Localization: On Division Rings and Tilting Modules Javier S´anchez Serd`a Mem`oriapresentada per a aspirar al grau de doctor en Matem`atiques Certifico que la present Mem`oria ha estat realitzada per en Javier S´anchez Serd`a, sota la direcci´ode la Dra. Dolors Herbera Espinal. Bellaterra, Juny 2008 Firmat: Dra. Dolors Herbera Espinal “And here we are again the door is closed behind us and the long road lies ahead where do we go from here?” Fates Warning, Leave the past behind Introduction Let R be a ring. Suppose that R embeds in a division ring E (i.e. E is a not necessarily commutative field). By the division ring of fractions of R inside E we mean the embedding R,→ E(R) where E(R) denotes the intersection of all subdivision rings of E that contain R. When R is a commutative ring, the answer to whether R is embeddable in a division ring is well known: (a) Existence: R has a division ring of fractions Q(R) if and only if R is a domain, i.e. R is a nonzero ring such that xy = 0 implies that x = 0 or y = 0. (b) Uniqueness: If λ: R,→ Q(R) is a division ring of fractions of R, given any embedding ψ : R,→ E in a division ring E, there exists a morphism of rings ψ¯: Q(R) → E such that ψλ¯ = ψ. In particular E(R) =∼ Q(R). (c) Form of the elements: The division ring of fractions Q(R) is constructed in an analogous way as the rationals from the integers. The elements of Q(R) are of the form s−1r for some r ∈ R and s ∈ R \{0}. Moreover, we have a rule to decide when two fractions −1 −1 s1 r1, s2 r2 represent the same element of E (iff s1r2 = s2r1). This situation extends to the non-commutative setting provided that R is a left (right) Ore domain [Ore31], but in general the picture is not like that. Certainly, being a domain is a necessary condition for the ring R to be embeddable in a division ring, but it is not sufficient as A.I. Mal’cev showed [Mal37]. Necessary and sufficient conditions can be found in [Coh95], but they are difficult to verify for a given domain R and, in addition, the proof of the existence of the division ring is not constructive. There are also many (non-Ore) domains R with more than one division ring of frac- tions [Fis71] (or see also Chapter 7). For example, for any field k, the free k-algebra khXi on a set X of cardinality at least two. Furthermore, if R,→ D is a division ring of fractions of R, the elements of D can be built up from the elements of R in stages, using addition, subtraction, multiplication, and division by nonzero elements. It can happen that the subset {s−1r ∈ D | r ∈ R, s ∈ R \{0}} is not a division ring, and it may not be possible to simplify expressions like uv−1w + xy−1z or −1 −1 −1 (w2 − x2(w1 − x1y1 z1) z2) . This leads to the concept of inversion height. The inversion height of a rational expression is the maximum number of nested inversions that occur in it. −1 −1 −1 −1 −1 For example, the inversion height of uv w+xy z is one and of (w2−x2(w1−x1y1 z1) z2) is three. The inversion height of an element of D is the least inversion height of the rational expressions of elements of R that represent f. Observe that the inversion height of an element is not so easy to compute. It depends of course on the nature of the ring R, but there are also rational identities. To illustrate this point, consider elements x, y ∈ R. At first sight the element f = (x−1 + (y−1 − x)−1)−1 ∈ D seems to be of inversion height 3. However, by Hua’s identity, f = x − xyx, and therefore is of inversion height zero. The inversion height of a division ring of fractions R,→ D is the supremum of the inversion height of the elements of D (it may be infinite). Notice that if R is a left (right) Ore domain, R,→ D is of inversion height 1. ii Introduction Therefore, description of division rings of fractions turns out to be a difficult task. About two thirds of this work is concerned with the problem of the uniqueness of division rings of fractions and the form in which the elements of a division ring of fractions can be expressed. We proceed to outline our main results in this direction. Let G be a group and k a division ring. One can construct a crossed product group ring kG, for example the usual group ring k[G]. We deal with the problem of embedding kG in