Example 23.2.3. Let p be a prime number. Let A be a ring such that every integer n not divisible by p is invertible, i.e., A is a \mathbf{Z}_{(p)}-algebra. Then I = pA has a canonical divided power structure. Namely, given x = pa \in I we set
The reader verifies immediately that p^ n/n! \in p\mathbf{Z}_{(p)} for n \geq 1 (for instance, this can be derived from the fact that the exponent of p in the prime factorization of n! is \left\lfloor n/p \right\rfloor + \left\lfloor n/p^2 \right\rfloor + \left\lfloor n/p^3 \right\rfloor + \ldots ), so that the definition makes sense and gives us a sequence of maps \gamma _ n : I \to I. It is a straightforward exercise to verify that conditions (1) – (5) of Definition 23.2.1 are satisfied. Alternatively, it is clear that the definition works for A_0 = \mathbf{Z}_{(p)} and then the result follows from Lemma 23.4.2.
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