## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularities — Niigata–Toyama 2007, J.-P. Brasselet, S. Ishii, T. Suwa and M. Vaquie, eds. (Tokyo: Mathematical Society of Japan, 2009), 341 - 361

### Standard bases and algebraic local cohomology for zero dimensional ideals

Shinichi Tajima, Yayoi Nakamura, and Katsusuke Nabeshima

#### Abstract

Zero-dimensional ideals in the formal power series and the associated vector space consisting of algebraic local cohomology classes are considered in the context of Grothendieck local duality. An algorithmic strategy for computing relative Čech cohomology representations of the algebraic local cohomology classes are described. A new algorithmic method for computing standard bases of a given zero-dimensional ideal is derived by using algebraic local cohomology and the Grothendieck local duality.

#### Article information

**Dates**

Received: 13 April 2008

Revised: 11 July 2008

First available in Project Euclid:
28 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543448025

**Digital Object Identifier**

doi:10.2969/aspm/05610341

**Mathematical Reviews number (MathSciNet)**

MR2604090

**Zentralblatt MATH identifier**

1194.13020

**Subjects**

Primary: 13D45: Local cohomology [See also 14B15] 32C37: Duality theorems 13J05: Power series rings [See also 13F25] 32A27: Local theory of residues [See also 32C30]

**Keywords**

Standard bases algebraic local cohomology Grothendieck duality

#### Citation

Tajima, Shinichi; Nakamura, Yayoi; Nabeshima, Katsusuke. Standard bases and algebraic local cohomology for zero dimensional ideals. Singularities — Niigata–Toyama 2007, 341--361, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05610341. https://projecteuclid.org/euclid.aspm/1543448025