5,543 research outputs found
Projective Systems of Noncommutative Lattices as a Pregeometric Substratum
We present an approximation to topological spaces by {\it noncommutative}
lattices. This approximation has a deep physical flavour based on the
impossibility to fully localize particles in any position measurement. The
original space being approximated is recovered out of a projective limit.Comment: 30 pages, Latex. To appear in `Quantum Groups and Fundamental
Physical Applications', ISI Guccia, Palermo, December 1997, D. Kastler and M.
Rosso Eds., (Nova Science Publishers, USA
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
Imaginaries and definable types in algebraically closed valued fields
The text is based on notes from a class entitled {\em Model Theory of
Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and
retains the flavor of class notes. It includes an exposition of material from
\cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the
model completion of the theory of valued fields, and the classification of
imaginary sorts. The latter is given a new proof, based on definable types
rather than invariant types, and on the notion of {\em generic
reparametrization}. I also try to bring out the relation to the geometry of
\cite{HL} - stably dominated definable types as the model theoretic incarnation
of a Berkovich point
Resolutions and Cohomologies of Toric Sheaves. The affine case
We study equivariant resolutions and local cohomologies of toric sheaves for
affine toric varieties, where our focus is on the construction of new examples
of decomposable maximal Cohen-Macaulay modules of higher rank. A result of
Klyachko states that the category of reflexive toric sheaves is equivalent to
the category of vector spaces together with a certain family of filtrations.
Within this setting, we develop machinery which facilitates the construction of
minimal free resolutions for the smooth case as well as resolutions which are
acyclic with respect to local cohomology functors for the general case. We give
two main applications. First, over the polynomial ring, we determine in
explicit combinatorial terms the Z^n-graded Betti numbers and local cohomology
of reflexive modules whose associated filtrations form a hyperplane
arrangement. Second, for the non-smooth, simplicial case in dimension d >= 3,
we construct new examples of indecomposable maximal Cohen-Macaulay modules of
rank d - 1.Comment: 39 pages, requires packages ams*, enumerat
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