Modulus sheaves with transfers

TL;DR

This paper extends the theory of modulus pairs to include schemes with effective Cartier divisors, developing sheaf theories and motives that connect ramification, rigid geometry, and singularities.

Contribution

It generalizes the theory of modulus pairs and develops new sheaf and homotopy theories for these pairs, linking ramification and irregular singularities.

Findings

Established sheaf theories for modulus pairs across various topologies.

Extended the Suslin-Voevodsky correspondence to modulus pairs with Noetherian interior.

Developed a homotopy theory and motives with modulus over general bases.

Abstract

We generalise Kahn, Miyazaki, Saito, Yamazaki's theory of modulus pairs to pairs $(X, D)$ consisting of a qcqs scheme $X$ equipped with an effective Cartier divisor $D$ representing a ramification bound. We develop theories of sheaves on such pairs for modulus versions of the Zariski, Nisnevich, \'etale, fppf, and qfh-topologies. We extend the Suslin-Voevodsky theory of correspondances to modulus pairs, under the assumption that the interior $U = X \setminus D$ is Noetherian. The resulting point of view highlights connections to (Raynaud-style) rigid geometry, and potentially provides a setting where wild ramification can be compared with irregular singularities. This framework leads to a homotopy theory of modulus pairs $\underline{M}H(X,D)$ and a theory of motives with modulus $\underline{M}DM^{eff}(X,D)$ over a general base $(X, D)$. For example, the case where $X$ is the spectrum of a rank one valuation ring (of mixed or equal characteristic) equipped with a choice $D$ of pseudo-uniformiser is allowed.