arXiv subjects

Raymond T. Hoobler

Publications and source records attributed to Raymond T. Hoobler.

4 recordsLinked to original sources

arXiv2014

The Witt Ring of a Curve with Good Reduction over a Non-dyadic Local Field

In this work, we present a generalization to varieties and sheaves of the fundamental ideal of the Witt ring of a field by defining a sheaf of fundamental ideals $\tilde{I}$ and a sheaf of Witt rings $\tilde{W}$ in the obvious way. The Milnor conjecture then relates the associated graded of $\tilde{W}$ to Milnor K-theory and so allows the classical invariants of a bilinear space over a field to be extended to our setting using étale cohomology. As an application of these results, we calculate the Witt ring of a smooth curve with good reduction over a non-dyadic local field.

math.AG↗

arXiv2012

The Witt Ring of a Smooth Projective Curve over a Finite Field

In this paper we calculate the Witt ring W(C) of a smooth geometrically connected projective curve C over a finite field of characteristic different from 2. We view W(C) as a subring of W(k(C)) where k(C) is the function field of C. We show that the triviality of the Clifford algebra of a bilinear space over C gives the main relation. The calculation is then completed using classical results for bilinear spaces over fields.

math.AG↗

arXiv2010

The Differential Brauer Group

Let $A$ be a ring equipped with a derivation $δ$. We study differential Azumaya $A$ algebras, that is, Azumaya $A$ algebras equipped with a derivation that extends $δ$. We calculate the differential automorphism group of the trivial differential algebra, $M_{n}(A)$ with coordinatewise differentiation. We introduce the $δ$-flat Grothendieck topology to show that any differential Azumaya $A$ algebra is locally isomorphic to a trivial one and then construct, as in the non-differential setting, the embedding of the differential Brauer group into $H^{2}(A_{δ-pl},G_{m,δ}) $. We conclude by showing that the differential Brauer group coincides with the usual Brauer group in the affine setting.

math.AG↗

arXiv2005

The Merkuriev-Suslin theorem for any semi-local ring

We introduce here a method which uses etale neighborhoods to extend results from smooth semi-local rings to arbitrary semi-local rings A by passing to the henselization of a smooth presentation of A. The technique is used to show that etale cohomology of A agrees with Galois cohomology, the Merkuriev-Suslin theorem holds for A, and to describe torsion in K_2(A).

math.AG↗