Purity for Barsotti-Tate groups in some mixed characteristic situations

2018/09/13 by Gabber, Ofer, Vasiu, Adrian
#11G10 #11G18 #14F30 #14G35 #14K10 #14K15 #14K99 #14L05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #and 14L15

paper · doi:10.48550/arxiv.1809.05141

Abstract

Let p be a prime. Let R be a regular local ring of dimension d≥ 2 whose completion is isomorphic to C(k)[[x₁,…,x_d]]/(h), with C(k) a Cohen ring with the same residue field k as R and with h∈ C(k)[[x₁,…,x_d]] such that its reduction modulo p does not belong to the ideal (x₁^p,…,x_d^p)+(x₁,…,x_d)^2p-2 of k[[x₁,…,x_d]]. We extend a result of Vasiu-Zink (for d=2) to show that each Barsotti-Tate group over Frac(R) which extends to every local ring of Spec(R) of dimension 1, extends uniquely to a Barsotti-Tate group over R. This result corrects in many cases several errors in the literature. As an application, we get that if Y is a regular integral scheme such that the completion of each local ring of Y of residue characteristic p is a formal power series ring over some complete discrete valuation ring of absolute ramification index e≤ p-1, then each Barsotti-Tate group over the generic point of Y which extends to every local ring of Y of dimension 1, extends uniquely to a Barsotti-Tate group over Y.

Related

• Potentially Good Reduction of Barsotti-Tate Groups
• Purity of level m stratifications
• Canonical Barsotti-Tate Groups of Finite Level
• Exterior Powers of Barsotti-Tate Groups
• On Abelian extensions in mixed characteristic and ramification in codimension one