Let [Formula: see text] be a finite-dimensional representation of the group [Formula: see text] of [Formula: see text] matrices with complex coefficients and determinant one. Let [Formula: see text] be the algebra of [Formula: see text]-invariant polynomials on [Formula: see text]. We present a calculation of the Hilbert series [Formula: see text] as well as formulas for the first four coefficients of the Laurent expansion of [Formula: see text] at [Formula: see text].

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Let V be a finite-dimensional representation of the complex circle C × determined by a weight vector a ∈ Z n . We study the Hilbert series Hilba(t) of the graded algebra C[V ] C × a of polynomial C × -invariants in terms of the weight vector a of the C × -action. In particular, we give explicit formulas for Hilba(t) as well as the first four coefficients of the Laurent expansion of Hilba(t) at t = 1. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomial that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of C[V ] C × a in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.

It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case the authors consider almost-free abelian groups and assume the existence of large canonical, free subgroups. Nevertheless, there exist ℵ 1 -separable torsion-free groups G of size ℵ 1 with a basic subgroup B of rank ℵ 1 such that all subgroups of G disjoint from B are also free, but the groups G are still not free. What else can we say about G? The other paper deals with Kaplansky's test problems (which are excellent indicators that the objects defy classification). The authors are able to construct very free abelian groups and verify the test problems for them by a careful choice of particular elements of their endomorphism rings.Accordingly, we want to investigate and construct ℵ n -free R-modules M (with n an arbitrary, but fixed natural number) over a domain R with End R M = R for the first time more systematically and uniformly. Recall that M is ℵ n -free if every subset of size < ℵ n is contained in a pure, free submodule of M. The requirement End R M = R implies that M is indecomposable, hence complicated. (We will also allow that End R M is a prescribed R-algebra, as in the title of this paper.)By now it is folklore to construct such modules M using additional set-theoretic axioms, most notably Jensen's ♦-principle. In this case the freeness condition can even be strengthened (see [6] and many examples in [9]). However, if we insist on proving this result in ordinary ZFC, then the known arguments fail: The classical constructions from the fundamental paper by Corner [2] do not apply because they are based on pure submodules of p-adic completions of free A-modules, which are never even ℵ 1 -free. If we use Shelah's Black Box instead of Jensen's ♦-principle, then the constructed modules M are still ℵ 1 -free, but always fail to be even ℵ 2 -free (see [4]). Thus we must develop new methods, which are presented for the first time in Sections 2 to 6, to achieve the desired result (Main Theorem 7.6). With these methods we provide a useful tool for a wide range of problems concerning ℵ n -free structures which can then be attacked.

Abstract. Let R be a commutative, indecomposable ring with identity and (P, ≤) a partially ordered set. Let F I(P ) denote the finitary incidence algebra of (P, ≤) over R. We will show that, in most cases, local automorphisms of F I(P ) are actually R-algebra automorphisms. In fact, the existence of local automorphisms which fail to be R-algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals. We will discuss local automorphisms of cartesian products as a special case in preparation of the general result.