Finite flat commutative group schemes over complete discrete valuation rings III: classification, tangent spaces, and semistable reduction of Abelian varieties
| dc.creator | Bondarko, M. V. | |
| dc.date | 2004-12-29 | |
| dc.date | 2005-12-09 | |
| dc.date.accessioned | 2026-07-07T06:39:13Z | |
| dc.date.available | 2026-07-07T06:39:13Z | |
| dc.description | We classify group schemes in terms of their Cartier modules. We also prove the equivalence of different definitions of the tangent space and the dimension for these group schemes; in particular, the minimal dimension of a formal group law that contains $S$ as a closed subgroup is equal to the minimal number of generators for the affine algebra of $S$. As an application the following reduction criteria for Abelian varieties are proved. Let $K$ be a mixed characteristic local field, let its residue field have characteristic $p$, $L$ be a finite extension of $K$, let $\mathfrak{O}_K\subset\mathfrak{O}_L$ be their rings of integers. Let $e$ be the absolute ramification index of $L$, $s=[\log_p(pe/(p-1))]$, $e_0$ be the ramification index of $L/K$, $l=2s+v_p(e_0)+1$. For a finite flat commutative $\mathfrak{O}_L$-group scheme $H$ we denote the $\mathfrak{O}_L$-dual of the module $J/J^2$ by $TH$. Here $J$ is the augmentation ideal of the affine algebra of $H$. Let $V$ be an $m$-dimensional Abelian variety over $K$. Suppose that $V$ has semistable reduction over $L$. \begin{theor} $V$ has semistable reduction over $K$ if and only if for some group scheme $H$ over $\mathfrak{O}_K$ there exist embeddings of $H_K$ into $\operatorname{Ker}[p^{l}]_{V,K}$, and of $(\mathfrak{O}_L/p^l\mathfrak{O}_L)^m$ into $TH_\ol$. \end{theor} This criterion has a very nice-looking version in the ordinary reduction case. \begin{theor} $V$ has ordinary reduction over $K$ if and only if for some $H_K\subset \operatorname{Ker}[p^{l}]_{V,K}$ and $M$ unramified over $K$ we have $H_M\cong (μ_{p^{l},M})^m$. Here $μ$ denotes the group scheme of roots of unity.\end{theor} | |
| dc.identifier | https://arxiv.org/abs/math/0412521 | |
| dc.identifier | http://arxiv.org/abs/math/0412521 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100998 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14L15, 14L05, 14G20, 11G10, 11S31 | |
| dc.title | Finite flat commutative group schemes over complete discrete valuation rings III: classification, tangent spaces, and semistable reduction of Abelian varieties | |
| dc.type | text |