Divided Power Structure

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 	o 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)) = rac{(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} = rac{(mn)!}{(m!)^n n!}

is an integer.

For convenience of notation,

\gamma_n(x)

is often written as

x^{

{Link without Title} } 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.

EXAMPLES

If ''A'' is an algebra over the rational numbers Q, then every ideal ''I'' has a unique divided power structure where $\gamma_n(x) = rac{1}{n!} \cdot x^n$ . (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 $\gamma_n(x) = rac{1}{n!} x^n$ 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 
angle

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

:

} \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 	o \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 .

REFERENCES

Pierre Berthelot and Arthur Ogus, ''Notes on Crystalline Cohomology.'' Annals of Mathematics Studies. Princeton University Press, 1978.

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Divided Power Structure