental techniques or even
then.
In order to illustrate this we give a somewhat typical example. Let f : V -+ W be an algebraic map between irreducible varieties over yields, by restriction, a continuous map sets of real pOints. We assume that means that
~'l(~)
X = f
w(~)
... \'i1(~)
This
between the
is Zariski dense in W which
contains non singular points or, equivalently, that the
function field lR(W) -1
f~: V(~)
~.
is formally real. The generic fibre X of f, Le.
(n) with n the generic paint of 1'1 (regarding V and W as schemes),
is an algebraic scheme over the function field m(W) tains a lot of information about f and f N . to study X, since the field
~(W)
of W, which con-
But it may be too difficult
is usually very complicated. In alge-
braic geometry one often replaces X by the algebraic variety from X by extension of the base field N(W)
X obtained
to the algebraic closure C
of lR(W) . It is much easier to study the "geometric generic fibre" X
VI
instead of X, and still one may hope to extract relevant information
X.
about f from
But in sernialgebraic geometry thi
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