Jean-Jacques Sansuc

Principal homogeneous spaces under flasque tori: Applications

In this paper, we define flasque tori and flasque resolutions of tori over an arbitrary base scheme (Sect. 1) and we establish the basic cohomological properties of flasque tori over a regular scheme (Sect. 2). These properties are then used in a systematic and sometimes biased manner in the study of various problems, which will now be briefly listed. In Section 3, an alternative approach to R-equivalence upon tori [S] is given. Section 4 studies the behaviour of the first and second cohomology groups of arbitrary tori over a regular local ring, when going over to the fraction field. Applications to the representation of elements by norm forms and quadratic forms are described in Sections 5 and 6. In Sections 7 and 8, we study the behaviour of the group of sections of a torus, and of the first cohomology group of a group of multiplicative type when going over from a local ring to its residue class field, or when going over from a discretely valued field to its completion. We thus recover and generalize results of Saltman [30, 311 on the Grunwald-Wang theorem and its relation with the Noether problem [35]. In Section 9, Formanek’s description [17] of the centre of the generic division ring as the function field of a certain torus provides a different route to two results of Saltman [31, 321. Let us now give more details on the contents of this paper. If U is an open set of an integral regular scheme X, the restriction map 148 OO21-8693/87

Spectral Properties of Non-Selfadjoint Hill’s Operators with Smooth Potentials

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La

The aim of the present paper is to describe some spectral properties of Hill’s operators (1) with π-periodic complex-valued potentials q(x) belonging to the Sobolev space where n ≥ 0 is a fixed integer. The space W 0 2 is identified with ℒ 2[0, π], no restrictions being imposed at the boundary points 0 and π. A complete spectral parametrization of potentials q ∈ W 0 2 was given in our previous paper [8].