AI 中文总结
本文研究正特征下同维简单奇点的奇异范畴，推广Hua与Keller的相关定理，确定有理二重点奇异范畴为标准的条件，完善了正特征下奇点范畴的相关理论。
AI 中文摘要
我们研究代数闭域上正特征下同维简单奇点的奇异范畴，证明与特征零情形类似，当且仅当底层奇点解析同构时，这些范畴作为三角范畴不等价。与特征零情形不同，正特征下的简单奇点无法仅通过其奇异范畴的Auslander-Reiten箭图区分。为解决此问题，我们将Hua和Keller的一个定理推广到正特征，该定理断言特征零下孤立超曲面奇点的dg奇异范畴的第0个Hochschild上同调同构于定义多项式的Tyurina代数。此外作为应用，我们确定了有理二重点（即二维简单奇点）的奇异范畴为标准的条件，证明该范畴为标准当且仅当定义多项式是加权齐次的。
英文摘要
We study the singularity categories of simple singularities of the same dimension over an algebraically closed field of positive characteristic, and show that, as in characteristic zero, these categories are not equivalent as triangulated categories unless the underlying singularities are analytically isomorphic. In contrast to the characteristic zero case, simple singularities in positive characteristic cannot be distinguished solely from the Auslander-Reiten quivers of their singularity categories. To address this, we extend to positive characteristics a theorem by Hua and Keller, which asserts that the 0th Hochschild cohomology of the dg singularity category of an isolated hypersurface singularity in characteristic zero is isomorphic to the Tyurina algebra of the defining polynomial. Furthermore as an application, we determine the condition for the singularity category of a rational double point (i.e., a simple singularity of dimension two) to be standard. We prove that such a category is standard if and only if the defining polynomial is weighted homogeneous.
