Year: 2021
Type: preprint
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, étale, 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.
Source: arXiv (Cornell University)
Authors: Kelly, Shane, Miyazaki, Hiroyasu
Institutions: Tokyo Institute of Technology, RIKEN
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3177449554