12,181 research outputs found
Simultaneous similarity and triangularization of sets of 2 by 2 matrices
Let be a finite or infinite sequence of
matrices with entries in an integral domain. We show that, except
for a very special case, is (simultaneously) triangularizable if
and only if all pairs are triangularizable, for . We also provide a simple numerical criterion for
triangularization.
Using constructive methods in invariant theory, we define a map (with the
minimal number of invariants) that distinguishes simultaneous similarity
classes for non-commutative sequences over a field of characteristic .
We also describe canonical forms for sequences of matrices over
algebraically closed fields, and give a method for finding sequences with a
given set of invariants.Comment: 22 page
Vector bundles and torsion free sheaves on degenerations of elliptic curves
In this paper we give a survey about the classification of vector bundles and
torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on
singular curves of arithmetic genus one can be studied using the technique of
matrix problems or via Fourier-Mukai transforms, both methods are discussed
here. Moreover, we include new proofs of some classical results about vector
bundles on elliptic curves.Comment: 39 pages, 5 figure
On periodicity in bounded projective resolutions
Let A be a self-injective algebra over an algebraically closed field k. We
show that if an A-module M of complexity one has an open orbit in the variety
of d-dimensional A-modules, then M is periodic. As a corollary we see that any
simple A-module of complexity one must be periodic. In the course of the proof,
we also show that modules with open orbits are preserved by stable equivalences
of Morita type between self-injective algebras
Flexible varieties and automorphism groups
Given an affine algebraic variety X of dimension at least 2, we let SAut (X)
denote the special automorphism group of X i.e., the subgroup of the full
automorphism group Aut (X) generated by all one-parameter unipotent subgroups.
We show that if SAut (X) is transitive on the smooth locus of X then it is
infinitely transitive on this locus. In turn, the transitivity is equivalent to
the flexibility of X. The latter means that for every smooth point x of X the
tangent space at x is spanned by the velocity vectors of one-parameter
unipotent subgroups of Aut (X). We provide also different variations and
applications.Comment: Final version; to appear in Duke Math.
- …