论文标题
派生的拉格曼人和派生的舒尔功能者
Derived Grassmannians and derived Schur functors
论文作者
论文摘要
本文开发了两种理论，分别是衍生的拉格曼人（和旗帜方案）的几何理论以及衍生的schur（和weyl）函数的代数理论，并建立了它们的联系，这是对Borel-Weil-Weil-Bott Theorem的衍生概括。更具体地： （1）衍生的拉格曼人和国旗方案的理论是衍生的项目衍生理论的自然扩展[Arxiv：2202.11636]，并将Grothendieck的草个羊奶和吊羊方案的理论推广到复杂的案例。我们建立了他们的基本特性，并研究其中的各种自然形态。 （2）在$ \ mathrm {gl} _n（\ Mathbb {z}）$中，将衍生的Schur和Weyl函数的理论扩展了Schur和Weyl模块函数的经典理论 - 将其扩展到复杂的情况。我们表明，这些函子具有出色的函数属性，并满足了经典公式的概括，例如库奇的分解公式，直接和-Richardson规则。我们还从派生的对称能力的情况下将各种结果推广到派生的Schur函子，例如Illusie-Lurie的Décalage同构。 （3）这两种理论通过Borel-Weil-Bott Theorem的派生版本连接，该版本概括了经典的Borel-Weil-Bott Theorem定理，并计算出衍生的旗帜方案在派生的Schur函数方面，在派生的Schur函数上，当复合物具有完美的示例示例时，在派生的旗帜方案上的派生推送，并计算出衍生的旗帜方案。
This paper develops two theories, the geometric theory of derived Grassmannians (and flag schemes) and the algebraic theory of derived Schur (and Weyl) functors, and establishes their connection, a derived generalization of the Borel-Weil-Bott theorem. More specifically: (1) The theory of derived Grassmannians and flag schemes is the natural extension of the theory of derived projectivizations [arXiv:2202.11636] and generalizes Grothendieck's theory of Grassmannians and flag schemes of sheaves to the case of complexes. We establish their fundamental properties and study various natural morphisms among them. (2) The theory of derived Schur and Weyl functors extends the classical theory of Schur and Weyl module functors studied in $\mathrm{GL}_n(\mathbb{Z})$-representation theory to the case of complexes. We show that these functors have excellent functorial properties and satisfy derived generalizations of classical formulae such as Cauchy's decomposition formula, direct-sum decomposition formula and Littlewood-Richardson rule. We also generalize various results from the case of derived symmetric powers to derived Schur functors, such as Illusie-Lurie's décalage isomorphism. (3) These two theories are connected by a derived version of the Borel-Weil-Bott theorem, which generalizes the classical Borel-Weil-Bott theorem and calculates the derived pushforwards of tautological perfect complexes on derived flag schemes in terms of derived Schur functors when the complexes have perfect-amplitude $\le 1$ and positive ranks.
