7,511 research outputs found
Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras
The assignment (nonstable K_0-theory), that to a ring R associates the monoid
V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices
with only finitely nonzero entries over R, extends naturally to a functor. We
prove the following lifting properties of that functor: (1) There is no functor
F, from simplicial monoids with order-unit with normalized positive
homomorphisms to exchange rings, such that VF is equivalent to the identity.
(2) There is no functor F, from simplicial monoids with order-unit with
normalized positive embeddings to C*-algebras of real rank 0 (resp., von
Neumann regular rings), such that VF is equivalent to the identity. (3) There
is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be
lifted, with respect to the functor V, by exchange rings and by C*-algebras of
real rank 1, but not by semiprimitive exchange rings, thus neither by regular
rings nor by C*-algebras of real rank 0. By using categorical tools from an
earlier paper (larders, lifters, CLL), we deduce that there exists a unital
exchange ring of cardinality aleph three (resp., an aleph three-separable
unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence
2, such that V(R) is the positive cone of a dimension group and V(R) is not
isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0
or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea
An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals
A construction of the real number system based on almost homomorphisms of the
integers Z was proposed by Schanuel, Arthan, and others. We combine such a
construction with the ultrapower or limit ultrapower construction, to construct
the hyperreals out of integers. In fact, any hyperreal field, whose universe is
a set, can be obtained by such a one-step construction directly out of
integers. Even the maximal (i.e., On-saturated) hyperreal number system
described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can
be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via
a one-step construction by means of a definable ultrapower (modulo a suitable
definable class ultrafilter).Comment: 17 pages, 1 figur
Some Concepts of Modern Algebraic Geometry: Point, Ideal and Homomorphism
Starting from classical algebraic geometry over the complex numbers (as it
can be found for example in Griffiths and Harris it was the goal of these
lectures to introduce some concepts of the modern point of view in algebraic
geometry. Of course, it was quite impossible even to give an introduction to
the whole subject in such a limited time. For this reason the lectures and now
the write-up concentrate on the substitution of the concept of classical points
by the notion of ideals and homomorphisms of algebras.Comment: 36 pages. This is a write-up of lectures given at the ``Kleine
Herbstschule 93'' of the Graduiertenkolleg ``Mathematik im Bereich Ihrer
Wechselwirkungen mit der Physik'' at the Ludwig-Maximilians-Universitaet
Muenche
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