On the Non-Cohen–Macaulayness of Certain Factorial Closures

Abstract

Let R = K[x, y, z] denote the polynomial ring in three variables over an arbitrary field K. We study the factorial closure B(E) of certain R-modules E of projective dimension 1, called monomial modules. By an explicit computation, we derive an example of an algebra which is a factorial non-Cohen–Macaulay ring if the characteristic of the basic field char k is two. To this end, we study examples of monomial modules E such that the factorial closure B(E) is generated by elements of degree at most 3 over the symmetric algebra Sym(E). In order to do that it will be necessary to understand—at least partially—the third component B(E)₃ of the factorial closure. This work continues the investigations of Imtiaz and Schenzel [7] were the case of monomial modules was described such that the factorial closure B(E) is generated in degrees at most two.

Key Words:

• Factorial closure
• Monomial curve
• Monomial module
• Symmetric algebra

2010 Mathematics Subject Classification:

• Primary: 13A30
• Secondary: 13C14

Notes

Communicated by I. Swanson.