979 research outputs found
Topology on cohomology of local fields
Arithmetic duality theorems over a local field are delicate to prove if
. In this case, the proofs often exploit topologies
carried by the cohomology groups for commutative finite type
-group schemes . These "\v{C}ech topologies", defined using \v{C}ech
cohomology, are impractical due to the lack of proofs of their basic
properties, such as continuity of connecting maps in long exact sequences. We
propose another way to topologize : in the key case ,
identify with the set of isomorphism classes of objects of the
groupoid of -points of the classifying stack and invoke
Moret-Bailly's general method of topologizing -points of locally of finite
type -algebraic stacks. Geometric arguments prove that these "classifying
stack topologies" enjoy the properties expected from the \v{C}ech topologies.
With this as the key input, we prove that the \v{C}ech and the classifying
stack topologies actually agree. The expected properties of the \v{C}ech
topologies follow, which streamlines a number of arithmetic duality proofs
given elsewhere.Comment: 36 pages; final version, to appear in Forum of Mathematics, Sigm
A closedness theorem and applications in geometry of rational points over Henselian valued fields
We develop geometry of algebraic subvarieties of over arbitrary
Henselian valued fields . This is a continuation of our previous article
concerned with algebraic geometry over rank one valued fields. At the center of
our approach is again the closedness theorem that the projections are definably closed maps. It enables application
of resolution of singularities in much the same way as over locally compact
ground fields. As before, the proof of that theorem uses i.a. the local
behavior of definable functions of one variable and fiber shrinking, being a
relaxed version of curve selection. But now, to achieve the former result, we
first examine functions given by algebraic power series. All our previous
results will be established here in the general settings: several versions of
curve selection (via resolution of singularities) and of the \L{}ojasiewicz
inequality (via two instances of quantifier elimination indicated below),
extending continuous hereditarily rational functions as well as the theory of
regulous functions, sets and sheaves, including Nullstellensatz and Cartan's
theorems A and B. Two basic tools applied in this paper are quantifier
elimination for Henselian valued fields due to Pas and relative quantifier
elimination for ordered abelian groups (in a many-sorted language with
imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications
of the closedness theorem are piecewise continuity of definable functions,
H\"{o}lder continuity of definable functions on closed bounded subsets of
, the existence of definable retractions onto closed definable subsets
of , and a definable, non-Archimedean version of the Tietze--Urysohn
extension theorem. In a recent preprint, we established a version of the
closedness theorem over Henselian valued fields with analytic structure along
with some applications.Comment: This paper has been published in Journal of Singularities 21 (2020),
233-254. arXiv admin note: substantial text overlap with arXiv:1704.01093,
arXiv:1703.08203, arXiv:1702.0784
Isometry groups of proper metric spaces
Given a locally compact Polish space X, a necessary and sufficient condition
for a group G of homeomorphisms of X to be the full isometry group of (X,d) for
some proper metric d on X is given. It is shown that every locally compact
Polish group G acts freely on GxY as the full isometry group of GxY with
respect to a certain proper metric on GxY, where Y is an arbitrary locally
compact Polish space with (card(G),card(Y)) different from (1,2). Locally
compact Polish groups which act effectively and almost transitively on complete
metric spaces as full isometry groups are characterized. Locally compact Polish
non-Abelian groups on which every left invariant metric is automatically right
invariant are characterized and fully classified. It is demonstrated that for
every locally compact Polish space X having more than two points the set of
proper metrics d such that Iso(X,d) = {id} is dense in the space of all proper
metrics on X.Comment: 24 page
Eigenvalues for double phase variational integrals
We study an eigenvalue problem in the framework of double phase variational
integrals and we introduce a sequence of nonlinear eigenvalues by a minimax
procedure. We establish a continuity result for the nonlinear eigenvalues with
respect to the variations of the phases. Furthermore, we investigate the growth
rate of this sequence and get a Weyl-type law consistent with the classical law
for the -Laplacian operator when the two phases agree.Comment: 42 pages, typos corrected, final version, to appear in Ann. Mat. Pura
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