Abstract:

This thesis uses the idea of geometrization in two contexts. The first part is devoted to the Cartier transform developed by Ogus and Vologodsky. We strengthen their main result by weakening the assumptions and by allowing certain reasonable stacks as inputs, obtaining, in particular, corollaries in the logarithmic setting. The second part studies the category of F-gauges over the formal spectrum of the Witt vectors of a perfect field of positive characteristic. We identify the full subcategory of F-gauges with Hodge-Tate weights in the range from 0 to p-2 with the category of Fontaine-Laffaille modules satisfying the analogous weight constraint.