Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Deformation subspaces of $p$-divisible groups as formal Lie groups associated to $p$-divisible groups


Author: Adrian Vasiu
Journal: J. Algebraic Geom. 20 (2011), 1-45
DOI: https://doi.org/10.1090/S1056-3911-2010-00571-1
Published electronically: September 9, 2010
MathSciNet review: 2729274
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Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$ which is not isoclinic. Let $\mathcal {D}$ (resp. $\mathcal {D}_{k}$) be the formal deformation space of $D$ over $\text {Spf}(W(k))$ (resp. over $\text {Spf}(k)$). We use axioms to construct formal subschemes $\mathcal {G}_{k}$ of $\mathcal {D}_{k}$ that: (i) have canonical structures of formal Lie groups over $\text {Spf}(k)$ associated to $p$-divisible groups over $k$, and (ii) give birth, via all geometric points $\text {Spf}(K)\to \mathcal {G}_{k}$, to $p$-divisible groups over $K$ that are isomorphic to $D_{K}$. We also identify when there exist formal subschemes $\mathcal {G}$ of $\mathcal {D}$ which lift $\mathcal {G}_{k}$ and which have natural structures of formal Lie groups over $\text {Spf}(W(k))$ associated to $p$-divisible groups over $W(k)$. Applications to Traverso (ultimate) stratifications are included as well.


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Adrian Vasiu
Affiliation: Department of Mathematical Sciences, Binghamton University, P.O. Box 6000, Binghamton, New York 13902-6000
Email: adrian@math.binghamton.edu

Received by editor(s): April 23, 2008
Received by editor(s) in revised form: November 12, 2009
Published electronically: September 9, 2010
Additional Notes: This project was partially supported by the NSF grant DMS #0900967.