6,112 research outputs found
When does NIP transfer from fields to henselian expansions?
Let be an NIP field and let be a henselian valuation on . We ask
whether is NIP as a valued field. By a result of Shelah, we know that
if is externally definable, then is NIP. Using the definability of
the canonical -henselian valuation, we show that whenever the residue field
of is not separably closed, then is externally definable. In the case
of separably closed residue field, we show that is NIP as a pure valued
field.Comment: 8 pages. Contains an unconditional version of the main theorem (even
in case the residue field is separably closed
Selected methods for the classification of cuts, and their applications
We consider four approaches to the analysis of cuts in ordered abelian groups
and ordered fields, their interconnection, and various applications. The
notions we discuss are: ball cuts, invariance group, invariance valuation ring,
and cut cofinality
A definable henselian valuation with high quantifier complexity
We give an example of a parameter-free definable henselian valuation ring
which is neither definable by a parameter-free -formula nor by
a parameter-free -formula in the language of rings. This
answers a question of Prestel.Comment: 6 page
Finite burden in multivalued algebraically closed fields
We prove that an expansion of an algebraically closed field by arbitrary
valuation rings is NTP, and in fact has finite burden. It fails to be
NIP, however, unless the valuation rings form a chain. Moreover, the incomplete
theory of algebraically closed fields with valuation rings is decidable.Comment: 40 page
Ideal theory of infinite directed unions of local quadratic transforms
Let be a regular local ring of dimension at least 2. Associated to each
valuation domain birationally dominating , there exists a unique sequence
of local quadratic transforms of along this valuation domain. We
consider the situation where the sequence is infinite,
and examine ideal-theoretic properties of the integrally closed local domain . Among the set of valuation overrings of , there
exists a unique limit point for the sequence of order valuation rings of
the . We prove the existence of a unique minimal proper Noetherian
overring of , and establish the decomposition . If is
archimedian, then the complete integral closure of has the form
, where is the rank valuation overring of .Comment: Final version, to appear in J. of Algebr
Complete ideals defined by sign conditions and the real spectrum of a two-dimensional local ring
This paper is about the local geometry of a real surfaces. It introduces
machinery for studying families of subsets which are determined by conditions
which are similar to base conditions, but also involve
positivity/non-negativity. The methods used are the real spectrum and Zariski's
theory of complete ideals.Comment: 12 pages, TeX version 3.
Real closed valued fields with analytic structure
We show quantifier elimination theorems for real closed valued fields with
separated analytic structure and overconvergent analytic structure in their
natural one-sorted languages and deduce that such structures are weakly
o-minimal. We also provide a short proof that algebraically closed valued
fields with separated analytic structure (in any rank) are -minimal.Comment: 10 pages. Any comments welcome
Defining coarsenings of valuations
We study the question which henselian fields admit definable henselian
valuations (with or without parameters). We show that every field which admits
a henselian valuation with non-divisible value group admits a
parameter-definable (non-trivial) henselian valuation. In equicharacteristic
, we give a complete characterization of henselian fields admitting a
parameter-definable (non-trivial) henselian valuation. We also obtain partial
characterization results of fields admitting 0-definable (non-trivial)
henselian valuations. We then draw some Galois-theoretic conclusions from our
results.Comment: 19 page
(Non)Vanishing results on local cohomology of valuation rings
We examine local cohomology in the setting of valuation rings. The novelty of
this investigation stems from the fact that valuation rings are usually
non-Noetherian, whereas local cohomology has been extensively developed mostly
in a Noetherian setting. We prove various vanishing results on local cohomology
for valuation rings of finite Krull dimension. These vanishing results stem
from a uniform bound on the global dimension of such rings. Our investigation
reveals differences in the sheaf theoretic definition of local cohomology, and
the algebraic definition in terms of a limit of certain Ext functors.Comment: Comments are welcome; latest edit corrects numerous typos and makes
the article consistent with the journal versio
Semigroups of valuations on local rings
In this paper the question of which semigroups are realizable as the
semigroup of values attained on a Noetherian local ring which is dominated by a
valuation is considered. We give some striking examples, indicating that there
may be no constraints on the semigroup beyond those known classically.Comment: 19 page
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