3,641 research outputs found
Homological dimensions for co-rank one idempotent subalgebras
Let be an algebraically closed field and be a (left and right)
Noetherian associative -algebra. Assume further that is either
positively graded or semiperfect (this includes the class of finite dimensional
-algebras, and -algebras that are finitely generated modules over a
Noetherian central Henselian ring). Let be a primitive idempotent of ,
which we assume is of degree if is positively graded. We consider the
idempotent subalgebra and the simple right
-module , where is the Jacobson radical
of , or the graded Jacobson radical of if is positively graded. In
this paper, we relate the homological dimensions of and , using the
homological properties of . First, if has no self-extensions of any
degree, then the global dimension of is finite if and only if that of
is. On the other hand, if the global dimensions of both and
are finite, then cannot have self-extensions of degree greater
than one, provided is finite dimensional.Comment: 24 page
Rank Two Sheaves on K3 Surfaces: A Special Construction
Let X be a K3 surface of degree 8 in P^5 with hyperplane section H. We
associate to it another K3 surface M which is a double cover of P^2 ramified on
a sextic curve C. In the generic case when X is smooth and a complete
intersection of three quadrics, there is a natural correspondence between M and
the moduli space M' of rank two vector bundles on X with Chern classes c_1=H
and c_2=4. We build on previous work of Mukai and others, giving conditions and
examples where M' is fine, compact, non-empty; and birational or isomorphic to
M. We also present an explicit calculation of the Fourier-Mukai transform when
X contains a line and has Picard number two.Comment: Fixed various minor errors and reworked some argument
A Small-Gain Theorem with Applications to Input/Output Systems, Incremental Stability, Detectability, and Interconnections
A general ISS-type small-gain result is presented. It specializes to a
small-gain theorem for ISS operators, and it also recovers the classical
statement for ISS systems in state-space form. In addition, we highlight
applications to incrementally stable systems, detectable systems, and to
interconnections of stable systems.Comment: 16 pages, no figure
- …