Resumen de Localization: on division rings and tilting modules

Javier Sánchez Serdá

• Let G be a locally indicable group, k a division ring, and kG a crossed-product group ring, In [Hug70], Ian Hughes proved that, up to kG-isomorphism, at most one division ring of fractions of kG satisfies a certain independence condition, now called Hughes freeness. This result was applied by others in work on division rings of fractions of group rings of free groups. We introduce concepts that illuminate Hughes' arguments, and we simplify the proof of the theorem.

  A group G is Hughes-free embeddable if for every division ring k, then every crossed-product group ring kG has a Hughes-free division ring of fractions. With the tools developed to prove the foregoing result, we prove [Hug72]. That is, the extension of Hughes-free embeddable groups is a Hughes-free embeddable group.

  In [Lew74] and [LL78] it is proved that the universal division ring of fractions of a crossed-product group ring of a division ring k over a free group G is the division ring of fractions of kG inside any of its Mal'cev-Neumann series ring. A simpler proof of this fact was given by [Reu99] in a less general situation. We extend Reutenauer's proof to show the result by J. Lewin and T. Lewin in its full generality.

  Let R be a ring with a division ring of fractions D. The inversion height of D is the number of nested inversions needed to express the elements of D from elements of R.Let X be a set of at least two elements and k a field. We show that the so-called JF-embeddings of the free algebra k in division rings D have inversion height at most two, and give examples of inversion height one and two for any X and k. We use these embeddings to obtain embeddings of the free group algebra in a division ring of inversion height one and two.In [Neu49] it was conjectured that if G is a free group on a set X of at least two elements and k is a field, then the embedding of the .group ring k[G] inside the Mal'cev-Neumann series ring is infinite. In [Reu96] this conjecture is proved for X an infinite set. We prove that this conjecture also holds for X a finite set of at least two elements by reducing the problem to the situation proved in [Reu96].

  We show that if f:R2>S is an injective ring epimorphism such that Tor_1AR(S,S)=0 and the projective dimension of S as a right R-module is lesser or equal than one, then the right R-module S+S/R (direct sum of S and S/R) is a tilting right R-module. We then study the case where f is a universal localization in the sense of Schofield [Sch85]. Using results from [CB91], we give applications to tame hereditary algebras and hereditary noetherian prime rings. In particular, we show that every tilting module over a Dedekind domain or over a classical maximal order arises in this way.

  References [CB91] W. W. Crawley-Boevey, Regular modules for tame hereditary algebras, Proc. London Math. Soc. (3) 62 (1991), no. 3, 490- 508.

  [Hug70] Ian Hughes, Division rings of fractions for group rings, Comm. Pure Appl. Math. 23 (1970), 181-188.

  [Hug72] Ian Hughes, Division rings of fractions for group rings II, Comm..Pure App|. Math. 25 (1972), 1-9.

  [Lew74] Jacques Lewin, Fields of fractions for group algebras of free groups, Trans. Amer. Math. Soc. 192 (1974), 339-346.

  [LL78] Jacques Lewin and Tekla Lewin, An embedding of the group algebra of torsion-free one-relator group in a field, J. Algebra 52 (1978), no. 1, 39-74.

  [Neu49] B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202-252.

  [Reu96] Christophe Reutenauer, Inversion height in free fields, Selecta Math. (N.S.) 2 (1996), no. 1, 93-109.

  [Reu99] Christophe Reutenauer, Malcev-Neumann series and the free field, Exposition. Math. 17 (1999), no. 5, 469-478.

  [Sch85] A. H. Schofield, Representation of rings over skew fields, London Mathematical Society Lecture Note Series, vol. 92, Cambridge University Press, Cambridge, 1985