The Stacks project

83.35 Fppf descent of complexes

In this section we pull some of the previously shown results together for fppf coverings of algebraic spaces and derived categories of quasi-coherent modules.

Lemma 83.35.1. Let $X$ be an algebraic space over a scheme $S$. Let $K, E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $a : U \to X$ be an fppf hypercovering. Assume that for all $n \geq 0$ we have

\[ \mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_{U_ n}}^ i(La_ n^*K, La_ n^*E) = 0 \text{ for } i < 0 \]

Then we have

  1. $\mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ i(K, E) = 0$ for $i < 0$, and

  2. there is an exact sequence

    \[ 0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K, E) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_0}}(La_0^*K, La_0^*E) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_1}}(La_1^*K, La_1^*E) \]

Proof. Write $K_ n = La_ n^*K$ and $E_ n = La_ n^*E$. Then these are the simplicial systems of the derived category of modules (Definition 83.14.1) associated to $La^*K$ and $La^*E$ (Lemma 83.14.2) where $a : U_{\acute{e}tale}\to X_{\acute{e}tale}$ is as in Section 83.32. Let us prove (2) first. By Lemma 83.34.4 we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K, E) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(La^*K, La^*E) \]

Thus the sequence looks like this:

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(La^*K, La^*E) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_0}}(K_0, E_0) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_1}}(K_1, E_1) \]

The first arrow is injective by Lemma 83.14.4. The image of this arrow is the kernel of the second by Lemma 83.14.5. This finishes the proof of (2). Part (1) follows by applying part (2) with $K[i]$ and $E$ for $i > 0$. $\square$

Lemma 83.35.2. Let $X$ be an algebraic space over a scheme $S$. Let $a : U \to X$ be an fppf hypercovering. Suppose given $K_0 \in D_\mathit{QCoh}(U_0)$ and an isomorphism

\[ \alpha : L(f_{\delta _1^1})^*K_0 \longrightarrow L(f_{\delta _0^1})^*K_0 \]

satisfying the cocycle condition on $U_1$. Set $\tau ^ n_ i : [0] \to [n]$, $0 \mapsto i$ and set $K_ n = Lf_{\tau ^ n_ n}^*K_0$. Assume $\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{U_ n}}(K_ n, K_ n) = 0$ for $i < 0$. Then there exists an object $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and an isomorphism $La_0^*K \to K$ compatible with $\alpha $.

Proof. We claim that the objects $K_ n$ form the members of a simplicial system of the derived category of modules (Definition 83.14.1) of the ringed simplicial site $U_{\acute{e}tale}$ of Section 83.32. The construction is analogous to the construction discussed in Descent, Section 35.3 from which we borrow the notation $\tau ^ n_ i : [0] \to [n]$, $0 \mapsto i$ and $\tau ^ n_{ij} : [1] \to [n]$, $0 \mapsto i$, $1 \mapsto j$. Given $\varphi : [n] \to [m]$ we define $K_\varphi : L(f_\varphi )^*K_ n \to K_ m$ using

\[ \xymatrix{ L(f_\varphi )^*K_ n \ar@{=}[r] & L(f_\varphi )^* L(f_{\tau ^ n_ n})^*K_0 \ar@{=}[r] & L(f_{\tau ^ m_{\varphi (n)}})^*K_0 \ar@{=}[r] & L(f_{\tau ^ m_{\varphi (n)m}})^* L(f_{\delta ^1_1})^*K_0 \ar[d]_{L(f_{\tau ^ m_{\varphi (n)m}})^*\alpha } \\ & K_ m \ar@{=}[r] & L(f_{\tau ^ m_ m})^*K_0 \ar@{=}[r] & L(f_{\tau ^ m_{\varphi (n)m}})^* L(f_{\delta ^1_0})^*K_0 } \]

We omit the verification that the cocycle condition implies the maps compose correctly (in their respective derived categories) and hence give rise to a simplicial systems of the derived category of modules1. Once this is verified, we obtain an object $K' \in D_\mathit{QCoh}(\mathcal{O}_{U_{\acute{e}tale}})$ such that $(K_ n, K_\varphi )$ is the system deduced from $K'$, see Lemma 83.14.6. Finally, we apply Lemma 83.34.4 to see that $K' = La^*K$ for some $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ as desired. $\square$

[1] This verification is the same as that done in the proof of Lemma 83.12.4 as well as in the chapter on descent referenced above. We should probably write this as a general lemma about fibred and cofibred categories over $\Delta $.

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 0DL8. Beware of the difference between the letter 'O' and the digit '0'.