The Stacks project

Lemma 76.21.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The cohomology sheaves of the complex $\mathop{N\! L}\nolimits _{X/Y}$ are quasi-coherent, zero outside degrees $-1$, $0$ and equal to $\Omega _{X/Y}$ in degree $0$.

Proof. By construction of the naive cotangent complex in Modules on Sites, Section 18.35 we have that $\mathop{N\! L}\nolimits _{X/Y}$ is a complex sitting in degrees $-1$, $0$ and that its cohomology in degree $0$ is $\Omega _{X/Y}$ (by our construction of $\Omega _{X/Y}$ in Section 76.7). The sheaf of differentials is quasi-coherent (by Lemma 76.7.4). To finish the proof it suffices to show that $H^{-1}(\mathop{N\! L}\nolimits _{X/Y})$ is quasi-coherent. This follows by checking ├ętale locally (allowed by Lemma 76.21.2 and Properties of Spaces, Lemma 66.29.6) reducing to the case of schemes (Lemma 76.21.3) and finally using the result in the case of schemes (More on Morphisms, Lemma 37.13.3). $\square$


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Your email address will not be published.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 0D0Y. Beware of the difference between the letter 'O' and the digit '0'.