Cohen-Macaulay fiber cones and defining ideal of Rees algebras of modules

Abstract

Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos in [41] to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the Cohen-Macaulay property of fiber cones of modules and to study the defining ideal of Rees algebras. This is possible as long as the Rees algebra of a given module E is a deformation of the Rees algebra of a generic Bourbaki ideal I of E. Our main technical result provides a deformation condition that in fact extends the applicability of generic Bourbaki ideals to situations not covered in [41].[51] M. Weaver, Rees algebras of ideals and modules over hypersurface and complete intersection rings, Ph.D. thesis, Purdue University, in preparation.

On Rees algebras of ideals and modules over hypersurface rings

“…We show that E admits a generic Bourbaki ideal I which is perfect of grade 2 in a hypersurface ring. Following the path laid out in [6] we show that, after a generic extension, R(E) is a deformation of R(I) and the defining ideals of these Rees rings are of a similar form. We then derive the defining equations of R(E) from those of R(I) which are known by Theorem 1.1.…”

“…As the connection between almost linear presentation within polynomial rings and linear presentation within hypersurface rings has been established for ideals, we extend this to modules by following the approach of Section 5 of [6]. We take all conventions and notation from [29] restated in Section 2 along with the construction of the generic Bourbaki ideal.…”

“…However, much progress has been made in recent years (see e.g. [29,6,7,27,22,10]). Many of the difficulties have been greatly reduced with the innovation of generic Bourbaki ideals.…”

“…With this, the structure of the defining ideal of R(E) can be deduced from the defining ideal of R(I) which is known from [24]. In [6] Costantini extended this by considering a module E of projective dimension one with almost linear presentation. It was shown that the module E admits a generic Bourbaki ideal I which is almost linearly presented as well and there is a similar relation between R(E) and R(I).…”