The Stacks project

Lemma 92.28.2. Let $S$ be a scheme. Let $X \to B$ and $Y \to B$ be morphisms of algebraic spaces over $S$. The object $E$ in ( satisfies $H^ i(E) = 0$ for $i = 0, -1$ and for a geometric point $(\overline{x}, \overline{y}) : \mathop{\mathrm{Spec}}(k) \to X \times _ B Y$ we have

\[ H^{-2}(E)_{(\overline{x}, \overline{y})} = \text{Tor}_1^ R(A, B) \otimes _{A \otimes _ R B} C \]

where $R = \mathcal{O}_{B, \overline{b}}$, $A = \mathcal{O}_{X, \overline{x}}$, $B = \mathcal{O}_{Y, \overline{y}}$, and $C = \mathcal{O}_{X \times _ B Y, (\overline{x}, \overline{y})}$.

Proof. The formation of the cotangent complex commutes with taking stalks and pullbacks, see Lemmas 92.18.9 and 92.18.3. Note that $C$ is a henselization of $A \otimes _ R B$. $L_{C/R} = L_{A \otimes _ R B/R} \otimes _{A \otimes _ R B} C$ by the results of Section 92.8. Thus the stalk of $E$ at our geometric point is the cone of the map $L_{A/R} \otimes C \to L_{A \otimes _ R B/R} \otimes C$. Therefore the results of the lemma follow from the case of rings, i.e., Lemma 92.15.2. $\square$

Comments (0)

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Your email address will not be published.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 09DM. Beware of the difference between the letter 'O' and the digit '0'.