7,931 research outputs found
Decomposition of semigroup algebras
Let A \subseteq B be cancellative abelian semigroups, and let R be an
integral domain. We show that the semigroup ring R[B] can be decomposed, as an
R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A].
In the case of a finite extension of positive affine semigroup rings we obtain
an algorithm computing the decomposition. When R[A] is a polynomial ring over a
field we explain how to compute many ring-theoretic properties of R[B] in terms
of this decomposition. In particular we obtain a fast algorithm to compute the
Castelnuovo-Mumford regularity of homogeneous semigroup rings. As an
application we confirm the Eisenbud-Goto conjecture in a range of new cases.
Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.Comment: 12 pages, 2 figures, minor revisions. Package may be downloaded at
http://www.math.uni-sb.de/ag/schreyer/jb/Macaulay2/MonomialAlgebras/html
Bianchi spaces and their 3-dimensional isometries as S-expansions of 2-dimensional isometries
In this paper we show that some 3-dimensional isometry algebras, specifically
those of type I, II, III and V (according Bianchi's classification), can be
obtained as expansions of the isometries in 2 dimensions. It is shown that in
general more than one semigroup will lead to the same result. It is impossible
to obtain the algebras of type IV, VI-IX as an expansion from the isometry
algebras in 2 dimensions. This means that the first set of algebras has
properties that can be obtained from isometries in 2 dimensions while the
second set has properties that are in some sense intrinsic in 3 dimensions. All
the results are checked with computer programs. This procedure can be
generalized to higher dimensions, which could be useful for diverse physical
applications.Comment: 23 pages, one of the authors is new, title corrected, finite
semigroup programming is added, the semigroup construction procedure is
checked by computer programs, references to semigroup programming are added,
last section is extended, appendix added, discussion of all the types of
Bianchi spaces is include
Nuclear dimension and Z-stability of non-simple C*-algebras
We investigate the interplay of the following regularity properties for
non-simple C*-algebras: finite nuclear dimension, Z-stability, and algebraic
regularity in the Cuntz semigroup. We show that finite nuclear dimension
implies algebraic regularity in the Cuntz semigroup, provided that known type I
obstructions are avoided. We demonstrate how finite nuclear dimension can be
used to study the structure of the central sequence algebra, by factorizing the
identity map on the central sequence algebra, in a manner resembling the
factorization arising in the definition of nuclear dimension.
Results about the central sequence algebra are used to attack the conjecture
that finite nuclear dimension implies Z-stability, for sufficiently non-type I,
separable C*-algebras. We prove this conjecture in the following cases: (i) the
C*-algebra has no purely infinite subquotients and its primitive ideal space
has a basis of compact open sets, (ii) the C*-algebra has no purely infinite
quotients and its primitive ideal space is Hausdorff. In particular, this
covers C*-algebras with finite decomposition rank and real rank zero. Our
results hold more generally for C*-algebras with locally finite nuclear
dimension which are (M,N)-pure (a regularity condition of the Cuntz semigroup).Comment: Rewrote abstract and introduction. Added a couple of results. Main
results unchange
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