symmetric monoidal (∞,1)-category of spectra
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.
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:
For each $x\in I$, we have $\gamma_{1}(x)=x$.
For each $x,y\in I$ and $n\geq 0$, we have
For each $\lambda\in A$, each $x\in I$ and $n\geq 0$, we have
For each $x\in I$ and each $m,n\geq 0$, we have
For each $x\in I$ and each $m\geq 0$, $n\geq 1$, we have
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.
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.
Genuine powers can be constructed in the expected way from the divided powers, and when $A$ is torsion free, the reverse is true:
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 $x^0:=1$).
It is true for $n=0$ and $n=1$ by definition. For $n \geq 2$, this follows by induction, since $n! \gamma_n(x) = (n-1)! \gamma_{n-1}(x) \cdot 1! \gamma_1(x) = x^{n-1} \cdot x$.
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 $\gamma_n(x) = x^n / n!$.
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:
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.
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.