Foundations of Algebraic Stacks

本项目旨在促进和积极发展代数堆栈的研究。代数堆栈被广泛使用，特别是在考虑模空间时，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性问题上。两个问题是：你能在多大程度上用最少的假设集来发展这个理论？在什么样的通用性下，我们可以发展代数堆栈的交点理论？两个几何开放问题是：哪些代数堆是商堆？域上有限型分离格式的Brauer群与上同调Brauer群是否相同？首席研究员将访问数学系并就这些问题发表演讲。主要研究者将解决一些研究问题，并与国际数学界的研究生、博士后和代数几何学者合作解决其他问题。作为提案的一个组成部分，首席研究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”。其目标是发展和记录理论，以缩小研究生教育与当前研究之间的差距。堆栈项目打算部分是一个高层次的教科书，部分是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.