A result on formal linearity
type: Item
instance of: scholarly article
title: A result on formal linearity
zbMATH Open document ID: 0974.14032
published in: Journal of Algebra
publication date: 2001-05-28T00:00:00Z
review text: Let \(\Lambda\) be a complete discrete valuation ring of mixed characteristic \((0, p)\) with an automorphism \(\tau\). Let \(X\) be the formal \(n\)-dimensional torus \(\widehat\mathbb{G}_m^n\) over \(\Lambda\) with coordinates \(q_1,\dots,q_n\), and let \(\Phi: X \rightarrow X\) be the \(\tau\)-linear morphism such that \(\Phi^*(q_i) = q_i^q\), where \(q=p^t\) for some integer \(t > 0\). Given a closed formal subscheme \(Z\) of \(X\) define \(\Phi(Z)\) to be the minimal closed subscheme of \(X\) such that the morphism \(Z \rightarrow X \rightarrow_\Phi X\) factors through \(\Phi(Z)\). Define inductively \(\Phi^a(Z)\) for a positive integer \(a\). The paper is concerned with the characterization of formal subtori of \(X\), and generalizations thereof, in terms of the map \(\Phi\). For example, the author proves the following theorem: Let \(Z\) be a closed formal subscheme of \(X\) that is equi-dimensional, reduced and flat over \(\text{Spf}(\Lambda)\). Then \(Z\) is a formal subtorus of \(X\) iff \(Z\) is irreducible and \(\Phi^a(Z) \subseteq Z\) for some \(a \geq 1\). This result is useful in the study of reduction of Shimura varieties because Serre-Tate theory identifies the deformations of an ordinary abelian variety to \(\text{Spf}(\Lambda)\) with a formal torus over \(\Lambda\) with canonical coordinates. The study of the closed subscheme along which certain endomorphisms of the special fibre deform leads to formal subtori [see \textit{B. Moonen}, Compos. Math. 114, No. 1, 3-35 (1998; Zbl 0960.14012)].
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1429476
zbMATH Keywords: formal linearity
zbMATH Keywords: formal subtorus
zbMATH Keywords: discrete valuation ring
zbMATH Keywords: Shimura varieties
author: Aise Johan de Jong
reviewed by: Eyal Z. Goren
MaRDI profile type: Publication
full work available at URL: jabr.1999.8186
OpenAlex ID: W2020606999
cites work: Q4267948
DOI: 10.1006/JABR.1999.8186
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description: scientific article; zbMATH DE number 1429476