# Divided power structure

{{ safesubst:#invoke:Unsubst||$N=Context |date=__DATE__ |$B=
{{#invoke:Message box|ambox}}
}}In mathematics, specifically commutative algebra, a **divided power structure** is a way of making expressions of the form meaningful even when it is not possible to actually divide by .

## 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 for *n*=0, 1, 2, ... such that:

For convenience of notation, is often written as 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.

## Examples

- If
*A*is an algebra over the rational numbers**Q**, then every ideal*I*has a unique divided power structure where . (The uniqueness follows from the easily verified fact that in general, .) Indeed, this is the example which motivates the definition in the first place.

- If
*A*is a ring of characteristic , where*p*is prime, and*I*is an ideal such that , then we can define a divided power structure on*I*where if*n*<*p*, and if . (Note the distinction between and the ideal generated by for ; the latter is always zero if a divided power structure exists, while the former is not necessarily zero.)

- If
*M*is an*A*-module, let denote the symmetric algebra of*M*over*A*. Then its dual has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a natural completion of (see below) if*M*has finite rank.

## 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 . 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

(The case of divided power polynomials is the special case in which *M* is a free module over *A* of finite rank.)

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.

## References

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}