Foundations of Algebraic Stacks

代数栈的基础

• 批准号: 1303247
• 负责人: Aise de Jong
• 金额: $ 18.12万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303247&HistoricalAwards=false
• 关键词: Foundations; Algebraic; Stacks; Foundations; Algebraic; Stacks

This project aims to stimulate and actively develop research on algebraic stacks. Algebraic stacks are widely used, especially for thinking about moduli spaces, and are one of the most important tools in algebraic geometry today. The research will focus on foundational questions and open questions related to global geometric problems. Two questions are: How far can you get developing the theory with a minimal set of hypotheses? In what generality can we develop intersection theory for algebraic stacks? Two geometric open problems are: Which algebraic stacks are quotient stacks? Are the Brauer group and the cohomological Brauer group of a separated scheme of finite type over a field the same?The principal investigator will visit mathematics departments and give talks to advertise these problems. The principal investigator will attack some of the research problems, and collaborate on others with graduate students, post-docs, and algebraic geometers in the international mathematical community. As an integral part of the proposal the principal investigator will maintain a web-based collaborative algebraic geometry project, called the "Stacks Project." The goal is to develop and document theory, with the aim of closing the gap between graduate education and current research. The stacks project intends to be partly a high level text book, partly a foundational reference on algebraic stacks.

该项目旨在刺激并积极开发有关代数堆栈的研究。代数堆栈被广泛使用，尤其是用于思考模量空间，并且是当今代数几何形状中最重要的工具之一。该研究将重点放在与全球几何问题有关的基本问题和开放问题上。两个问题是：您能以最少的假设来发展多远？我们可以在哪些一般性中开发代数堆栈的交点理论？两个几何开放问题是：哪些代数堆栈是商堆栈？ Brauer集团和同时的Brauer集团是否在一个领域相同的有限类型方案？主要调查员将访问数学部门并进行会谈以宣传这些问题。首席研究人员将攻击一些研究问题，并与国际数学社区中的研究生，DOCS和代数几何图形与其他研究人员合作。作为提案不可或缺的一部分，主要研究人员将维护一个基于网络的协作代数几何项目，称为“堆栈项目”。目的是开发和记录理论，目的是缩小研究生教育与当前研究之间的差距。堆栈项目打算部分成为一本高级教科书，部分是代数堆栈的基础参考。