无穷多元多项式环上的GL-等变模
GL-equivariant modules over polynomial rings in infinitely many variables
摘要：
考虑特征为零的域上可数无穷多个变量的多项式环，以及它的无限一般线性群G的自然作用。本文研究了这类环上的有限生成模的代数和同调性质，这些模都具有相容的G-作用。我们定义和证明有限性质的类似物的希尔伯特系列，系统的参数，深度，局部上同调，Koszul对偶和正则性。我们还表明，这一类是建立在一个更简单的，更组合，我们明确描述的类。我们的工作是受到最近文献中研究具有群作用的无限多项式环的有限性的启发。(For例如，Church，Ellenberg和Farb关于FI-模范畴的论文，它等价于我们的范畴。在这一过程中，我们从对称群的表示论中看到了与特征多项式的几个联系。文中给出了几个例子来说明我们所引入的不变量是明确的和可计算的。
英文摘要：
Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
