论文标题
三角形类别的对数动机
Triangulated categories of logarithmic motives over a field
论文作者
论文摘要
在这项工作中，我们从Fontaine，Illusie和Kato的意义上发展了对数方案的动机理论。我们的构造基于有限日志对应关系的概念，对日志方案的Nisnevich拓扑结构，以及通过$ \ overline {\ square} $参数化同型的基本思想，即相对于其紧缩的无限结构的投影线。我们表明，日志方案的Hodge共同体是$ \ edline {\ square} $ - 不变理论，在对数动机类别中代表。我们的类别与Voevodsky的动机类别和$ \ Mathbb {a}^{1} $ - 不变理论密切相关：假设奇异性，我们以完整的子类别组成了后者，其中包括$ \ \ \ mathbb {a}}^{a} {1} $ - 本地对象 - 与众类别的本地对象。基本属性，例如$ \ Overline {\ Square} $ - 同型不变性，Mayer-Vietoris供覆盖物，Gysin序列的类似物和Thom空间同构以及爆炸的配方以及一个投射的捆绑束公式证明了设置的鲁棒性。
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a $\overline{\square}$-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voevodsky's category of motives and $\mathbb{A}^{1}$-invariant theories: assuming resolution of singularities, we identify the latter with the full subcategory comprised of $\mathbb{A}^{1}$-local objects in the category of logarithmic motives. Fundamental properties such as $\overline{\square}$-homotopy invariance, Mayer-Vietoris for coverings, the analogs of the Gysin sequence and the Thom space isomorphism as well as a blow-up formula and a projective bundle formula witness the robustness of the setup.
