nLab divided power algebra

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.

Related concepts

• crystalline cohomology

References

Divided power algebras were originally introduced in

• Henri Cartan, Puissances divisées, Séminaire Henri Cartan, Tome 7 (1954–1955) no. 1, Exposé no. 7, 11 p. Numdam:SHC_1954–1955__7_1_A7_0.

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:

• Pierre Berthelot, Arthur Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, (ISBN:0-691-08218-9)

• Aise Johan de Jong et al., The Stacks Project, Chapter 09PD.

Recent works on divided power algebras include:

• Sacha Ikonicoff, Divided power algebras over an operad, Glasgow Math. J. 62 (2020) 477-517, doi:10.1017/S0017089519000223, pdf

• Sacha Ikonicoff, Divided power algebras and distributive laws, 2021, doi:10.48550/arXiv.2104.11736, pdf

It is related to differential categories in:

• Sacha Ikonicoff, Jean-Simon Pacaud Lemay, Cartesian Differential Comonads and New Models of Cartesian Differential Categories, 2021, doi:10.48550/arXiv.2108.04304, pdf

On divided powers:

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

See also:

• Wikipedia, Divided power structure

In relation to the sphere spectrum

• Sanath Devalapurkar, Divided powers and the sphere spectrum, blog post