Tensor Triangular Geometry and Applications

张量三角形几何及其应用

• 批准号: 0654397
• 负责人: Paul Balmer
• 金额: $ 14.39万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654397&HistoricalAwards=false
• 关键词: Tensor; Triangular; Geometry; Applications

``Tensor triangular geometry'' is a recent term coined by the PI for the ongoing unification by means of tensor triangulated categories of geometric aspects of very different areas of mathematics and theoretical physics, among which algebraic geometry, modular representation theory, stable homotopy theory, motivic theory, noncommutative geometry, string theory, and probably more in the future. The geometric foundation of this new theory is a construction proposed by the PI, called the ``spectrum of a tensor triangulated category'', which produces a locally ringed space. For example, the spectrum of the category of perfect complexes over a scheme recovers the scheme. Heuristically, this says that algebraic geometry embeds into tensor triangular geometry. In modular representation theory, the spectrum of the stable category recovers the ``projective support variety''. For the triangulated categories discovered more recently, the computation of the spectrum is one of the challenges proposed here by the PI, e.g. for mixed Tate motives or equivariant KK-theory of C*-algebras. This will establish unexpected bridges between these areas and algebraic geometry.The PI wants to extend well-understood concepts from the above numerous examples to tensor triangular geometry (e.g. invariants, like Chow groups, K-theory, Brauer groups, Dade groups,... etc). Then, he wants to prove general theorems at the astract level that can be applied to the various areas listed above to obtain new results. This ideal strategy has already been successfully started by the PI, like with filtrations by dimension of support yielding a new local-global spectral sequence for K-theory of singular varieties, or like in the joint work with Benson and Carlson on the Picard group, a standard invariant in algebraic geometry but a more subtle one in modular representation theory. The PI also applies such ideas to quadratic forms (via ``triangular Witt groups''), allowing many new results, like the proof of the 1980 Gersten-Witt Conjecture, and more computations to come.

“张量三角几何”是 PI 最近创造的一个术语，用于通过张量三角化类别不断统一数学和理论物理不同领域的几何方面，其中包括代数几何、模表示理论、稳定同伦理论、动机理论、非交换几何、弦理论，未来可能还会有更多。这一新理论的几何基础是 PI 提出的一种构造，称为“张量三角范畴的谱”，它产生一个局部环形空间。例如，一个方案上的完美复形的范畴的谱可以恢复该方案。启发式地，这表明代数几何嵌入到张量三角几何中。在模表示理论中，稳定范畴的谱恢复了“投影支持多样性”。对于最近发现的三角类别，频谱的计算是 PI 提出的挑战之一，例如对于混合 Tate 动机或 C* 代数的等变 KK 理论。这将在这些领域和代数几何之间建立意想不到的桥梁。PI 希望将上述众多示例中易于理解的概念扩展到张量三角几何（例如不变量，如 Chow 群、K 理论、Brauer 群、Dade 群等）。 。 ETC）。然后，他想在抽象层面证明一般定理，这些定理可以应用于上面列出的各个领域，以获得新的结果。这种理想的策略已经由 PI 成功启动，例如通过支持维数过滤为奇异簇 K 理论产生新的局部全局谱序列，或者像与 Benson 和 Carlson 在 Picard 群中的联合工作一样，代数几何中的一个标准不变量，但模表示论中的一个更微妙的不变量。 PI 还将这些想法应用于二次形式（通过“三角维特群”），从而获得许多新结果，例如 1980 年格斯顿-维特猜想的证明，以及更多的计算。