Modules with pure resolutions

ABSTRACT

Let R a standard graded algebra over a field k. In this paper, we give a relation in terms of graded Betti numbers, called the Herzog–Kühl equations, for a pure R-module M to satisfy the condition dim(R)−depth(R) = dim(M)−depth(M). When R is Cohen–Macaulay, we prove an analogous result characterizing all graded Cohen–Macaulay R-modules of finite projective dimension. Finally, as an application, we show that the property of R being Cohen–Macaulay is characterized by the existence of pure Cohen–Macaulay R-modules corresponding to any degree sequence of length at most depth(R).

KEYWORDS:

• Cohen–Macaulay defect
• graded Betti numbers
• Herzog–Kühl equations
• pure modules

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13A02
• 13C14
• 13D02