Divided power structure

In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form formula_1 meaningful even when it is not possible to actually divide by formula_2.
Definition.
Let "A" be a commutative ring with an ideal "I". A divided power structure (or PD-structure, after the French "puissances divisées") on "I" is a collection of maps formula_3 for "n" = 0, 1, 2, ... such that:
For convenience of notation, formula_18 is often written as formula_19 when it is clear what divided power structure is meant.
The term "divided power ideal" refers to an ideal with a given divided power structure, and "divided power ring" refers to a ring with a given ideal with divided power structure.
Homomorphisms of divided power algebras are ring homomorphisms that respects the divided power structure on its source and target.
Constructions.
If "A" is any ring, there exists a divided power ring
consisting of "divided power polynomials" in the variables
that is sums of "divided power monomials" of the form
with formula_30. Here the divided power ideal is the set of divided power polynomials with constant coefficient 0.
More generally, if "M" is an "A"-module, there is a universal "A"-algebra, called
with PD ideal
and an "A"-linear map
If "I" is any ideal of a ring "A", there is a universal construction which extends "A" with divided powers of elements of "I" to get a divided power envelope of "I" in "A".
Applications.
The divided power envelope is a fundamental tool in the theory of PD differential operators and crystalline cohomology, where it is used to overcome technical difficulties which arise in positive characteristic.
The divided power functor is used in the construction of co-Schur functors.