Shmuel Rosset

A new proof of the Amitsur-Levitski identity

We give a simple proof of the Amitsur-Levitzki identity by analysing the powers of matrices with “differential 1-forms” as entries. Using the fact that 2-forms are central the identity is seen to follow from the Cayley-Hamilton theorem.

Elements of trace zero that are not commutators

In 1936 Shoda proved (see [B]) that over a field k of characteristic zero a matrix of trace zero is a commutator ab ba in the full matrix algebra hln(k). (Henceforth the commutator will be denoted [a, b ] ) . Shodas theorem was extended to fields of positive characteristic by Albert and Nluckenhoupt in 1957 (see [I]) . It has recently been proved (by Amitsur and Rowen [2], and also but later by us (71) that in a central simple algebra an element of (reduced) trace zero is always a sum of two commutators. But it is not known whether Shodas theorem holds in complete generality, i.e. whether it is true that in a central simple algebra an element of trace zero is a commutator. It turns out that even over a commutative ring it is not easy to find a matrix of trace zero which is not a commutator. This is explainable, perhaps, by the fact, shown in the thesis of the first named author (see [6]), that a matrix of trace zero in Mn(R), where R is any commutative ring, is always a sum of two commutators. Here we give two examples of elements of trace zero that are not commutators. The first, in a full matrix algebra, generalizes an example of the above-mentioned thesis, and is a family of examples. We prove that if in a commutative ring R there are elements f , g , h that satisfy certain conditions (see lemma 1.1) then the matrix

Finite index and finite codimension

Abstract An old idea of M. Hall on finitely generated subgroups of free groups is developed. We show that it implies that such subgroups have “roots” which are normalizers of certain other subgroups. Similarly in free algebras or group rings of free groups over a field every finitely generated right ideal has a root, which is the unique maximal subalgebra that contains the ideal as an ideal of finite codimension. In analogy to the group case, it is an “idealizer” of another, related, ideal. We also define the “Hall index” of a subgroup of a free group and relate it to Howsons theorem.

GROWTH AND UNIQUENESS OF RANK

We prove that algebras of sub-exponential growth and, more generally, rings with a sub-exponential “growth structure” have the unique rank property. In the opposite direction the proof shows that if the rank is not unique one gets lower bounds on the exponent of growth. Fixing the growth exponent it shows that an isomorphism between free modules of greatly differing ranks can only be implemented by matrices with entries of logarithmically proportional high degrees.

An invariant of free central extensions

If G is a free abelian finitely generated group, the ‘most-general’ 2-cocycle on G with trivial action is the map G × G →f Λ2G, defined as follows. Let e1,…,en be a basis of G and f the bilinear map satisfying f(ei, ej) = ei ∧ ej if i < j and =0 if i ≥ j. We show that KαG has global dimension 1 where K is the field of fractions of the group ring C[Λ2G] and α ϵ H2(G, K∗) is represented by the map above. More generally for every finitely generated group we define an invariant ξ(G) and gl.dim(KαG) = 1 is equivalent to ξ(G) = 1 for G free abelian. We also show that if G1,…, Gn are non-commutative free groups, then ξ(G1 × ∣ × Gn) = n. In general, if G is not a torsion group, 1 ≤ ξ(G) ≤ cdC(G).

Some non-finitely presented Lie algebras

Let L be a free Lie algebra over a field k, I a non-trivial proper ideal of L, n > 1 an integer. The multiplicator H2(LIn, k) of LIn is not finitely generated, and so in particular, LIn is not finitely presented, even when LI is finite dimensional.

The higher lower central series

The Baer invariants Γn(G) of a group are central extensions of the elementsγn(G) of the lower central series. We show that the inclusionsγn+1 ⊂γn can be lifted to functor morphisms Γn+1→Γn and a canonical Lie algebra, analogous to Lazard’s Lie algebra, can be constructed which is explicitly computable. This is applied in various ways.

The Euler-Goldie rank of some virtually polycyclic groups

Let k be skew field and f a virtually polycyclic (VP) group, i.e., r contains a polycyclic subgroup of finite index. The Euler characteristic of kT modules is defined as follows. Let F be a poly-infinite-cyclic (poly-Z) subgroup of finite index in IY There are many such subgroups (unless Iis finite, in which case there is only one). One knows (see [6]) that kT’ is a regular ring, it even has a finite global dimension, and that kZ-’ projective modules are stably free. Since kF (and, of course, kr) is Noetherian it follows that every finitely generated (Eg.) kr’ module has a finite free resolution, i.e., a finite resolution by finitely generated free kr modules. In particular if M is an f.g. kT module let

Remarks on a principle of localization

We prove that the Brauer class of a crossed product is a sum of symbols iff its “local” components are. Analogously we show that a solution of the “Goldie rank conjecture” would follow from the “local” statements; an extension of a result of Cliff-Sehgal is an easy corollary.

Locally inner automorphisms of algebras

The connection between automorphisms of Azumaya algebras and the Picard group of the center has been noticed by Rosenberg-Zelinsky (RZ) [8], following a remark by Auslander-Goldman [l]. This connection has been generalized, using the Morita context, first by Bass [4] and more recently by Frohlich [5]. In my thesis I suggested another way of proving the result of [8]. This is done rather mechanically, by passing to the spectrum of the center of the given Azumaya algebra and taking the exact cohomology sequence resulting from an exact sequence of sheaves of units. This is described, in more detail, further on. However, this way of proving the RZ theorem shows two things: