Divided power structure

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 $n!$.

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:

1. $\gamma _{0}(x)=1$ for $x\in I$, while $\gamma _{n}(x)\in I$ for n > 0.
2. $\gamma _{n}(x+y)=\sum _{i=0}^{n}\gamma _{n-i}(x)\gamma _{i}(y)$ for $x,y\in I$.
3. $\gamma _{n}(\lambda x)=\lambda ^{n}\gamma _{n}(x)$ for $\lambda \in A,x\in I$.
4. $\gamma _{m}(x)\gamma _{n}(x)=((m,n))\gamma _{m+n}(x)$ for $x\in I$, where $((m,n))={\frac {(m+n)!}{m!n!}}$ is an integer.
5. $\gamma _{n}(\gamma _{m}(x))=C_{n,m}\gamma _{mn}(x)$ 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.

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, $x^{n}=n!\gamma _{n}(x)$.) Indeed, this is the example which motivates the definition in the first place.

• If A is a ring of characteristic $p>0$, 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 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 I and an A-linear map $M\to I$. (The case of divided power polynomials is the special case in which M is a free module over A of finite rank.)

• 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 a divided power ring; in fact, it is canonically isomorphic to $\Gamma _{A}({\check {M}})$ if M has 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.