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A Dieudonné Theory for $p$-Divisible Groups

2018; Mathematical Society of Japan; Linguagem: Inglês

10.2969/aspm/03010139

2433-8915

Thomas Zink,

Advanced Topology and Set Theory

Let k be a perfect field of characteristic p > 0. We denote by W(k) the ring of Witt vectors.Let us denote by ( -----, F(, ( E W(k) the Frobenius automorphism of the ring W(k).A Dieudonne module over k is a finitely generated free W(k)-module M equipped with an F_ linear map F : M ___, M such that pM C FM.By a classical theorem of Dieudonne ( compare Grothendieck [ G]) the category of p-divisible formal groups over k is equivalent to the category of Dieudonne modules over k.In this paper we will prove a totally similiar result for p-divisible groups over a complete noetherian local ring R with residue field k if either p > 2, or if pR = 0.For formal p-divisible groups (i.e.without etale part) this is done in [Z2].We will now give a description of our result.Let R be as above but assume firstly that R is artinian.The maximal ideal of R will be denoted by m.The most important point is that we do not work with the Witt ring W(R) but with a subring W(R) C W(R).This subring is characterized by the following properties: It is functorial in R. It is stable by the Frobenius endomorphism F and by the Verschiebungsurjective, and its kernel consists exactly of the Witt vectors in W(m) with only finitely many non-zero components.The ring W(R) is a non-noetherian local ring with residue class field k.It is separated and complete as a local ring.If R is an arbitrary complete local ring as above we set W(R) lim W(R/mn).