Divided power structure

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:

$\gamma_0(x) = 1$ and $\gamma_1(x) = x$ for $x \in I$ , while $\gamma_n(x) \in I$ for n > 0.
$\gamma_n(x + y) = \sum_{i=0}^n \gamma_{n-i}(x) \gamma_i(y)$ for $x, y \in I$ .
$\gamma_n(\lambda x) = \lambda^n \gamma_n(x)$ for $\lambda \in A, x \in I$ .
$\gamma_m(x) \gamma_n(x) = ((m, n)) \gamma_{m+n}(x)$ for $x \in I$ , where $((m, n)) = \frac{(m+n)!}{m! n!}$ is an integer.
$\gamma_n(\gamma_m(x)) = C_{n, m} \gamma_{mn}(x)$ for $x \in I$ , where $C_{n, m} = \frac{(mn)!}{(m!)^n n!}$ is an integer.

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.

Examples

$\mathbb{Z}\langle{x}\rangle:=\mathbb{Z}[x,\frac{x^2}{2},\ldots,\frac{x^n}{n!},\ldots]\subset \mathbb{Q}[x]$ is a divided power algebra, it is the free divided power algebra over $\mathbb{Z}$ on one generator.

If A is an algebra over the rational numbers Q, then every ideal I has a unique divided power structure where $\gamma_n(x) = \frac{1}{n!} \cdot x^n$ . (The uniqueness follows from the easily verified fact that in general, $x^n = n! \gamma_n(x)$ .) Indeed, this is the example which motivates the definition in the first place.

If A is a ring of characteristic $p > 0$ , where p is prime, and I is an ideal such that $I^p = 0$ , then we can define a divided power structure on I where $\gamma_n(x) = \frac{1}{n!} x^n$ if n < p, and $\gamma_n(x) = 0$ if $n \geq p$ . (Note the distinction between $I^p$ and the ideal generated by $x^p$ for $x \in I$ ; 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 $S^\cdot M$ denote the symmetric algebra of M over A. Then its dual $(S^\cdot M) \check{~} = Hom_A(S^\cdot M, A)$ has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a natural completion of $\Gamma_A(\check{M})$ (see below) if M has finite rank.

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

c x_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.

References

Berthelot, Pierre; Ogus, Arthur (1978). Notes on Crystalline Cohomology. Annals of Mathematics Studies. Princeton University Press. Zbl 0383.14010.
Hazewinkel, Michiel (1978). Formal Groups and Applications. Pure and applied mathematics, a series of monographs and textbooks 78. Elsevier. p. 507. ISBN 0123351502. Zbl 0454.14020.