Homomorphisms and duality for torsion-free modules (Q2738741)

The title of this article is intentionally reminiscent of \textit{R. B. Warfield}, jun.'s 1968 paper ``Homomorphisms and duality for torsion-free Abelian groups'' [Math. Z. 107, 189-200 (1968; Zbl 0169.03602)]. In his ground breaking paper, Warfield used \(\Hom\) and tensor functors to determine when a group was isomorphic to certain \(\Hom\) and tensor groups, thus proving a type of representation theorem for finite rank torsion-free Abelian groups. The present article surveys newer techniques which better accommodate generalizations of Warfield's work to modules over integral domains. Olberding also attempts to fill in some of the gaps in the theory that still remain. The rationale for doing this is to reveal more about the structure of torsion-free modules over integral domains, and to explore interesting and canonical classes of integral domains in these settings. Examples include 2-generator rings, stable rings, generalized Dedekind domains, strongly discrete Prüfer domains, and almost maximal Prüfer domains.NEWLINENEWLINENEWLINEMore specifically, this article explores in what settings a torsion-free \(R\)-module \(G\) might be isomorphic to NEWLINE\[NEWLINE\Hom_R(\Hom_R(G,X),X),\qquad X\otimes_{R(X)}\Hom_{R(X)}(X,G),\quad\text{or}\quad\Hom_{R(X)}(X,X\otimes_{R(X)}G)NEWLINE\]NEWLINE where \(X\) is a rank one module, and when \(\bigwedge^kG\) is isomorphic to \(\Hom_R(\bigwedge^{n-k}G,\bigwedge^nG)\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].