A NOTE ON FROBENIUS DIVIDED MODULES IN MIXED CHARACTERISTICS

作者：Berthelot Pierre^*

来源：Bulletin de la Societe Mathematique de France, 2012, 140(3): 441-458.

DOI：10.24033/bsmf.2632

• D-modules
• Frobenius morphism
• descent theory
• deformation theory

If X is a smooth scheme over a perfect field of characteristic p, and if D-X((infinity)) is the sheaf of differential operators on X [7], it is well known that giving an action of D-X((infinity)) on an O-X-module E is equivalent to giving an infinite sequence of O-X-modules descending E via the iterates of the Frobenius endomorphism of X [5]. We show that this result can be generalized to any infinitesimal deformation f : X -%26gt; S of a smooth morphism in characteristic p, endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that p belongs to an ideal of definition. In [12], dos Santos used this result to lift D-X((infinity))-modules from characteristic p to characteristic 0 with control of the differential Galois group.

• 出版日期2012