Remark 23.6.4. We can also adjoin a set (possibly infinite) of exterior or divided power generators in a given degree $d > 0$, rather than just one as in Examples 23.6.2 and 23.6.3. Namely, following Remark 23.5.2: for $(A,\gamma )$ as above and a set $J$, let $A\langle T_ j:j\in J\rangle $ be the directed colimit of the algebras $A\langle T_ j:j\in S\rangle $ over all finite subsets $S$ of $J$. It is immediate that this algebra has a unique divided power structure, compatible with the given structure on $A$ and on each generator $T_ j$.

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