For a prime number p > 2, we give a direct proof of Breuil’s classification of finite flat group schemes killed by p over the valuation ring of a p-adic field with perfect residue field. As application we establish a correspondence between finite flat group schemes and Faltings’s strict modules which respects associated Galois modules via the Fontaine-Wintenberger field-of-norms functor

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

Let R be a right GF-closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorensntein flat dimensnion of R/I as a right R-module and the Gorensntein injective dimensnnion of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorensntein ring R, ...

It is proved that every commutative ring whose RD-injective modules are Σ-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian (respectively simple) module is RD-injective, is given. These results can be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively th...