7,706 research outputs found
A computational approach to the D-module of meromorphic functions
Let be a divisor in . We present methods to compare the
-module of the meromorphic functions to some
natural approximations. We show how the analytic case can be treated with
computations in the Weyl algebra.Comment: 11 page
Noncommutative curves and noncommutative surfaces
In this survey article we describe some geometric results in the theory of
noncommutative rings and, more generally, in the theory of abelian categories.
Roughly speaking and by analogy with the commutative situation, the category
of graded modules modulo torsion over a noncommutative graded ring of
quadratic, respectively cubic growth should be thought of as the noncommutative
analogue of a projective curve, respectively surface. This intuition has lead
to a remarkable number of nontrivial insights and results in noncommutative
algebra. Indeed, the problem of classifying noncommutative curves (and
noncommutative graded rings of quadratic growth) can be regarded as settled.
Despite the fact that no classification of noncommutative surfaces is in sight,
a rich body of nontrivial examples and techniques, including blowing up and
down, has been developed.Comment: Suggestions by many people (in particular Haynes Miller and Dennis
Keeler) have been incorporated. The formulation of some results has been
improve
Liaison classes of modules
We propose a concept of module liaison that extends Gorenstein liaison of
ideals and provides an equivalence relation among unmixed modules over a
commutative Gorenstein ring. Analyzing the resulting equivalence classes we
show that several results known for Gorenstein liaison are still true in the
more general case of module liaison. In particular, we construct two maps from
the set of even liaison classes of modules of fixed codimension into stable
equivalence classes of certain reflexive modules. As a consequence, we show
that the intermediate cohomology modules and properties like being perfect,
Cohen-Macaulay, Buchsbaum, or surjective-Buchsbaum are preserved in even module
liaison classes. Furthermore, we prove that the module liaison class of a
complete intersection of codimension one consists of precisely all perfect
modules of codimension one
Hochster duality in derived categories and point-free reconstruction of schemes
For a commutative ring , we exploit localization techniques and point-free
topology to give an explicit realization of both the Zariski frame of (the
frame of radical ideals in ) and its Hochster dual frame, as lattices in the
poset of localizing subcategories of the unbounded derived category .
This yields new conceptual proofs of the classical theorems of Hopkins-Neeman
and Thomason. Next we revisit and simplify Balmer's theory of spectra and
supports for tensor triangulated categories from the viewpoint of frames and
Hochster duality. Finally we exploit our results to show how a coherent scheme
can be reconstructed from the tensor triangulated structure
of its derived category of perfect complexes.Comment: v5:Minoir typos corrected the proof of tensor nilpotence is made
totally point-free and self-contained; some simplifications and expository
improvements; section on preliminaries shortened; 50pp. To appear in Trans.
AM
Alexander duality and Stanley depth of multigraded modules
We apply Miller's theory on multigraded modules over a polynomial ring to the
study of the Stanley depth of these modules. Several tools for Stanley's
conjecture are developed, and a few partial answers are given. For example, we
show that taking the Alexander duality twice (but with different "centers") is
useful for this subject. Generalizing a result of Apel, we prove that Stanley's
conjecture holds for the quotient by a cogeneric monomial ideal.Comment: 18 pages. We have removed Lemma 2.3 of the previous version, since
the proof contained a gap. This deletion does not affect the main results,
while we have revised argument a little (especially in Sections in 2 and 3
Algebraic computation of some intersection D-modules
Let be a complex analytic manifold, a locally
quasi-homogeneous free divisor, an integrable logarithmic connection with
respect to and the local system of the horizontal sections of on
. In this paper we give an algebraic description in terms of of the
regular holonomic D-module whose de Rham complex is the intersection complex
associated with . As an application, we perform some effective computations
in the case of quasi-homogeneous plane curves.Comment: 18 page
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