Extensions of rank one ϕ Sample Clauses

Extensions of rank one ϕ modules. We are interested in studying extensions of Kisin modules. In particular, after fixing a subset X X {0, . . . , f — 1} we will study extensions of N = M(h0, . . . , hf−1; a) by P = M(r0 — h0, . . . , rf−1 — hf−1; b). Here r0, . . . , rf−1 are non-negative integers and the value of hi depends on the subset J as follows:  i h = ,ri if i ∈ J, ,0 otherwise. We use a structure theorem of extensions of Kisin modules ( [GLS15, Proposition 5.1.3]). By this result we can choose bases for these extensions such that we have explicit descriptions for the action of ϕ on these bases. This will allow us later to pin down the extensions, and to count the possible extensions. These results are crucial ingredients for the main results of this section. The results of the structure theorem depend on whether or not there exist non- zero maps from N to P. In [GLS15], the authors give a very explicit condition for this existence:
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