Example 23.6.2 (Adjoining odd 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 odd integer. In this setting we can adjoin a variable $T$ of degree $d$ to $A$. Namely, set

$A\langle T \rangle = A \oplus AT$

with grading given by $A\langle T \rangle _ m = A_ m \oplus A_{m - d}T$. We claim there is a unique divided power structure on $A\langle T \rangle$ compatible with the given divided power structure on $A$. Namely, we set

$\gamma _ n(x + yT) = \gamma _ n(x) + \gamma _{n - 1}(x)yT$

for $x \in A_{even, +}$ and $y \in A_{odd}$.

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