Lemma 23.6.6 (09PK)—The Stacks project

Lemma 23.6.6. Let $(A, \text{d}, \gamma )$ and $(B, \text{d}, \gamma )$ be as in Definition 23.6.5. Let $f : A \to B$ be a map of differential graded algebras compatible with divided power structures. Assume

$H_ k(A) = 0$ for $k > 0$, and
$f$ is surjective.

Then $\gamma $ induces a divided power structure on the graded $R$-algebra $H(B)$.

Proof. Suppose that $x$ and $x'$ are homogeneous of the same degree $2d$ and define the same cohomology class in $H(B)$. Say $x' - x = \text{d}(w)$. Choose a lift $y \in A_{2d}$ of $x$ and a lift $z \in A_{2d + 1}$ of $w$. Then $y' = y + \text{d}(z)$ is a lift of $x'$. Hence

\[ \gamma _ n(y') = \sum \gamma _ i(y) \gamma _{n - i}(\text{d}(z)) = \gamma _ n(y) + \sum \nolimits _{i < n} \gamma _ i(y) \gamma _{n - i}(\text{d}(z)) \]

Since $A$ is acyclic in positive degrees and since $\text{d}(\gamma _ j(\text{d}(z))) = 0$ for all $j$ we can write this as

\[ \gamma _ n(y') = \gamma _ n(y) + \sum \nolimits _{i < n} \gamma _ i(y) \text{d}(z_ i) \]

for some $z_ i$ in $A$. Moreover, for $0 < i < n$ we have

\[ \text{d}(\gamma _ i(y) z_ i) = \text{d}(\gamma _ i(y))z_ i + \gamma _ i(y)\text{d}(z_ i) = \text{d}(y) \gamma _{i - 1}(y) z_ i + \gamma _ i(y)\text{d}(z_ i) \]

and the first term maps to zero in $B$ as $\text{d}(y)$ maps to zero in $B$. Hence $\gamma _ n(x')$ and $\gamma _ n(x)$ map to the same element of $H(B)$. Thus we obtain a well defined map $\gamma _ n : H_{2d}(B) \to H_{2nd}(B)$ for all $d > 0$ and $n > 0$. We omit the verification that this defines a divided power structure on $H(B)$. $\square$

The Stacks project

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