705 research outputs found
Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
In this paper we establish an equivalence between the category of graded
D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W
and the triangulated category of singularities of the fiber of W over zero. The
main result is a theorem that shows that the graded triangulated category of
singularities of the cone over a projective variety is connected via a fully
faithful functor to the bounded derived category of coherent sheaves on the
base of the cone. This implies that the category of graded D-branes of type B
in Landau-Ginzburg models with homogeneous superpotential W is connected via a
fully faithful functor to the derived category of coherent sheaves on the
projective variety defined by the equation W=0.Comment: 26 pp., LaTe
Higher Descent Data as a Homotopy Limit
We define the 2-groupoid of descent data assigned to a cosimplicial
2-groupoid and present it as the homotopy limit of the cosimplicial space
gotten after applying the 2-nerve in each cosimplicial degree. This can be
applied also to the case of -groupoids thus providing an analogous
presentation of "descent data" in higher dimensions.Comment: Appeared in JHR
The branching structure of diffusion-limited aggregates
I analyze the topological structures generated by diffusion-limited
aggregation (DLA), using the recently developed "branched growth model". The
computed bifurcation number B for DLA in two dimensions is B ~ 4.9, in good
agreement with the numerically obtained result of B ~ 5.2. In high dimensions,
B -> 3.12; the bifurcation ratio is thus a decreasing function of
dimensionality. This analysis also determines the scaling properties of the
ramification matrix, which describes the hierarchy of branches.Comment: 6 pages, 1 figure, Euro-LaTeX styl
Cohomology Groups of Deformations of Line Bundles on Complex Tori
The cohomology groups of line bundles over complex tori (or abelian
varieties) are classically studied invariants of these spaces. In this article,
we compute the cohomology groups of line bundles over various holomorphic,
non-commutative deformations of complex tori. Our analysis interpolates between
two extreme cases. The first case is a calculation of the space of
(cohomological) theta functions for line bundles over constant, commutative
deformations. The second case is a calculation of the cohomologies of
non-commutative deformations of degree-zero line bundles.Comment: 24 pages, exposition improved, typos fixe
Tensor product of dualizing complexes over a field
Let be a field, and let be two locally noetherian -schemes
(respectively -formal schemes) with dualizing complexes and
respectively. We show that (respectively its derived
completion) is a dualizing complex over if and only if
is locally noetherian of finite Krull dimension.Comment: 13 pages, final version, to appear in the Journal of Commutative
Algebr
Unruffled extensions and flatness over central subalgebras
A condition on an affine central subalgebra of a noetherian algebra
of finite Gelfand-Kirillov dimension, which we call here \emph{unruffledness},
is shown to be equivalent in some circumstances to the flatness of as a
-module. Unruffledness was studied by Borho and Joseph in work on enveloping
algebras of complex semisimple Lie algebras, and we discuss applications of our
result to enveloping algebras, as well as beginning the study of this condition
for more general algebras
A variant of the Mukai pairing via deformation quantization
We give a new method to prove a formula computing a variant of Caldararu's
Mukai pairing \cite{Cal1}. Our method is based on some important results in the
area of deformation quantization. In particular, part of the work of Kashiwara
and Schapira in \cite{KS} as well as an algebraic index theorem of Bressler,
Nest and Tsygan in \cite{BNT},\cite{BNT1} and \cite{BNT2} are used. It is hoped
that our method is useful for generalization to settings involving certain
singular varieties.Comment: 8 pages. Comments and suggestions welcom
The homotopy theory of dg-categories and derived Morita theory
The main purpose of this work is the study of the homotopy theory of
dg-categories up to quasi-equivalences. Our main result provides a natural
description of the mapping spaces between two dg-categories and in
terms of the nerve of a certain category of -bimodules. We also prove
that the homotopy category is cartesian closed (i.e. possesses
internal Hom's relative to the tensor product). We use these two results in
order to prove a derived version of Morita theory, describing the morphisms
between dg-categories of modules over two dg-categories and as the
dg-category of -bi-modules. Finally, we give three applications of our
results. The first one expresses Hochschild cohomology as endomorphisms of the
identity functor, as well as higher homotopy groups of the \emph{classifying
space of dg-categories} (i.e. the nerve of the category of dg-categories and
quasi-equivalences between them). The second application is the existence of a
good theory of localization for dg-categories, defined in terms of a natural
universal property. Our last application states that the dg-category of
(continuous) morphisms between the dg-categories of quasi-coherent (resp.
perfect) complexes on two schemes (resp. smooth and proper schemes) is
quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect)
on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm.
8.15 is new. Minor corrections. Final version, to appear in Inventione
Formality theorems for Hochschild complexes and their applications
We give a popular introduction to formality theorems for Hochschild complexes
and their applications. We review some of the recent results and prove that the
truncated Hochschild cochain complex of a polynomial algebra is non-formal.Comment: Submitted to proceedings of Poisson 200
Topological self-similarity on the random binary-tree model
Asymptotic analysis on some statistical properties of the random binary-tree
model is developed. We quantify a hierarchical structure of branching patterns
based on the Horton-Strahler analysis. We introduce a transformation of a
binary tree, and derive a recursive equation about branch orders. As an
application of the analysis, topological self-similarity and its generalization
is proved in an asymptotic sense. Also, some important examples are presented
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