scholarly journals Hilbert Coefficients of Integrally Closed Ideals

1995 ◽  
Vol 176 (2) ◽  
pp. 638-652 ◽  
Author(s):  
S. Itoh
Keyword(s):  
Hilbert Coefficients ◽  
Integrally Closed ◽  
Integrally Closed Ideals

scholarly journals THE STRUCTURE OF THE SALLY MODULE OF INTEGRALLY CLOSED IDEALS

2016 ◽  
Vol 227 ◽  
pp. 49-76 ◽  
Author(s):  
KAZUHO OZEKI ◽  
MARIA EVELINA ROSSI
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Hilbert Coefficients ◽  
Integrally Closed ◽  
Numerical Invariants ◽  
Minimal Reduction ◽  
Integrally Closed Ideals ◽  
Extremal Value

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen–Macaulay local ring $A$ satisfying the equality $\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\text{e}_{0}(I)$ and $\text{e}_{1}(I)$ denote the first two Hilbert coefficients of $I,$ respectively, the multiplicity and the Chern number of $I.$ This almost extremal value of $\text{e}_{1}(I)$ with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.

scholarly journals On the Ideal Case of a Conjecture of Huneke and Wiegand

2019 ◽  
Vol 62 (3) ◽  
pp. 847-859 ◽  
Author(s):  
Olgur Celikbas ◽  
Shiro Goto ◽  
Ryo Takahashi ◽  
Naoki Taniguchi
Keyword(s):  
Free Module ◽  
Affirmative Answer ◽  
Noetherian Rings ◽  
Ideal Case ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Integrally Closed ◽  
Integrally Closed Ideals ◽  
Torsion Free ◽  
The Ideal

AbstractA conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

scholarly journals Birational extensions in dimension two and integrally closed ideals

1988 ◽  
Vol 115 (2) ◽  
pp. 481-500 ◽  
Author(s):  
Craig Huneke ◽  
Judith D Sally
Keyword(s):  
Integrally Closed ◽  
Integrally Closed Ideals
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