Table Of Content1791
Lecture Notes in Mathematics
Editors:
J.–M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
Heidelberg
NewYork
Barcelona
HongKong
London
Milan
Paris
Tokyo
Manfred Knebusch
Digen Zhang
Manis Valuations
and Pru¨fer Extensions I
A New Chapter in Commutative Algebra
1 3
Authors
ManfredKnebusch
DigenZHANG
DepartmentofMathematics
UniversityofRegensburg
Universita¨tstr.31
93040Regensburg
Germany
e-mail:
[email protected]
[email protected]
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Knebusch, Manfred:
Manis valuations and Prüfer extensions / Manfred Knebusch ; Digen Zhang. -
Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ;
Paris ; Tokyo : Springer
1. A new chapter in commutative algebra. - 2002
(Lecture notes in mathematics ; 1791)
ISBN 3-540-43951-X
MathematicsSubjectClassification(2000):
primary13A15,13A18,13F05,secondary13B02,14A05,14P10,14P05
ISSN0075-8434
ISBN3-540-43951-xSpringer-VerlagBerlinHeidelbergNewYork
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Contents
Introduction 1
Summary 7
Chapter I: Basics on Manis valuations and Pru¨fer
extensions 9
§1 Valuations on rings 10
§2 Valuation subrings and Manis pairs 22
§3 Weakly surjective homomorphisms 32
§4 More on weakly surjective extensions 41
§5 Basic theory of Pru¨fer extensions 46
§6 Examples of Pru¨fer extensions and convenient
ring extensions 57
§7 Principal ideal results 73
Chapter II: Multiplicative ideal theory 83
§1 Multiplicative properties of regular modules 84
§2 Characterizing Pru¨fer extensions by the behavior
of their regular ideals 93
§3 Describing a Pru¨fer extension by its lattice
of regular ideals 105
§4 Tight extensions 109
§5 Distributive submodules 119
§6 Transfer theorems 123
§7 Polars and factors in a Pru¨fer extension 129
§8 Decomposition of regular modules 134
§9 Pru¨fer overmodules 140
§10 Bezout extensions 144
§11 The Pru¨fer extensions of a noetherian ring 159
§12 Invertible hulls for modules over noetherian rings 169
Chapter III: PM-valuations and valuations
of weaker type 177
§1 The PM-overrings in a Pru¨fer extension 178
§2 Regular modules in a PM-extension 182
§3 More ways to characterize PM-extensions,
and a look at BM-extensions 186
§4 Tight valuations 198
§5 Existence of various valuation hulls 205
§6 Inside and outside the Manis valuation hull 214
§7 The TV-hull in a valuative extension 222
§8 Principal valuations 228
§9 Descriptions of the PM-hull 233
§10 Composing valuations with ring homomorphisms 240
§11 Transfer of valuations 243
Appendix A(toI,§4andI,§5):
Flatepimorphisms251
Appendix B(toII,§2):
Arithmeticalrings252
Appendix C(toIII,§6):
AdirectproofoftheexistenceofManisvaluationhulls255
References257
Index263
Introduction
We feel that new chapters of commutative algebra are needed to cope
with the extensions and derivates of algebraic geometry over promi-
nent fields like R, Q ,... and abstract versions of such geometries.
p
In particular, the real algebra associated with abstract semialgebraic
geometry, say the theory of real closed spaces (cf. [Sch], [Sch1]), is
still very much in its infancy. The present book is meant as a con-
tribution to such a commutative algebra.
Typically,thecommutativeringsnaturallyoccurringingeometriesas
abovearenotnoetherian, butincompensationhaveotherproperties,
which make them manageable. These properties may look strange
from the viewpoint of classical commutative algebra with its center
in polynomial rings over fields, but can be quite beautiful for an eye
accustomed to them.
Valuations seem to play a much more dominant role here than in
classical commutative algebra. For example, if F is any ordered
field, then all the convex subrings of F are Krull valuation rings
with quotient field F for a trivial reason (cf. e.g. [BCR, p.249], [KS,
p.55]), and this fact is of primary importance in real algebra. Many
of the mysteries of real algebra seem to be related to the difficulty of
controlling the value group and the residue class field of such convex
rings. They are almost never noetherian.