division rings when G is a locally indicable group, that is, a group such that every nontrivial finitely generated subgroup H of G has a normal subgroup N with H/N infinite cyclic. These groups form a large class: locally free groups, torsion-free abelian groups, orderable groups, torsion-free one-relator groups and extensions of groups in these classes are locally indicable groups. Observe that if 1 6= H ≤ G is finitely generated and the class of t generates H/N, then the powers of t are kN-linearly independent. An embedding kG ,→ D, with D a division ring, is Hughes-free if the powers of t are D(kN)-linearly independent for all 1 6= H ≤ G finitely generated, N C H such that H/N is infinite cyclic and t such that the coset of t generates H/N, that is, we can extend the kN-linear independence of t to the division ring D(kN) generated by kN inside D. An important example of Hughes-free embedding goes back to A.I. Mal’cev [Mal48] and B.H. Neumann [Neu49a]. They independently showed: for each ordered group (G, <), division ring k and crossed product group ring kG, the embedding kG ,→ k((G, <)) is a Hughes-free embedding, where k((G, <)) denotes the Mal’cev-Neumann series ring, i.e. X k((G, <)) = {γ = agg | supp γ is well-ordered}. g∈G In particular, this holds for (locally) free groups. A locally indicable group G is Hughes-free embeddable if kG has a Hughes-free division ring of fractions for every division ring k and every crossed product group ring kG. Orderable groups are examples of Hughes-free embeddable groups. We give new proofs of two results by I. Hughes, [Hug70] and [Hug72]. More precisely, we show: Hughes’ Theorem I. Let k be a division ring and G a locally indicable group. If kG ,→ D1 and kG ,→ D2 are two Hughes-free division ring of fractions, then there exists a (unique) isomorphism ϕ: D1 → D2 D1 o7 * ooo kG t ϕ OO OO' D2 making the diagram commutative. Hughes’ Theorem II. Suppose that G is a locally indicable group with a normal subgroup L such that G/L is locally indicable. If both L and G/L are Hughes-free embeddable, then G is Hughes-free embeddable. The proof of Hughes’ Theorem I and the machinery developed to show both results is a joint work with W. Dicks and D. Herbera [DHS04]. Hughes’ Theorem I has played an important role in the study of division rings of fractions of the free algebra and of crossed product group rings of free groups. It has been used by J. Lewin in [Lew74] to describe the universal division ring of fractions of the free algebra and of the crossed product group ring of a free group as the division ring of fractions of kG inside the Mal’cev-Neumann series rings. Also, P. Linnell [Lin00], [Lin93] made use of it to prove that the division ring of fractions that arises from group von Neumann algebras of free groups is the same as the universal division iii ring of fractions of the group ring of a free group. Because of these important results and the fact that, in words of P. Linnell [Lin06, p. 52], “the proof given by Hughes in [Hug70] is extremely condensed, and ... it is difficult to follow”, a new proof was needed. We also give an easier proof of the aforementioned result by J. Lewin following closely the work by C. Reutenauer [Reu99]. Hughes’ Theorem II gives a large supply of Hughes-free embeddings and partial positive answers to Mal’cev problem of whether the group ring of a right orderable group is embeddable in a division ring [MMC83, Question 1.6]. Moreover, it allows us to define a division ring coproduct, and it is also useful for computing the inversion height of some embeddings of rings in division rings. In this direction, we close a conjecture by B.H. Neumann [Neu49a, p. 215]. More precisely, we prove Theorem. Let G be a free group on a set X of at least two elements and k a field. Let < be a total order on G such that (G, <) is an ordered group. Then the embedding kG ,→ k((G, <)) is of infinite inversion height. This result was showed by C. Reutenauer [Reu96] for X an infinite set. Indeed, he proved that the inversion height of the entries of the inverse of an n × n generic matrix (i.e. a matrix of the form (xij) where the xij’s are distinct noncommuting variables in X) equals n with respect to khXi ,→ k((G, <)).