On Rees algebras of ideals and modules over hypersurface rings

Abstract

The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of hypersurface rings and produce a minimal generating set for their defining ideals. Then, using generic Bourbaki ideals, we study Rees algebras of modules with projective dimension one over hypersurface rings. We describe the defining ideal of such algebras and determine Cohen-Macaulayness and other invariants.

The equations of Rees algebras of height three Gorenstein ideals in hypersurface rings

“…The search for a set of minimal generators of J , the defining equations of R(I), has become a fundamental problem and has been studied to great extent in recent years (see e.g. [25,19,24,29,16,30,9,15,5,18,3,32]).…”

“…These ideals and their Rees algebras have been studied under a multitude of various assumptions (see e.g. [24,25,1,5,16,22,23,32]). Furthermore, perfect Gorenstein ideals of grade three and their Rees algebras have been a topic of great interest in recent years.…”

“…As the Rees ring is the algebraic realization of the blowup of Spec(R) along V (I), altering the ring is reflected by the blowup of a different scheme. There has been recent success in the way of determining the equations defining Rees algebras of perfect ideals with grade two in hypersurface rings in [32]. Expanding upon this, we consider perfect Gorenstein ideals of grade three in these rings and study the defining equations of their Rees algebras.…”

“…However, in the setting above, the Jacobian dual is insufficient and such a matrix must be altered. Repeating the construction in [32], we introduce a modified Jacobian dual matrix. A recursive algorithm of gcd-iterations is then developed in order to produce the equations of J .…”

“…A recursive algorithm of gcd-iterations is then developed in order to produce the equations of J . This iterative procedure is similar to the methods used in [1,5,32].…”