Localization and elliptic motivic relations

Abstract

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.