Computing finite and infinite free resolutions with Pommaret-like bases

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation Pub Date : 2025-04-28 DOI:10.1016/j.jsc.2025.102454

Amir Hashemi , Matthias Orth , Werner M. Seiler

引用次数: 0

Abstract

Free resolutions are an important tool in algebraic geometry for the structural analysis of modules over polynomial rings and their quotient rings. Minimal free resolutions are unique up to isomorphism and induce homological invariants in the form of Betti numbers. It is known that Pommaret bases of ideals in the polynomial ring induce finite free resolutions and that the Castelnuovo-Mumford regularity and projective dimension can be read off directly from the Pommaret basis. In this article, we generalize this construction to Pommaret-like bases, which are generally smaller. We apply Pommaret-like bases also to infinite resolutions over quotient rings. Over Clements–Lindström rings, we derive bases for the free modules in the resolution using only the Pommaret-like basis. Finally, restricting to monomial ideals in a non-quotient polynomial ring, we derive an explicit formula for the differential of the induced resolution.

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用类波马雷基底计算有限和无限自由分辨率

自由分辨是代数几何中用于分析多项式环及其商环上模的结构的重要工具。最小自由分辨率在同构范围内是唯一的，并以贝蒂数的形式导出同调不变量。已知多项式环中理想的Pommaret基可以导出有限自由分辨率，Castelnuovo-Mumford正则性和投影维数可以直接从Pommaret基中读出。在本文中，我们将这种构造推广到类pommaret基，它通常更小。我们也将类波马雷基应用于商环上的无限分辨率。在Clements-Lindström环上，我们仅使用Pommaret-like基为分辨率中的自由模块导出基。最后，在非商多项式环的单项式理想条件下，导出了诱导分辨的微分的显式公式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.