Motivic Cohomology and K-Theory

动机上同调和 K 理论

• 批准号: 1406502
• 负责人: Charles Weibel
• 金额: $ 15.7万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406502&HistoricalAwards=false
• 关键词: Motivic; Cohomology; Theory

The research project will try to settle old conjectures about the structure of K-theory and Motives, and clarify recently discovered relationships between disparate areas of algebra, algebraic geometry, and algebraic topology. The motive of an algebraic variety is built from geometric relationships, and is designed to be the universal motif for cohomological invariants (hence the name), so its structure reveals aspects of geometry. In contrast, the K-theory of a ring or variety is built from relationships involving linear algebra using algebraic topology, so its structure reveals aspects of algebra and topology. Clarifying the relationships between these constructs will help our understanding of many phenomena that are currently understand only poorly.One part of the project is to find clean proofs of major pieces of the proof of the recently-verified Bloch-Kato conjecture. This includes trying to: find a general degree formula which combines various degree formulas due to Rost, in terms of algebraic cobordism theory; establish an equivalence between models of the symmetric powers of a normal variety; and understand motivic cohomology operations better. These all play a role in the proof. This part of the project will be invaluable in training young researchers around the world. A related part of the project is to understand how Voevodsky's slice filtration is related to algebraic cobordism and other motivic spectra. The goal is to verify several of the slice conjectures, especially in finite characteristic, using homological algebra and algebraic Thom spaces. Part of this will be joint work with graduate students, contributing to their training. The project will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques and the relationship to cyclic homology and de Rham-Witt theory. This should shed some light on classes of singularities related to du Bois singularities. The final part of the project is to formulate an archimedean cohomology for a motive over a number field and relate it to Serre's local factors for the L-functions of the motive. This generalizes work of Connes and Consani, and uses cyclic homology for schemes.

该研究项目将试图解决关于k理论和动机结构的旧猜想，并澄清最近发现的代数、代数几何和代数拓扑等不同领域之间的关系。代数变量的动机是建立在几何关系之上的，并被设计为上同调不变量的通用母题（因此得名），因此它的结构揭示了几何的各个方面。相比之下，环或变体的k理论是利用代数拓扑从涉及线性代数的关系中建立起来的，因此其结构揭示了代数和拓扑的各个方面。澄清这些构念之间的关系将有助于我们理解目前知之甚少的许多现象。该项目的一部分是为最近验证的Bloch-Kato猜想的主要证明找到清晰的证据。这包括：从代数共数论的角度，找到一个综合了Rost的各种次公式的一般次公式；建立正态变量的对称幂模型之间的等价性；更好地理解动机上同运算。这些都在证明中发挥了作用。该项目的这一部分在培训世界各地的年轻研究人员方面将是无价的。该项目的一个相关部分是了解Voevodsky的切片过滤是如何与代数共坐标和其他动力谱相关联的。目的是验证几个片猜想，特别是在有限特征，使用同构代数和代数Thom空间。其中一部分将与研究生合作，为他们的培训做出贡献。该项目还将研究奇点之间的关系，它的k -理论和它的cdh-上同调，使用最近发展的上同调技术和循环同调和德拉姆-维特理论的关系。这应该会对与杜波依斯奇点相关的奇点类有所启发。项目的最后一部分是在数域上建立动机的阿基米德上同，并将其与动机的l -函数的Serre局部因子联系起来。推广了cones和Consani的工作，并对方案使用了循环同调。