Patching and admissibility over two-dimensional complete local domains

Abstract

We develop a patching machinery over the field E = K((X, Y )) of formal power series in two variables over an infinite field K. We apply this machinery to prove that if K is separably closed and G is a finite group of order not divisible by char(E), then there exists a G-crossed product algebra with center E if and only if the Sylow subgroups of G are abelian of rank at most 2. MR Classification: 12E30, 16S35 Introduction Complete local domains play an important role in commutative algebra and algebraic geometry, and their algebraic properties were already described in 1946 by Cohen’s The second author was partially supported by the Israel Science Foundation (grant No. 343/07), and by an ERC grant. structure theorem. The Galois theoretic properties of their quotient fields were extensively studied over the past two decades. The pioneering work in this line of research is due to Harbater [Ha87], who introduced the method of patching to prove that if R is a complete local domain with quotient field K, then every finite group occurs as a Galois group over K(x). This result was strengthened by Pop [Po96], and in a different language, by Haran-Jarden [HaJ98], who showed that if moreover R is of dimension 1, then every finite split embedding problem over K(x) is solvable. The first step towards higher dimension was made by Harbater-Stevenson [HaS05], who essentially showed that if R is a complete local domain of dimension 2, then every finite split embedding problem over Quot(R) has |R| independent solutions. That is, the absolute Galois group of Quot(R) is semi-free of rank |R| (see [BHH10] for details on this notion). This result was later generalized by Pop [Po10] and by the second author [Pa10], who showed that if K is the quotient field of a complete local domain of dimension exceeding 1, then Gal(K) is semi-free. Despite the major progress made in the study of Galois theory over these fields, little is known about the structure of division algebras over them. A step in that direction was recently made by Harbater-Hartmann-Krashen [HHK10]. In that work, the authors consider a question relating Galois theory and Brauer theory over a field E – which groups are admissible over E ? That is, which finite groups occur as a Galois group of an adequate Galois extension F/E (recall that an extension F/E is called adequate if F is a maximal subfield in an E-central division algebra). Equivalently, for which groups G there is a G-crossed product division algebra with center E. Note that for E as above and a finite extension F/E, the above maximality requirement can be omitted since any F which is a subfield of an E-division algebra is also a maximal subfield of some E-division algebra (see Remark 3.9). This question was first considered by Schacher over global fields. In [Sch68], Schacher proved that any Q-admissible group has metacyclic Sylow subgroups and conjectured the converse. Admissibility was studied extensively over global fields (for example: [Ste82], [Son83], [SS92], [Lid96], [Fei04], [Nef10]), function fields and fields of Laurent series (for example: [FSS92], [FS95a], [FS95b]).