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Arithmetic of linear algebraic groups over 2-dimensional geometric fields

Let K be the field of fractions of a two-dimensional, excellent, henselian, local domain with algebraically closed residue field of characteristic zero. In this introduction, we shall refer to such a field as a two-dimensional, strictly henselian field (indeed a henselian local ring with a separably closed residue field is called a strictly henselian local ring). An example of such a field is the field of fractions C((X,Y )) of the formal power series ring C[[X,Y ]]. In the paper [CTOP], Ojanguren and two of the authors investigated quadratic forms over such a field K, and established properties analogous to those familiar over number fields. An analogue of the local-global principle for the Brauer group (going back to work of M. Artin) was also established. In the present paper, we show that most well-known properties of linear algebraic groups and of their homogeneous spaces over totally imaginary number fields have counterparts for such groups over a two-dimensional strictly henselian field. To prove this, we recall (§1) that both types of fields share a number of properties : (i) their cohomological dimension is two ; (ii) index and exponent of central simple algebras coincide ; (iii) for any semisimple simply connected group G over K, we have H(K,G) = 1 ; (iv) on fields of either type, there is a natural set of rank one discrete valuations (i.e. with values in Z), with respect to which one can take completions and then investigate such natural questions as the local-global principle and weak approximation for homogeneous spaces of linear algebraic groups. We recall how properties (i) and (ii) are enough to deduce (iii) for groups without E8-factors. Properties (i), (ii) and (iii) are satified by other fields of interest, such as the Laurent series field l((t)) over a field l (of characteristic zero) of cohomological dimension one (such fields occur as completions of two-dimensional strictly henselian fields). For function fields in two variables over an algebraically closed field of characteristic zero, (i) holds and (ii) is known for algebras of 2-primary or 3-primary index. In §2, we show that if a field K satifies properties (i) and (ii), then for the standard isogeny 1→ μ→ G → G → 1