On Gauss sums over dedekind domains

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Arithmetic modules over generalized Dedekind domains

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

Chevalley groups of polynomial rings over Dedekind domains

Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank {\geq 2}. We prove thatG(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])\quad\text{for any}\ n% \geq 1.In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for {G=\mathrm{SL}_{N},\mathrm{Sp}_{2N}}. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.