# Ring with divided powers

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

5') 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].

How to Cite This Entry:
Ring with divided powers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_with_divided_powers&oldid=16426