Ring with divided powers

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Let be a commutative ring with unit, and let be an augmented -algebra, i.e. there is given a homomorphism of -algebras . A divided power structure on (or, more precisely, on the augmentation ideal ) is a sequence of mappings

such that

1) ;

2) ;

3) ;

4) ;

5) ;

where in 3) and

In case is a graded commutative algebra over with , these requirements are augmented as follows (and changed slightly):

6) ,

with 5) replaced by


Given an -module , an algebra with divided powers is constructed as follows. It is generated (as an -algebra) by symbols , , and between these symbols the following relations are imposed:

This satisfies 1)–5). The augmentation sends to (). If one assigns to the degree , a graded commutative algebra is obtained with , which satisfies 1)–4), 5'), 6).

If is a -algebra, divided powers can always be defined as . The relations 1)–5) can be understood as a way of writing down the interrelations between such "divided powers" (such as the one resulting from the binomial theorem) without having to use division by integers.

A divided power sequence in a co-algebra is a sequence of elements satisfying

Divided power sequences are used in the theories of Hopf algebras and formal groups (cf. Formal group; Hopf algebra), [a1][a3]. Rings with divided powers occur in algebraic topology (where they provide a natural setting for power cohomology operations), [a4], [a5], and the theory of formal groups [a3], [a2].


[a1] N. Roby, "Les algèbres à puissances divisées" Bull. Soc. Math. France , 89 (1965) pp. 75–91
[a2] M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978)
[a3] P. Cartier, "Exemples d'hyperalgèbres" , Sem. S. Lie 1955/56 , 3 , Secr. Math. Univ. Paris (1957)
[a4] E. Thomas, "The generalized Pontryagin cohomology operations and rings with divided powers" , Amer. Math. Soc. (1957)
[a5] S. Eilenberg, S. MacLane, "On the groups , II" Ann. of Math. , 60 (1954) pp. 49–189
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