Foundations of Algebraic Stacks

该项目旨在促进和积极发展对代数堆栈的研究。代数堆栈被广泛使用，特别是在考虑模空间时，它是当今代数几何中最重要的工具之一。研究将集中在与全球几何问题相关的基础问题和开放问题上。两个问题是：用最少的一组假设，你能在多大程度上发展理论？在什么一般情况下，我们可以发展代数堆栈的交集理论？两个几何公开问题是：哪些代数堆栈是商堆栈？域上有限型分离格式的Brauer群和上同调Brauer群是相同的吗？主要的研究人员将访问数学系并发表演讲来宣传这些问题。首席研究员将解决一些研究问题，并在其他问题上与国际数学界的研究生、博士后和代数几何学家合作。作为提案的一个组成部分，首席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”。目标是发展和记录理论，目的是缩小研究生教育和当前研究之间的差距。堆栈项目打算部分作为一本高级教科书，部分作为代数堆栈的基础参考。

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.