Mathematical Sciences: Research in Algebra and Algebraic Number Theory

这个项目是关于动机和动机上同源的研究。首席研究员将继续他对动机和通知上同调的新定义的各种性质的研究。给定代数闭域k上的任何拟投射方案X，X的Suslin复形应该给出X作为阿贝尔群的派生范畴中的一个对象的混合动机。首席研究人员已经证明，这是对X的一个非常自然的定义，k上的任意曲线，他将研究曲面的情况。此外，他还将调查他对Suslin复合体的广义定义在多大程度上可以证明满足所需的公理。这项研究涉及到代数K-理论。在广义上，代数K-理论涉及基和向量空间等线性代数概念的演化。这项工作对数论和代数几何具有重大意义，并有望在数学的许多不同领域之间建立令人兴奋的联系。

This project is concerned with research on motives and motivic cohomology. The principal investigator will continue his research in the various properties of his new definitions of motives and notivic cohomology. Given any quasiprojective scheme X over an algebraically closed field k, the Suslin complex of X should give the mixed motive of X as an object in the derived category of abelian groups. The principal investigator has already shown that this is a very natural definition for X an arbitrary curve over k, and he will investigate the case of surfaces. In addition, he will investigate the extent to which his generalized definition of the Suslin complex can be shown to satisfy the desired axioms. This research is concerned with algebraic K-theory. In a broad sense algebraic K-theory concerns th evolution of concepts from linear algebra such as basis and vector space. This work has significant implications for number theory and algebraic geometry, and promises to make exciting connections between a number of different areas in mathematics.