Example 23.6.3 (Adjoining even variable). Let $R$ be a ring. Let $(A, \gamma )$ be a strictly graded commutative graded $R$-algebra endowed with a divided power structure as in the definition above. Let $d > 0$ be an even integer. In this setting we can adjoin a variable $T$ of degree $d$ to $A$. Namely, set

$A\langle T \rangle = A \oplus AT \oplus AT^{(2)} \oplus AT^{(3)} \oplus \ldots$

with multiplication given by

$T^{(n)} T^{(m)} = \frac{(n + m)!}{n!m!} T^{(n + m)}$

$A\langle T \rangle _ m = A_ m \oplus A_{m - d}T \oplus A_{m - 2d}T^{(2)} \oplus \ldots$

We claim there is a unique divided power structure on $A\langle T \rangle$ compatible with the given divided power structure on $A$ such that $\gamma _ n(T^{(i)}) = T^{(ni)}$. To define the divided power structure we first set

$\gamma _ n\left(\sum \nolimits _{i > 0} x_ i T^{(i)}\right) = \sum \prod \nolimits _{n = \sum e_ i} x_ i^{e_ i} T^{(ie_ i)}$

if $x_ i$ is in $A_{even}$. If $x_0 \in A_{even, +}$ then we take

$\gamma _ n\left(\sum \nolimits _{i \geq 0} x_ i T^{(i)}\right) = \sum \nolimits _{a + b = n} \gamma _ a(x_0)\gamma _ b\left(\sum \nolimits _{i > 0} x_ iT^{(i)}\right)$

where $\gamma _ b$ is as defined above.

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