For a local field of characteristic p > 0, K, the combination of local class field theory and Artin-Schreier-Witt theory yield what is known as the Schmid-Witt symbol. The symbol encodes interesting data about the ramification theory of p-extensions of K and we can, for example, use it to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula f...

This note corrects an error in the char(K) = 2 case of the author’s computation of the bordism groups of K-Witt spaces for the field K. A complete computation is provided for the unoriented bordism group. For the oriented bordism groups, a nearly complete computation is provided as well a discussion of the difficulty of resolving a remaining ambiguity in dimensions equivalent to 2 mod 4. Recall...

we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

We give a concrete description of the category of étale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for certain analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of comm...

We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples. We show that any such generalized Witt algebra is a semisimple, indecomposable Lie algebra which does not contain any abelian Lie subalgebras of dimension greater than one. We develop an inva...

We present algorithms for square classes, quadratic forms and Witt classes of quadratic forms over the eld of rational functions of one variable over the reals. The algorithms are capable of: nding the unique representative of a square class, deciding if a given function is a square or a sum of squares and deciding if a quadratic form is isotropic or hyperbolic. Moreover we propose a representa...

Peter D. Witt, MD; Daniel T. Cohen, BA; Susan B. Mallory, MD; from the Departments of Surgery, Plastic and Reconstructive (Drs Witt and Mr Cohen), Internal Medicine (Dermatology) (Dr Mallory), and Pediatrics (Dr Mallory), Washington University School of Medicine, St Louis Children’s Hospital, and the Department of Plastic and Reconstructive Surgery, Shriners Hospital for Children (Dr Witt), St ...

Valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields, see, for example [BR]. In [MW], the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting. While it seems desirable to extend these results further, the techniques used for rings...

The Grothendieckand Wittring of orthogonal representations of a finite group is defined and studied. The main application (only indicated) is the reduction of the computation of Wall's various L-groups for a finite group n to those subgroups of n, which are a semidirect product of a cyclic group y of odd order with a 2-group /?, such that any element in ft acts on y either by the identity or by...

We find that a compatible graded left-symmetric algebra structure on the Witt algebra induces an indecomposable module of the Witt algebra with 1-dimensional weight spaces by its left multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebra structures on the Witt algebra are classified. All of them are simple and they in...