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divided power algebra

Contents

Contents

Idea

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.

Definition

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.

Remark

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.

Properties

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

Proposition

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).

Proof

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.

Proposition

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!.

Proof

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.

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

Review:

See also:

In relation to the sphere spectrum

Last revised on July 12, 2021 at 14:39:32. See the history of this page for a list of all contributions to it.