Super-stretched and graded countable Cohen-Macaulay type

We define what it means for a Cohen-Macaulay ring to be super-stretched and show that Cohen-Macaulay rings of graded countable Cohen-Macaulay type are super-stretched. We use this result to show that rings of graded countable Cohen-Macaulay type, and positive dimension, have possible h-vectors (1), (1, n), or (1, n, 1). Further, one-dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In the non-Gorenstein case, rings of graded countable Cohen-Macaulay type of dimension larger than 2 are shown to be of minimal multiplicity. (C) 2013 Elsevier Inc. All rights reserved.