19,782 research outputs found
Improving the bounds of the Multiplicity Conjecture: the codimension 3 level case
The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and
lower bounds for the multiplicity of a Cohen-Macaulay algebra in terms of
the shifts appearing in the modules of the minimal free resolution (MFR) of
. All the examples studied so far have lead to conjecture (see and
) that, moreover, the bounds of the MC are sharp if and only if has
a pure MFR. Therefore, it seems a reasonable - and useful - idea to seek
better, if possibly {\it ad hoc}, bounds for particular classes of
Cohen-Macaulay algebras. In this work we will only consider the codimension 3
case. In the first part we will stick to the bounds of the MC, and show that
they hold for those algebras whose -vector is that of a compressed algebra.
In the second part, we will (mainly) focus on the level case: we will construct
new conjectural upper and lower bounds for the multiplicity of a codimension 3
level algebra , which can be expressed exclusively in terms of the
-vector of , and which are better than (or equal to) those provided by
the MC. Also, our bounds can be sharp even when the MFR of is not pure.
Even though proving our bounds still appears too difficult a task in general,
we are already able to show them for some interesting classes of codimension 3
level algebras : namely, when is compressed, or when its -vector
ends with . Also, we will prove our lower bound when
begins with , where , and our upper bound when
ends with , where .Comment: 22 pages. A few (non-substantial) changes. To appear in J. of Pure
and Appl. Algebr
Boundary quotients and ideals of Toeplitz C*-algebras of Artin groups
We study the quotients of the Toeplitz C*-algebra of a quasi-lattice ordered
group (G,P), which we view as crossed products by a partial actions of G on
closed invariant subsets of a totally disconnected compact Hausdorff space, the
Nica spectrum of (G,P). Our original motivation and our main examples are drawn
from right-angled Artin groups, but many of our results are valid for more
general quasi-lattice ordered groups. We show that the Nica spectrum has a
unique minimal closed invariant subset, which we call the boundary spectrum,
and we define the boundary quotient to be the crossed product of the
corresponding restricted partial action. The main technical tools used are the
results of Exel, Laca, and Quigg on simplicity and ideal structure of partial
crossed products, which depend on amenability and topological freeness of the
partial action and its restriction to closed invariant subsets. When there
exists a generalised length function, or controlled map, defined on G and
taking values in an amenable group, we prove that the partial action is
amenable on arbitrary closed invariant subsets. Our main results are obtained
for right-angled Artin groups with trivial centre, that is, those with no
cyclic direct factor; they include a presentation of the boundary quotient in
terms of generators and relations that generalises Cuntz's presentation of O_n,
a proof that the boundary quotient is purely infinite and simple, and a
parametrisation of the ideals of the Toeplitz C*-algebra in terms of subsets of
the standard generators of the Artin group.Comment: 26 page
Essential dimension of simple algebras with involutions
Let be integers with and \cat{Alg}_{n,m} the class
of central simple algebras of degree and exponent dividing . In this
paper, we find new, improved upper bounds for the essential dimension and
2-dimension of \cat{Alg}_{n,2}. In particular, we show that
\ed_{2}(\cat{Alg}_{16,2})=24 over a field of characteristic different
from 2.Comment: Sections 1 and 3 are rewritte
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