23,176 research outputs found
Noncommutative localization in noncommutative geometry
The aim of these notes is to collect and motivate the basic localization
toolbox for the geometric study of ``spaces'', locally described by
noncommutative rings and their categories of one-sided modules.
We present the basics of Ore localization of rings and modules in much
detail. Common practical techniques are studied as well. We also describe a
counterexample for a folklore test principle. Localization in negatively
filtered rings arising in deformation theory is presented. A new notion of the
differential Ore condition is introduced in the study of localization of
differential calculi.
To aid the geometrical viewpoint, localization is studied with emphasis on
descent formalism, flatness, abelian categories of quasicoherent sheaves and
generalizations, and natural pairs of adjoint functors for sheaf and module
categories. The key motivational theorems from the seminal works of Gabriel on
localization, abelian categories and schemes are quoted without proof, as well
as the related statements of Popescu, Watts, Deligne and Rosenberg.
The Cohn universal localization does not have good flatness properties, but
it is determined by the localization map already at the ring level. Cohn
localization is here related to the quasideterminants of Gelfand and Retakh;
and this may help understanding both subjects.Comment: 93 pages; (including index: use makeindex); introductory survey, but
with few smaller new result
Segal's multisimplicial spaces
Some sufficient conditions on a simplicial space guaranteeing that
were given by Segal. We give a generalization of this
result for multisimplicial spaces. This generalization is appropriate for the
reduced bar construction, providing an -fold delooping of the classifying
space of a category.Comment: 12 page
Excitation spectrum of the Lieb-Liniger model
We study the integrable model of one-dimensional bosons with contact
repulsion. In the limit of weak interaction, we use the microscopic
hydrodynamic theory to obtain the excitation spectrum. The statistics of
quasiparticles changes with the increase of momentum. At lowest momenta good
quasiparticles are fermions, while at higher momenta they are Bogoliubov
bosons, in accordance with recent studies. In the limit of strong interaction,
we analyze the exact solution and find exact results for the spectrum in terms
of the asymptotic series. Those results undoubtedly suggest that fermionic
quasiparticle excitations actually exist at all momenta for moderate and strong
interaction, and also at lowest momenta for arbitrary interaction. Moreover, at
strong interaction we find highly accurate analytical results for several
relevant quantities of the Lieb-Liniger model.Comment: seven pages and two figure
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