# nLab divided power algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A divided power algebra is a commutative ring $A$ together with an ideal $I$ and a collection of operations $\{\gamma_{n}\colon I\to A\}_{n\in\mathbb{N}}$ which behave like operations of taking divided powers $x\mapsto x^{n}/n!$ in power series.

## Definition

A divided power algebra is a triple $(A,I,\gamma)$ with

• $A$ a commutative ring (with identity);

• $I$ an ideal of $A$;

• $\gamma=\{\gamma_{n}\colon I\to I\}_{n\geq 1}$ an indexed set of functions (of underlying sets);

where we additionally adopt the convention $\gamma_0(x) = 1$ (which is usually not in $I$), and this data is required to satisfy the following conditions:

1. For each $x\in I$, we have $\gamma_{1}(x)=x$.

2. For each $x,y\in I$ and $n\geq 0$, we have

$\gamma_{n}(x+y)=\sum_{k=0}^{n}\gamma_{n-k}(x)\gamma_{k}(y),$
3. For each $\lambda\in A$, each $x\in I$ and $n\geq 0$, we have

$\gamma_{n}(\lambda x)=\lambda^{n}\gamma_{n}(x).$
4. For each $x\in I$ and each $m,n\geq 0$, we have

$\gamma_{n}(x)\gamma_{m}(x)=\frac{(n+m)!}{n!m!}\gamma_{n+m}(x).$
5. For each $x\in I$ and each $m\geq 0$, $n\geq 1$, we have

$\gamma_{m}(\gamma_{n}(x))=\frac{(n m)!}{(n!)^{m}m!}\gamma_{n m}(x).$

For a given $(A,I)$, a divided power structure on $(A,I)$ is a $\gamma$ making $(A, I, \gamma)$ a divided power algebra.

If $A$ is an $R$-algebra for a ring $R$, we call it a divided power $R$-algebra or PD-$R$-algebra.

###### Remark

Some sources include $\gamma_0$ in the data rather than as convention. Some sources give the data as $\gamma_n : I \to A$ typing while including $\gamma_n(x) \in I$ for $n \geq 1$ as an axiom.

## Properties

Genuine powers can be constructed in the expected way from the divided powers, and when $A$ is torsion free, the reverse is true:

###### Proposition

If $(A,I,\gamma)$ is a divided power algebra, then $n! \gamma_n(x) = x^n$ for every $x \in I$ and $n \geq 0$ (taking $x^0:=1$).

###### Proof

It is true for $n=0$ and $n=1$ by definition. For $n \geq 2$, this follows by induction, since $n! \gamma_n(x) = (n-1)! \gamma_{n-1}(x) \cdot 1! \gamma_1(x) = x^{n-1} \cdot x$.

###### Proposition

If $A$ is a commutative, torsion free ring with an ideal $I$ such that $x^n$ is an $(n!)$-th multiple for every $x \in I$ and $n \geq 0$, then $(A,I)$ has a unique divided power structure, and it is given by $\gamma_n(x) = x^n / n!$.

###### Proof

The hypotheses imply the quotients $x^n / n!$ are unique and well-defined, and any divided power structure on $(A,I)$ must be given by that formula. It’s straightforward to check the definition does give a divided power algebra.

So in the torsion free case, the divided power algebras are precisely of the motivating form. In positive characteristic, though, examples can be somewhat more exotic.

Divided power algebras were originally introduced in

Their theory was further developed in Pierre Berthelot‘s PhD thesis (in the context of crystalline cohomology), which was later published as:

• Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique $p \gt 0$, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin, 1974. (doi:10.1007/BFb0068636, MR 0384804)

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