AI 中文总结
该研究利用局部化等推导动机框架，将苏斯林互律推广到任意基上光滑概形，还细化了模单位杯积关系，推广到整系数、整基及非满水平结构等场景。
AI 中文摘要
我们观察到，苏斯林互反律的动机类似物（以及类似的零次陈述）是局部化（加上纯性/某些六函子形式体系）的形式结果。特别地，Kriz提出的域上光滑概形的苏斯林互律陈述，是任意基上高 Chow 群局部化的推论；我们写下了在$\boldsymbol{\text{A}^1}$-不变动机上同调中，对任意基上光滑概形给出带系数此类关系的框架。作为应用，我们细化了模单位上杯积间的一些关系，使其在系数和基上均为整性：首先，我们模仿 Busuioc--Park--Patashnick--Stevens 针对满水平$N$椭圆概形的（有理系数、复解析）论证，利用初等互律陈述将结果推广到整基和整系数。随后，我们将该构造及所得关系细化到动机层的框架中；特别地，这在非满水平结构下，以及在任意光滑全局商层上，给出了类似的关系。
英文摘要
We observe that the motivic analogue of Suslin reciprocity (and similar degree-zero statements) is a formal consequence of localization (plus purity/some six functor formalism). In particular, the statement of Suslin reciprocity for smooth schemes over fields due to Kriz is a corollary of localization for higher Chow groups, over any base; we write down the framework yielding such relations with coefficients for schemes smooth over any base in $\A^1$-invariant motivic cohomology. As an application, we refine some relations between cup products of modular units to be integral in coefficients and in the base: first, we imitate the (rational-coefficients, complex-analytic) Busuioc--Park--Patashnick--Stevens argument for full-level-$N$ elliptic schemes, extending the result to integral bases and coefficients using the elementary reciprocity statement. We then refine the construction and resulting relations to the setting of motivic sheaves; in particular, this gives analogous relations at non-full level structure, as well as over any smooth global quotient stack.
