233,001 research outputs found
Boolean Witt vectors and an integral Edrei-Thoma theorem
A subtraction-free definition of the big Witt vector construction was
recently given by the first author. This allows one to define the big Witt
vectors of any semiring. Here we give an explicit combinatorial description of
the big Witt vectors of the Boolean semiring. We do the same for two variants
of the big Witt vector construction: the Schur Witt vectors and the -typical
Witt vectors. We use this to give an explicit description of the Schur Witt
vectors of the natural numbers, which can be viewed as the classification of
totally positive power series with integral coefficients, first obtained by
Davydov. We also determine the cardinalities of some Witt vector algebras with
entries in various concrete arithmetic semirings.Comment: Final version. To appear in Selecta Mat
Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras
In this paper we construct ternary -Virasoro-Witt algebras which
-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie
and Zachos using enveloping algebra techniques. The ternary
Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a
parameter and are not Nambu-Lie algebras for all but finitely many values of
this parameter. For the parameter values for which the ternary Virasoro-Witt
algebras are Nambu-Lie, the corresponding ternary -Virasoro-Witt algebras
constructed in this article are also Hom-Nambu-Lie because they are obtained
from the ternary Nambu-Lie algebras using the composition method. For other
parameter values this composition method does not yield Hom-Nambu Lie algebra
structure for -Virasoro-Witt algebras. We show however, using a different
construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and
Zachos, as well as the general ternary -Virasoro-Witt algebras we construct,
carry a structure of ternary Hom-Nambu-Lie algebra for all values of the
involved parameters
Witt groups of sheaves on topological spaces
This paper investigates the Witt groups of triangulated categories of sheaves
(of modules over a ring R in which 2 is invertible) equipped with
Poincare-Verdier duality. We consider two main cases, that of perfect complexes
of sheaves on locally compact Hausdorff spaces and that of cohomologically
constructible complexes of sheaves on polyhedra. We show that the Witt groups
of the latter form a generalised homology theory for polyhedra and continuous
maps. Under certain restrictions on the ring R, we identify the constructible
Witt groups of a finite simplicial complex with Ranicki's free symmetric
L-groups. Witt spaces are the natural class of spaces for which the rational
intersection homology groups have Poincare duality. When the ring R is the
rationals we show that every Witt space has a natural L-theory, or Witt,
orientation and we identify the constructible Witt groups with the 4-periodic
colimit of the bordism groups of Witt spaces. This allows us to interpret
Goresky and Macpherson's L-classes of singular spaces as stable homology
operations from the constructible Witt groups to rational homology.Comment: 38 pages, reformatted, minor corrections and changes as suggested by
referee. To appear in Commentarii Mathematici Helvetici no. 8
The basic geometry of Witt vectors, II: Spaces
This is an account of the algebraic geometry of Witt vectors and arithmetic
jet spaces. The usual, "p-typical" Witt vectors of p-adic schemes of finite
type are already reasonably well understood. The main point here is to
generalize this theory in two ways. We allow not just p-typical Witt vectors
but those taken with respect to any set of primes in any ring of integers in
any global field, for example. This includes the "big" Witt vectors. We also
allow not just p-adic schemes of finite type but arbitrary algebraic spaces
over the ring of integers in the global field. We give similar generalizations
of Buium's formal arithmetic jet functor, which is dual to the Witt functor. We
also give concrete geometric descriptions of Witt spaces and arithmetic jet
spaces and investigate whether a number of standard geometric properties are
preserved by these functors.Comment: Final versio
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