The Stacks project

Lemma 23.6.11. Let $R$ be a commutative ring. Suppose that $(A, \text{d}, \gamma )$ and $(B, \text{d}, \gamma )$ are as in Definition 23.6.5. Let $\overline{\varphi } : H_0(A) \to H_0(B)$ be an $R$-algebra map. Assume

  1. $A$ is a graded divided power polynomial algebra over $R$.

  2. $H_ k(B) = 0$ for $k > 0$.

Then there exists a map $\varphi : A \to B$ of differential graded $R$-algebras compatible with divided powers that lifts $\overline{\varphi }$.

Proof. The assumption means that $A$ is obtained from $R$ by successively adjoining some set of polynomial generators in degree zero, exterior generators in positive odd degrees, and divided power generators in positive even degrees. So we have a filtration $R \subset A(0) \subset A(1) \subset \ldots $ of $A$ such that $A(m + 1)$ is obtained from $A(m)$ by adjoining generators of the appropriate type (which we simply call “divided power generators”) in degree $m + 1$. In particular, $A(0) \to H_0(A)$ is a surjection from a (usual) polynomial algebra over $R$ onto $H_0(A)$. Thus we can lift $\overline{\varphi }$ to an $R$-algebra map $\varphi (0) : A(0) \to B_0$.

Write $A(1) = A(0)\langle T_ j:j\in J\rangle $ for some set $J$ of divided power variables $T_ j$ of degree $1$. Let $f_ j \in B_0$ be $f_ j = \varphi (0)(\text{d}(T_ j))$. Observe that $f_ j$ maps to zero in $H_0(B)$ as $\text{d}T_ j$ maps to zero in $H_0(A)$. Thus we can find $b_ j \in B_1$ with $\text{d}(b_ j) = f_ j$. By the universal property of divided power polynomial algebras from Lemma 23.5.1, we find a lift $\varphi (1) : A(1) \to B$ of $\varphi (0)$ mapping $T_ j$ to $f_ j$.

Having constructed $\varphi (m)$ for some $m \geq 1$ we can construct $\varphi (m + 1) : A(m + 1) \to B$ in exactly the same manner. We omit the details. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 23.6: Tate resolutions

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09PP. Beware of the difference between the letter 'O' and the digit '0'.