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
where in 3) and
In case is a graded commutative algebra over with , these requirements are augmented as follows (and changed slightly):
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|
Ring with divided powers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_with_divided_powers&oldid=16426