Divided power structure

Examples

• The free divided power algebra over ${\displaystyle \mathbb{Z}}$ on one generator:

${\displaystyle \mathbb{Z}⟨x⟩:=\mathbb{Z}\left[x,\frac{x^2}{2},\dots ,\frac{x^n}{n!},\dots \right]\subset \mathbb{Q}[x].}$

• If A is an algebra over ${\displaystyle \mathbb{Q},}$ then every ideal I has a unique divided power structure where ${\displaystyle {\gamma }_n(x)=\frac{1}{n!}\cdot x^n.}$ Indeed, this is the example which motivates the definition in the first place.

• If M is an A-module, let ${\displaystyle S^{\bullet }M}$ denote the symmetric algebra of M over A. Then its dual ${\displaystyle (S^{\bullet }M{)}^{\vee }={\text{Hom}}_A(S^{\bullet }M,A)}$ has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a natural completion of ${\displaystyle {\mathrm{\Gamma }}_A(\check{M})}$ (see below) if M has finite rank.

Constructions

If A is any ring, there exists a divided power ring

${\displaystyle A⟨x_1,x_2,\dots ,x_n⟩}$

consisting of divided power polynomials in the variables

${\displaystyle x_1,x_2,\dots ,x_n,}$

that is sums of divided power monomials of the form

${\displaystyle c{x}_1^{[i_1]}{x}_2^{[i_2]}\cdots x_n^{[i_n]}}$

with ${\displaystyle c\in A}$. 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

${\displaystyle {\mathrm{\Gamma }}_A(M),}$

with PD ideal

${\displaystyle {\mathrm{\Gamma }}_{+}(M)}$

and an A-linear map

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

(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.