Powers, Exponentials and Crystals

Other variants (for example requiring that for each xfl, there be an n depending on x such that y (x) = 0 for i n) are possible, but we shall use the condition (1.1). The definition implies (taking k=N, i =... =i = 1) 

Remark (1.4) We shall use (sometimes implicitly) the following properties of r(M) which are proved by Roby [26, 27]. 

2) There is a unique system of divided powers on the augmentation 

To check that exp and log are inverse we can clearly reduce to the universal case where A = P ( ), the completion being taken with 

M by a quasi-coherent (3c-module. Divided powers on I are given 

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