1,103 research outputs found
A Goodwillie-type Theorem for Milnor K-Theory
Goodwillie's rational isomorphism between relative algebraic K-theory and
relative cyclic homology, together with the lambda decomposition of cyclic
homology, illustrates the close relationships among algebraic K-theory, cyclic
homology, and differential forms. In this paper, I prove a Goodwillie-type
theorem for relative Milnor -theory, working over a very general class of
commutative rings, defined via the stability criterion of Van der Kallen. Early
results of Van der Kallen and Bloch are special cases. The result likely
generalizes in terms of de Rahm-Witt complexes, by weakening some invertibility
assumptions, but the class of rings considered is already more than
sufficiently general for the intended applications. The main motivation for
this paper arises from applications to the infinitesimal theory of Chow groups,
first pointed out by Bloch in the 1970's, and prominent in recent work of Green
and Griffiths. Related results and geometric applications are discussed in the
final section.Comment: 34 page
Quantum Drinfeld Hecke Algebras
We consider finite groups acting on quantum (or skew) polynomial rings.
Deformations of the semidirect product of the quantum polynomial ring with the
acting group extend symplectic reflection algebras and graded Hecke algebras to
the quantum setting over a field of arbitrary characteristic. We give necessary
and sufficient conditions for such algebras to satisfy a Poincare-Birkhoff-Witt
property using the theory of noncommutative Groebner bases. We include
applications to the case of abelian groups and the case of groups acting on
coordinate rings of quantum planes. In addition, we classify graded
automorphisms of the coordinate ring of quantum 3-space. In characteristic
zero, Hochschild cohomology gives an elegant description of the
Poincare-Birkhoff-Witt conditions.Comment: 29 pages. Last example corrected; some indices in the last theorem
were accidentally transposed and now appear in correct orde
Properties of Lyubeznik numbers under localization and polarization
We exhibit a global bound for the Lyubeznik numbers of a ring of prime
characteristic. In addition, we show that for a monomial ideal, the Lyubeznik
numbers of the quotient rings of its radical and its polarization are the same.
Furthermore, we present examples that show striking behavior of the Lyubeznik
numbers under localization. We also show related results for generalizations of
the Lyubeznik numbers.Comment: 17 page
The realization of input-output maps using bialgebras
The theory of bialgebras is used to prove a state space realization theorem for input/output maps of dynamical systems. This approach allows for the consideration of the classical results of Fliess and more recent results on realizations involving families of trees. Two examples of applications of the theorum are given
An explicit KO-degree map and applications
The goal of this note is to study the analog in unstable -homotopy theory of the unit map from the motivic sphere spectrum to the
Hermitian K-theory spectrum, i.e., the degree map in Hermitian K-theory. We
show that "Suslin matrices", which are explicit maps from odd dimensional split
smooth affine quadrics to geometric models of the spaces appearing in Bott
periodicity in Hermitian K-theory, stabilize in a suitable sense to the unit
map. As applications, we deduce that for ,
which can be thought of as an extension of Matsumoto's celebrated theorem
describing of a field. These results provide the first step in a program
aimed at computing the sheaf for .Comment: 36 Pages, Final version, to appear Journal of Topolog
Gr\"obner-Shirshov bases for Lie algebras over a commutative algebra
In this paper we establish a Gr\"{o}bner-Shirshov bases theory for Lie
algebras over commutative rings. As applications we give some new examples of
special Lie algebras (those embeddable in associative algebras over the same
ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963)
\cite{Conh}). In particular, Cohn's Lie algebras over the characteristic
are non-special when . We present an algorithm that one can check
for any , whether Cohn's Lie algebras is non-special. Also we prove that any
finitely or countably generated Lie algebra is embeddable in a two-generated
Lie algebra
- …