The Stacks project

Lemma 91.16.7. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\mathcal{I}$ of square zero. Let $K_0 \in D^-(\mathcal{O})$. A lift of $K_0$ is a pair $(K, \alpha _0)$ consisting of an object $K$ in $D^-(\mathcal{O})$ and an isomorphism $\alpha _0 : K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}_0 \to K_0$ in $D(\mathcal{O}_0)$.

  1. Given a lift $(K, \alpha )$ the group of automorphism of the pair is canonically the cokernel of a map

    \[ \mathop{\mathrm{Ext}}\nolimits ^{-1}_{\mathcal{O}_0}(K_0, K_0) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_0}(K_0, K_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I}) \]
  2. If there is a lift, then the set of isomorphism classes of lifts is principal homogenenous under $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_0}(K_0, K_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I})$.

Proof. An automorphism of $(K, \alpha )$ is a map $\varphi : K \to K$ in $D(\mathcal{O})$ with $\varphi \otimes _\mathcal {O} \text{id}_{\mathcal{O}_0} = \text{id}$. This is the same thing as saying that

\[ K \xrightarrow {\varphi - \text{id}} K \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}_0 \]

is zero. We conclude the group of automorphisms is the cokernel of a map

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(K, K_0[-1]) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(K, K_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I}) \]

by the distinguished triangle

\[ K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I} \to K \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}_0 \to (K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I})[1] \]

in $D(\mathcal{O})$ and Derived Categories, Lemma 13.4.2. To translate into the groups in the lemma use adjunction of the restriction functor $D(\mathcal{O}_0) \to D(\mathcal{O})$ and $- \otimes _\mathcal {O} \mathcal{O}_0 : D(\mathcal{O}) \to D(\mathcal{O}_0)$. This proves (1).

Proof of (2). Assume that $K_0 = K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}_0$ in $D(\mathcal{O})$. By Lemma 91.16.6 the map sending a lift $(K', \alpha _0)$ to the obstruction $o(\alpha _0)$ to lifting $\alpha _0$ defines a canonical injective map from the set of isomomorphism classes of pairs to $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_0}(K_0, K_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I})$. To finish the proof we show that it is surjective. Pick $\xi : K_0 \to (K_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I})[1]$ in the $\mathop{\mathrm{Ext}}\nolimits ^1$ of the lemma. Choose a bounded above complex $\mathcal{F}^\bullet $ of flat $\mathcal{O}$-modules representing $K$. The map $\xi $ can be represented as $t \circ s^{-1}$ where $s : \mathcal{K}_0^\bullet \to \mathcal{F}_0^\bullet $ is a quasi-isomorphism and $t : \mathcal{K}_0^\bullet \to \mathcal{F}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I}[1]$ is a map of complexes. By Lemma 91.16.5 we can assume there exists a quasi-isomorphism $\mathcal{G}^\bullet \to \mathcal{F}^\bullet $ of complexes of $\mathcal{O}$-modules such that $\mathcal{G}_0^\bullet \to \mathcal{F}_0^\bullet $ factors through $s$ up to homotopy. We may and do replace $\mathcal{G}^\bullet $ by a bounded above complex of flat $\mathcal{O}$-modules (by picking a qis from such to $\mathcal{G}^\bullet $ and replacing). Then we see that $\xi $ is represented by a map of complexes $t : \mathcal{G}_0^\bullet \to \mathcal{F}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I}[1]$ and the quasi-isomorphism $\mathcal{G}_0^\bullet \to \mathcal{F}_0^\bullet $. Set

\[ \mathcal{H}^ n = \mathcal{F}^ n \times _{\mathcal{F}_0^ n} \mathcal{G}_0^ n \]

with differentials

\[ \mathcal{H}^ n \to \mathcal{H}^{n + 1},\quad (f^ n, g_0^ n) \mapsto (d(f^ n) + t(g_0^ n), d(g_0^ n)) \]

This makes sense as $\mathcal{F}_0^{n + 1} \otimes _{\mathcal{O}_0} \mathcal{I} = \mathcal{F}^{n + 1} \otimes _\mathcal {O} \mathcal{I} = \mathcal{I}\mathcal{F}^{n + 1} \subset \mathcal{F}^{n + 1}$. We omit the computation that shows that $\mathcal{H}^\bullet $ is a complex of $\mathcal{O}$-modules. By construction there is a short exact sequence

\[ 0 \to \mathcal{F}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I} \to \mathcal{H}^\bullet \to \mathcal{G}_0^\bullet \to 0 \]

of complexes of $\mathcal{O}$-modules. Exactly as in the proof of Lemma 91.16.4 one shows that this sequence induces an isomorphism $\alpha _0 : \mathcal{H}^\bullet \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}_0 \to \mathcal{G}_0^\bullet $ in $D(\mathcal{O}_0)$. In other words, we have produced a pair $(\mathcal{H}^\bullet , \alpha _0)$. We omit the verification that $o(\alpha _0) = \xi $; hint: $o(\alpha _0)$ can be computed explicitly in this case as we have maps $\mathcal{H}^ n \to \mathcal{F}^ n$ (not compatible with differentials) lifting the components of $\alpha _0$. This finishes the proof. $\square$


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