The Stacks project

Lemma 23.6.12. Let $R$ be a commutative ring. Let $S$ and $T$ be commutative $R$-algebras. Then there is a canonical structure of a strictly graded commutative $R$-algebra with divided powers on

\[ \operatorname {Tor}_*^ R(S, T). \]

Proof. Choose a factorization $R \to A \to S$ as above. Since $A \to S$ is a quasi-isomorphism and since $A_ d$ is a free $R$-module, we see that the differential graded algebra $B = A \otimes _ R T$ computes the Tor groups displayed in the lemma. Choose a surjection $R[y_ j:j\in J] \to T$. Then we see that $B$ is a quotient of the differential graded algebra $A[y_ j:j\in J]$ whose homology sits in degree $0$ (it is equal to $S[y_ j:j\in J]$). By Lemma 23.6.7 the differential graded algebras $B$ and $A[y_ j:j\in J]$ have divided power structures compatible with the differentials. Hence we obtain our divided power structure on $H(B)$ by Lemma 23.6.6.

The divided power algebra structure constructed in this way is independent of the choice of $A$. Namely, if $A'$ is a second choice, then Lemma 23.6.11 implies there is a map $A \to A'$ preserving all structure and the augmentations towards $S$. Then the induced map $B = A \otimes _ R T \to A' \otimes _ R T' = B'$ also preserves all structure and is a quasi-isomorphism. The induced isomorphism of Tor algebras is therefore compatible with products and divided powers. $\square$

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