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Definition 60.2.2. Let $(A, I, \gamma )$ be a divided power ring. Let $A \to B$ be a ring map. Let $J \subset B$ be an ideal with $IB \subset J$. The divided power algebra $(D, \bar J, \bar\gamma )$ constructed in Lemma 60.2.1 is called the divided power envelope of $J$ in $B$ relative to $(A, I, \gamma )$ and is denoted $D_ B(J)$ or $D_{B, \gamma }(J)$.

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