Let us stay a little with real algebra. If F is a formally real field, it
is well-known that the intersection H of the real valuation rings of F
is a Pru¨fer domain, and that H has quotient field F. {A valuation
ring is called real if its residue class field is formally real.} H is the
so called real holomorphy ring of F, cf. [B, §2], [S], [KS, Chap.III
§12]. If F is the function field k(V) of an algebraic variety V over
a real closed field k (e.g. k = R), suitable overrings of H in R can
tell us a lot about the algebraic and the semialgebraic geometry of
V(k).
These rings, of course, are again Pru¨fer domains. A very interesting
and – in our opinion – still mysterious role is played by some of these
rings which are related to the orderings of higher level of F, cf. e.g.
[B2], [B3]. Here we meet a remarkable phenomenon. For orderings of
level 1 (i.e. orderings in the classical sense) the usual procedure is to
M.KnebuschandD.Zhang:LNM1791,pp.1–6,2002.
(cid:1)c Springer-VerlagBerlinHeidelberg2002
2 Introduction
observe first that the convex subrings of ordered fields are valuation
rings, and then to go on to Pru¨fer domains as intersections of such
valuation rings, cf. e.g. [B], [S], [KS]. But for higher levels, up to
now, the best method is to directly construct a Pru¨fer domain A in F
from a “torsion preordering” of F, and then to obtain the valuation
rings necessary for analyzing the preordering as localizations Ap of
A, cf. [B2, p.1956 f], [B3]. Thus there is two way traffic between
valuations and Pru¨fer domains.
Up to now, less is done for the function field F = k(V) of an alge-
braic variety V over a p-adically closed field k (e.g. k = Q ). But
p
work of Kochen and Roquette (cf. §6 and §7 in the book [PR] by
Prestel and Roquette) gives ample evidence, that also here Pru¨fer
domains play a prominent role. In particular, every formally p-adic
field F contains a “p-adic holomorphy ring”, called the Kochen ring,
in complete analogy with the formally real case [PR, §6]. Actually,
the Kochen ring has been discovered and studied much earlier than
the real holomorphy ring ([Ko], [R1]).
If R is a commutative ring (with 1) and k is a subring of R, then we
can still define a real holomorphy ring H(R/k) consisting of those
elements a of R which can be bounded by elements of k on the
real spectrum of R (cf. [BCR], [B1], [KS]). {If R is a formally real
field F and k the prime ring of F, this ring coincides with the real
holomorphy ring H from above.} These rings H(R/k) have proven
to be very useful in real semialgebraic geometry. In particular, N.
SchwartzandM.Prechtelhaveusedthemforcompletingarealclosed
space and, more generally, to turn a morphism between real closed
spaces into a proper one in a universal way ([Sch, Chap V, §7], [Pt]).
The algebra of these holomorphy rings turns out to be particularly
good-natured if we assume that 1+ΣR2 ⊂ R∗, i.e. that all elements
1+a2 +···+a2 (n ∈ N,a ∈ R) are units in R. This is a natural
1 n i
condition in real algebra. The rings used by Schwartz and Prechtel,
consisting of abstract semialgebraic functions, fulfill the condition
automatically. More generally, if A is any commutative ring (always
−1
with 1) then the localization S A with respect to the multiplicative
2
set S = 1+ΣA is a ring R fulfilling the condition, and R has the
same real spectrum as A. Thus for many problems in real geometry
we may replace A by R.
Introduction 3
Now, V. Powers has proved that if 1 + ΣR2 ⊂ R∗, then the real
holomorphy ring H(R/k), with respect to any subring k, is an R-
∗)
Pru¨fer ring, as defined by Griffin in 1973 [G2]. More generally she
proved that if 1+ΣR2d ⊂ R∗, for some even number 2d, then every
subring A of R, containing the elements 1 with q ∈ ΣR2d, is R-
1+q
Pru¨fer ([P, Th.1.7], cf. also [BP]).
