The Stacks project

Lemma 57.6.3. Let $(A, I, \gamma ) \to (B, J, \delta )$ be a homomorphism of divided power rings. Let $(B(1), J(1), \delta (1))$ be the coproduct of $(B, J, \delta )$ with itself over $(A, I, \gamma )$, i.e., such that

\[ \xymatrix{ (B, J, \delta ) \ar[r] & (B(1), J(1), \delta (1)) \\ (A, I, \gamma ) \ar[r] \ar[u] & (B, J, \delta ) \ar[u] } \]

is cocartesian. Denote $K = \mathop{\mathrm{Ker}}(B(1) \to B)$. Then $K \cap J(1) \subset J(1)$ is preserved by the divided power structure and

\[ \Omega _{B/A, \delta } = K/ \left(K^2 + (K \cap J(1))^{[2]}\right) \]

canonically.

Proof. The fact that $K \cap J(1) \subset J(1)$ is preserved by the divided power structure follows from the fact that $B(1) \to B$ is a homomorphism of divided power rings.

Recall that $K/K^2$ has a canonical $B$-module structure. Denote $s_0, s_1 : B \to B(1)$ the two coprojections and consider the map $\text{d} : B \to K/K^2 +(K \cap J(1))^{[2]}$ given by $b \mapsto s_1(b) - s_0(b)$. It is clear that $\text{d}$ is additive, annihilates $A$, and satisfies the Leibniz rule. We claim that $\text{d}$ is a divided power $A$-derivation. Let $x \in J$. Set $y = s_1(x)$ and $z = s_0(x)$. Denote $\delta $ the divided power structure on $J(1)$. We have to show that $\delta _ n(y) - \delta _ n(z) = \delta _{n - 1}(y)(y - z)$ modulo $K^2 +(K \cap J(1))^{[2]}$ for $n \geq 1$. The equality holds for $n = 1$. Assume $n > 1$. Note that $\delta _ i(y - z)$ lies in $(K \cap J(1))^{[2]}$ for $i > 1$. Calculating modulo $K^2 + (K \cap J(1))^{[2]}$ we have

\[ \delta _ n(z) = \delta _ n(z - y + y) = \sum \nolimits _{i = 0}^ n \delta _ i(z - y)\delta _{n - i}(y) = \delta _{n - 1}(y) \delta _1(z - y) + \delta _ n(y) \]

This proves the desired equality.

Let $M$ be a $B$-module. Let $\theta : B \to M$ be a divided power $A$-derivation. Set $D = B \oplus M$ where $M$ is an ideal of square zero. Define a divided power structure on $J \oplus M \subset D$ by setting $\delta _ n(x + m) = \delta _ n(x) + \delta _{n - 1}(x)m$ for $n > 1$, see Lemma 57.3.1. There are two divided power algebra homomorphisms $B \to D$: the first is given by the inclusion and the second by the map $b \mapsto b + \theta (b)$. Hence we get a canonical homomorphism $B(1) \to D$ of divided power algebras over $(A, I, \gamma )$. This induces a map $K \to M$ which annihilates $K^2$ (as $M$ is an ideal of square zero) and $(K \cap J(1))^{[2]}$ as $M^{[2]} = 0$. The composition $B \to K/K^2 + (K \cap J(1))^{[2]} \to M$ equals $\theta $ by construction. It follows that $\text{d}$ is a universal divided power $A$-derivation and we win. $\square$


Comments (4)

Comment #4222 by Dario Weißmann on

Concerning the second paragraph of the proof. I suggest replacing "We will show this by induction ...." (till the end of the paragraph) by the shorter (and easier) direct calculation: The claim is clear for . Assume . Note that lies in for . Calculating modulo we have The claim follows.

Comment #4402 by Dario Weißmann on

Noticed two typos in the fix: "Calculating module..." and "This prove ..."

There are also:

  • 5 comment(s) on Section 57.6: Module of differentials

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 07HT. Beware of the difference between the letter 'O' and the digit '0'.