# 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.

## Categorically

We can define the concept of divided power in any symmetric monoidal category.

Let $\mathcal{C}$ be a symmetric monoidal category.

The $n^{th}$ divided power of an object $A$ is then defined as the equalizer of the following diagram:

where there is one arrow for every $\sigma \in \mathfrak{S}_{n}$. We write $\sigma$ for the natural transformation associated to $\sigma$, which is defined in the entry symmetric monoidal category.

The $n-th$ divided power $\Gamma_{n}(A)$ is thus described by the following equalizer diagram:

In the category of modules over some commutative ring, the $n-th$ divided power of a module $A$ is equivalently descibed as the space $(A^{\otimes n})^{\mathfrak{S}_{n}}$ of invariants under the action of $\mathfrak{S}_{n}$ on $A^{\otimes n}$ by permutation of the factors. Note that the $n-th$ symmetric power $S_{n}(A)$ of an object $A$ in a symmetric monoidal category is described as the coequalizer of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$. In a category of modules, it is equivalently described as the space $(A^{\otimes n})_{\mathfrak{S}_{n}}$ of coinvariants by the action of $\mathfrak{S}_{n}$. We then have the relation $(S_{n}(A^{*}))^{*} = \Gamma_{n}(A)$ which probably means that these powers can be interpreted as graded exponential modalities in a kind of graded differential linear logic. In characteristic 0, we have $\Gamma_{n}(A) \cong S_{n}(A)$. Nondegenerated models of such a logic in a category of vector spaces would thus be only in the case where the field is of positive characteristic.

## References

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)

Review:

Recent works on divided power algebras include:

It is related to differential categories in:

On divided powers:

• Luis Narváez Macarro?, Hasse-Schmidt derivations, divided powers and differential smoothness, 2009, doi:10.5802/aif.2513, pdf