Two simple invariants of a module are its rank and the codimension of the set of primes at which it fails to be locally free. In general these invariants are unrelated as can be seen by taking direct sums of ideals. However in the geometric context of extending a locally free sheaf given over an open set to its closure the modules that arise are reflexive and this naive example fails.

Restricting to regular rings and localizing at the non–free set of primes leads to the following problem: for a regular local ring A determine the ranks of those non–free reflexive A–modules which are locally free except at the maximal ideal.

Denote by m the maximal ideal of A, by k its residue field A/m, and by X the spectrum of A punctured at m. Call the modules in question X–bundles. The possible ranks of non–free X–bundles have been determined only when the dimension d of A is at most 5, and for these dimensions there are indecomposable X–bundles of all ranks. In the first section I review briefly some of the methods used for constructing X–bundles and in the second describe an approach to the problem of finding restrictions on the ranks especially in terms of their cohomology. Here the Syzygy Theorem of Evans and Griffith has interesting consequences [1, 3].