Tame characters and ramification of finite flat group schemes (Q2483154)

Let \(K\) be a complete discrete valuation field of mixed characteristic with residue field \(F\) of characteristic \(p\) and not necessarily perfect. Consider \({\mathcal G}:= \text{Spf}(B)\), a connected finite flat group scheme of \(p\)--power order over \({\mathcal O}_ K\), the ring of integers of \(K\). Let \(\big\{{\mathcal G}^ j\big\}_ {j>0}\) be the \(j\)--ramification filtration of \({\mathcal G}\) in the sense of Abbes and Saito, and let \({\mathcal G}^{j^ +}\) be the schematic closure of \({\mathcal G}^ {j^ +}(\bar{K}):=\bigcup_{j'>j}{\mathcal G}^ {j'}(\bar{K})\) in \({\mathcal G}\). Let \(G_ K:= \text{Gal\;} (\bar{K}/K)\) be the absolute Galois group of \(K\). The author proves that the \(G_ K\)-module \({\mathcal G}^ {j}(\bar{K})/{\mathcal G}^ {j^ +}(\bar{K})\) is tame and killed by \(p\). Also, if \(I_ K\) denotes the inertia subgroup of \(G_ K\), then the \(I_ K\)-module \({\mathcal G}^ {j}(\bar{K})/{\mathcal G}^ {j^ +}(\bar{K}) \otimes_{{\mathbb F}_ p}\bar{{\mathbb F}}_ p\) is the direct sum of fundamental characters of level \(j\). As corollaries of these results, it is obtained that the order of the image of the homomorphism \(I_ K\to \Aut({\mathcal G}(\bar{K}))\) is a power of \(p\) if and only if every jump \(j\) of the ramification filtration \(\big\{{\mathcal G}^ j\big\}_ {j>0}\) is an element of \({\mathbb Z}[1/p]\). In the last section, the author computes the conductor of a Raynaud \({\mathbb F}\)-vector space scheme over \({\mathcal O}_ K\), where \({\mathbb F}\) is a finite extension of the finite field \({\mathbb F}_ p\) of \(p\)--elements. For integers \(0\leq s_ 1,\ldots,s_ r\leq e\) where \(e\) is the absolute ramification index, let \({\mathcal G}(s_ 1,\ldots, s_ r)\) be the Raynaud \({\mathbb F}\)-vector space scheme over \({\mathcal O}_ K\) defined by \(T_ 1^ p=\pi ^ {s_ 1} T_ 2, T_ 2^ p=\pi^ {s_ 2}T_ 3,\ldots, T_ r^ p= \pi^{s_ r}T_ 1\) where \(\pi\) is a uniformizing parameter of \(K\). Set \(j_ k:=(p s_ k+ p^ 2 s_ {k-1}+\cdots+p^ k s_ 1+ p^{k+1} s_ r+p^ {k+2}s_{r-1}+\cdots+p^ r s_ {k+1})/(p^ r-1)\). Then the conductor \(c({\mathcal G}(s_ 1,\ldots,s_ r))\) is equal to \(\sup_ k j_ k\).