Definition 23.6.1. Let $R$ be a ring. Let $A = \bigoplus _{d \geq 0} A_ d$ be a graded $R$-algebra which is strictly graded commutative. A collection of maps $\gamma _ n : A_{even, +} \to A_{even, +}$ defined for all $n > 0$ is called a divided power structure on $A$ if we have

1. $\gamma _ n(x) \in A_{2nd}$ if $x \in A_{2d}$,

2. $\gamma _1(x) = x$ for any $x$, we also set $\gamma _0(x) = 1$,

3. $\gamma _ n(x)\gamma _ m(x) = \frac{(n + m)!}{n! m!} \gamma _{n + m}(x)$,

4. $\gamma _ n(xy) = x^ n \gamma _ n(y)$ for all $x \in A_{even}$ and $y \in A_{even, +}$,

5. $\gamma _ n(xy) = 0$ if $x, y \in A_{odd}$ homogeneous and $n > 1$

6. if $x, y \in A_{even, +}$ then $\gamma _ n(x + y) = \sum _{i = 0, \ldots , n} \gamma _ i(x)\gamma _{n - i}(y)$,

7. $\gamma _ n(\gamma _ m(x)) = \frac{(nm)!}{n! (m!)^ n} \gamma _{nm}(x)$ for $x \in A_{even, +}$.

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