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

with multiplication given by

and with grading given by

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

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

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

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