# 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 $x^{n}/n!$ meaningful even when it is not possible to actually divide by $n!$ .

## 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 $\gamma _{n}:I\to A$ for n=0, 1, 2, ... such that:

For convenience of notation, $\gamma _{n}(x)$ is often written as $x^{[n]}$ 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

$A\langle x_{1},x_{2},\ldots ,x_{n}\rangle$ consisting of divided power polynomials in the variables

$x_{1},x_{2},\ldots ,x_{n},$ that is sums of divided power monomials of the form

$cx_{1}^{[i_{1}]}x_{2}^{[i_{2}]}\cdots x_{n}^{[i_{n}]}$ with $c\in A$ . 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

$\Gamma _{A}(M),$ with PD ideal

$\Gamma _{+}(M)$ and an A-linear map

$M\to \Gamma _{+}(M).$ (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.