Rees Algebras Of Ideals Research Articles

On Rees algebras of ideals and modules over hypersurface rings

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.

On the Betti numbers and Rees algebras of ideals with linear powers

An ideal I subset mathbb {k}[x_1, ldots , x_n] is said to have linear powers if I^k has a linear minimal free resolution, for all integers k>0. In this paper, we study the Betti numbers of I^k, for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for d=2, 3 or n-1, and the product of all ideals generated by s variables, for s=n-1 or n-2. We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

An intrinsic definition of the Rees algebra of a module

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

Arithmetic Macaulayfication of projective schemes

In this paper, we study arithmetic Macaulayfication of projective schemes and Rees algebras of ideals. We discuss the existence of an arithmetic Macaulayfication for projective schemes. We give a simple neccesary and sufficient condition for nonsingular projective varieties to possess an arithmetic Macaulayfication (Theorem 1.5). We also show that this condition is sufficient in general, but give examples to show that it is not in general necessary. We further consider Rees algebras R λ ( I ) = R [ I λ t ] (truncated Rees algebras) associated to a homogeneous ideal I and show that they are Cohen–Macaulay for large λ in some important cases (Theorem 2.1 and Corollary 2.2.1).

Rees algebras of ideals with low codimension

For certain grade two perfect ideals, there is an expected description of the equations of the Rees algebra. In this paper, the Cohen–Macaulayness of the Rees algebra, numerical invariants of the ideal, and a condition on the minors of a presentation matrix of the ideal are shown to be related to the equations having this form.

Rees Algebras of Ideals Having Small Analytic Deviation

Rees Algebras of Ideals Having Small Analytic Deviation

Rees algebras of ideals having small analytic deviation

In this article we identify two large families of ideals of a Cohen-Macaulay (sometimes Gorenstein) local ring whose Rees algebras are Cohen-Macaulay. Our main results imply, for example, that if ( R , M ) (R,M) is a regular local ring and P P is a prime ideal of R R such that P n {P^n} is unmixed for all n ≥ 1 n \geq 1 , then the Rees algebra R [ P t ] R[Pt] is Cohen-Macaulay if either dim ⁡ ( R / P ) = 2 \dim (R/P) = 2 , or dim ⁡ ( R / P ) = 3 , R / P \dim (R/P) = 3,R/P is Cohen-Macaulay, and R / P R/P is integrally closed.