Crystalline Condition for Ainf–cohomology and Ramification Bounds

Pavel Coupek, Purdue University

Abstract

For a prime p > 2 and a smooth proper p–adic formal scheme X over OK where K is a p–adic field of absolute ramification degree e, we study a series of conditions (Crs), s ≥ 0 that partially control the GK–action on the image of the associated Breuil–Kisin prismatic cohomology RΓ∆ (X/S) inside the Ainf–prismatic cohomology RΓ∆ (XAinf/Ainf). The condition (Cr0) is a criterion for a Breuil–Kisin–Fargues GK–module to induce a crystalline representation used by Gee and Liu in [14, Appendix F], and thus leads to a proof of crystallinity of Hi ét(Xη, Qp) that avoids the crystalline comparison. The higher conditions (Crs) are used in an adaptation of a ramification bounds strategy of Caruso and Liu from [11]. As a result, we establish ramification bounds for the mod p representations Hi ét(Xη, Z/pZ) for arbitrary e and i, which extend or improve existing bounds in various situations.

Degree

Ph.D.

Advisors

Liu, Purdue University.

Subject Area

Mathematics

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