An R-Pru¨fer ring is related to Manis valuations on R in much the
same way a Pru¨fer domain is related to valuations of its quotient
field. Why shouldn’t we try to repeat the success story of Pru¨fer
domains and real valuations on the level of relative Pru¨fer rings and
Manis valuations? Already Marshall in his important paper [Mar]
has followed such a program. He worked with “Manis places” in a
ring R with 1 + ΣR2 ⊂ R∗, and related them to the points of the
real spectrum SperR.
We mention that Marshall’s notion of Manis places is slightly mis-
leading. According to his definition, these places do not correspond
to Manis valuations, but to a broader class of valuations which we
call “special valuations”, cf.I, §1 below. But then V. Powers (and in-
dependently one of us, D.Z.) observed that in the case 1+ΣR2 ⊂ R∗,
the places of Marshall in fact do correspond to the Manis valuations
of R [P]. {In Chapter I, §1 below we prove that every special valua-
tiononR isManisunderamuchweakerconditiononR, cf. Theorem
1.1.}
The program to study Manis valuations and relative Pru¨fer rings in
rings of real functions has gained new impetus and urgency from the
fact, that the theory of orderings of higher level has recently been
pushed from fields to rings, leading to real spectra of higher level.
These spectra in turn have already proven to be useful for ordinary
real semialgebraic geometry. We mention an opus magnum by Ralph
Berr [Be], where spectra of higher level are used in a fascinating way
to classify the singularities of real semialgebraic functions.
It seems that p-adic semialgebraic geometry is accessible as well.
L. Br¨ocker and H.-J. Schinke [BS] have brought the theory of p-adic
spectra to a rather satisfactory level by studying the “L-spectrum”
∗) §
The definition by Griffin needs a slight modification, cf. Def.1 in I, 5
below.
4 Introduction
L-specA of a commutative ring A with respect to a given non-
archimedian local field L (e.g. L = Q ). There seems to be no
p
major obstacle in sight which prevents us from defining and study-
ing rings of semialgebraic functions on a constructible (or even pro-
constructible) subset X of L-spec A. Here “semialgebraic” means
definable in a model-theoretic sense plus satisfying a suitable con-
tinuity condition. Relative Pru¨fer subrings of such rings should be
quite interesting.
The present book is devoted to the study of relative Pru¨fer rings
and Manis valuations, with an eye towards applications in real and
p-adic geometry.
Currently, there already exists a rich theory of “Pru¨fer rings with
zero divisors”, also started by Griffin [G1], cf. the books [LM], [Huc],
and the literature cited there. But this theory does not seem to be
taylored to geometric needs. A Pru¨fer ring with zero divisors A is
the same as an R-Pru¨fer ring with R = QuotA, the total quotient
ring of A. While this is a reasonable notion from the viewpoint of
ring theory, it may be artificial coming from a geometric direction.
A typical situation in real geometry is the following: R is the ring of
(continuous) semialgebraic functions on a semialgebraic set M over a
realclosedfieldk or, moregenerally, thesetofabstractsemialgebraic
functionsonaproconstructiblesubsetX ofarealspectrum(cf. [Sch],
[Sch1]). Although the ring R has very many zero divisors, we have
experience that in some sense R behaves nearly as well as a field,
cf. e.g. our notion of “convenient ring extensions” in I, §6 below.
Now, if A is a subring of R, then it is natural and interesting from
a geometric viewpoint to study the R-Pru¨fer rings B ⊃ A, while the
total quotient rings QuotA and QuotB seem to bear little geometric
relevance.
Except for a paper by P.L. Rhodes from 1991 [Rh], little seems to
∗)
be done on Pru¨fer extensions in general, and in the original pa-
per of Griffin the proofs of important facts [G2, Prop.6, Th.7] are
omitted. Moreover, the paper by Rhodes has a gap in the proof of
his main theorem. {[Rh, Th.2.1], condition (5b) is apparently not a
characterization of Pru¨fer extensions. Any algebraic field extension
∗)
The important work of Gr¨ater [Gr-Gr3] and of Alajbegovi´c and Moˇckoˇr
[Al-M] will be discussed in Part II of the book.
