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
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such that
1) ;
2) ;
3) ;
4) ;
5) ;
where in 3) and
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In case is a graded commutative algebra over
with
, these requirements are augmented as follows (and changed slightly):
6) ,
with 5) replaced by
5')
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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:
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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
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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].
References
[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 ![]() |
Ring with divided powers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_with_divided_powers&oldid=48573