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Crystalline Condition for Ainf-cohomology and Ramification Bounds
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 RΓΔ(X/S) inside the Ainf-prismatic cohomology RΓΔ(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, and thus leads to a proof of crystallinity of Hiet(XCK, Qp) that avoids the crystalline comparison. The higher conditions (Crs) are used in an adaptation of a ramification bounds strategy of Caruso and Liu. As a result, we establish ramification bounds for the mod p representations Hiet(XCK, Z/pZ) for arbitrary e and i, which extend or improve existing bounds in various situations.