Divided power structure

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In commutative algebra, a divided power structure is a way of making sense of expressions of the form $x n / n!$ which is meaningful even when it is not possible to actually divide by $n!$ .

1 Definition
2 Examples
3 Constructions
4 Applications
5 References

[edit] 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:

$γ 0 (x) = 1$ and $γ 1 (x) = x$ 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 $x, y \in I$ .
$γ n (λ x) = λ n γ n (x)$ for $\lambda \in A, x \in I$ .
$γ m (x)γ n (x) = ((m, n))γ m + n (x)$ for $x \in I$ , where $((m, n)) = \frac{(m+n)!}{m! n!}$ is an integer.
$γ n (γ m (x)) = C n, m γ m n (x)$ for $x \in I$ , where $C_{n, m} = \frac{(mn)!}{(m!)^n n!}$ is an integer.

For convenience of notation, $γ n (x)$ 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.

[edit] Examples

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, $x n = n!γ n (x)$ .) 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 $γ n (x) = 0$ 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.

[edit] 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

Γ A (M),

with PD ideal

Γ + (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.

[edit] 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.