Graded Transcendental Extensions of Graded Fields

We study transcendency properties for graded field extension and give an application to valued field extensions. 1. Introduction. An important tool to study rings with valuation is the so-called associated graded ring construction: to a valuation ring R, we can associate a ring gr(R) graded by the valuation group. This ring is often easier to study, and one tries to lift properties back from gr(R) to R. This principle has been recently applied to rings of differential operators (see [9]), the Brauer group (see, e.g., [8]), and to valuations on division algebras (see [1, 11, 12]). This has been one of the motivations to study graded rings, see [10] for a detailed discussion. In a sense, the easiest example of a graded ring is a graded field, this is a commutative graded ring in which every homogeneous element is invertible, and the terminology has been introduced in [13]. This note is a continuation of earlier work of the author (see [3, 4, 5, 6]), in which graded fields and graded division rings are studied with special emphasis on applications to valuation theory. The aim of this note is to introduce and study the notion of gr-transcendental graded field extension, at least in the case where the grading group is torsion-free abelian; application to valued field extensions leads to three different notions of transcendental extensions of valued fields. In Section 2, we recall some basic results on graded ring theory and on grad-ings on polynomial rings. We introduce the notions of gr-algebraically freeness and gr-transcendental extension in Section 3 and prove some elementary properties (see, e.g., Proposition 3.4). In Section 4, we look at two special cases: unramified graded field extensions, where the grading groups of both graded fields are the same, and totally ramified extensions, where the parts of degree zero of both extensions coincide. The transcendency can be described explicitly in both cases; combination of the two situations leads to the existence of a gr-transcendency basis in general (Proposition 4.5) and to the notion of gr-transcendency degree. In Section 5, we give a structure theorem for purely gr-transcendental graded field extensions of divisible type (Proposition 5.1);