nLab divided power algebra




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


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

where we additionally adopt the convention γ 0(x)=1\gamma_0(x) = 1 (which is usually not in II), and this data is required to satisfy the following conditions:

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

  2. For each x,yIx,y\in I and n0n\geq 0, we have

    γ n(x+y)= k=0 nγ nk(x)γ k(y), \gamma_{n}(x+y)=\sum_{k=0}^{n}\gamma_{n-k}(x)\gamma_{k}(y),
  3. For each λA\lambda\in A, each xIx\in I and n0n\geq 0, we have

    γ n(λx)=λ nγ n(x). \gamma_{n}(\lambda x)=\lambda^{n}\gamma_{n}(x).
  4. For each xIx\in I and each m,n0m,n\geq 0, we have

    γ n(x)γ m(x)=(n+m)!n!m!γ n+m(x). \gamma_{n}(x)\gamma_{m}(x)=\frac{(n+m)!}{n!m!}\gamma_{n+m}(x).
  5. For each xIx\in I and each m0m\geq 0, n1n\geq 1, we have

    γ m(γ n(x))=(nm)!(n!) mm!γ nm(x). \gamma_{m}(\gamma_{n}(x))=\frac{(n m)!}{(n!)^{m}m!}\gamma_{n m}(x).

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

If AA is an RR-algebra for a ring RR, we call it a divided power RR-algebra or PD-RR-algebra.


Some sources include γ 0\gamma_0 in the data rather than as convention. Some sources give the data as γ n:IA\gamma_n : I \to A typing while including γ n(x)I\gamma_n(x) \in I for n1n \geq 1 as an axiom.


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


If (A,I,γ)(A,I,\gamma) is a divided power algebra, then n!γ n(x)=x nn! \gamma_n(x) = x^n for every xIx \in I and n0n \geq 0 (taking x 0:=1x^0:=1).


It is true for n=0n=0 and n=1n=1 by definition. For n2n \geq 2, this follows by induction, since n!γ n(x)=(n1)!γ n1(x)1!γ 1(x)=x n1xn! \gamma_n(x) = (n-1)! \gamma_{n-1}(x) \cdot 1! \gamma_1(x) = x^{n-1} \cdot x.


If AA is a commutative, torsion free ring with an ideal II such that x nx^n is an (n!)(n!)-th multiple for every xIx \in I and n0n \geq 0, then (A,I)(A,I) has a unique divided power structure, and it is given by γ n(x)=x n/n!\gamma_n(x) = x^n / n!.


The hypotheses imply the quotients x n/n!x^n / n! are unique and well-defined, and any divided power structure on (A,I)(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.


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

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

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

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

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

In the category of modules over some commutative ring, the nthn-th divided power of a module AA is equivalently descibed as the space (A n) 𝔖 n(A^{\otimes n})^{\mathfrak{S}_{n}} of invariants under the action of 𝔖 n\mathfrak{S}_{n} on A nA^{\otimes n} by permutation of the factors. Note that the nthn-th symmetric power S n(A)S_{n}(A) of an object AA in a symmetric monoidal category is described as the coequalizer of the n!n! permutations A nA nA^{\otimes n} \rightarrow A^{\otimes n}. In a category of modules, it is equivalently described as the space (A n) 𝔖 n(A^{\otimes n})_{\mathfrak{S}_{n}} of coinvariants by the action of 𝔖 n\mathfrak{S}_{n}. We then have the relation (S n(A *)) *=Γ n(A)(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 Γ n(A)S n(A)\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.


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>0p \gt 0, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin, 1974. (doi:10.1007/BFb0068636, MR 0384804)


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

See also:

In relation to the sphere spectrum

Last revised on November 19, 2022 at 20:44:34. See the history of this page for a list of all contributions to it.