The Stacks project

73.24 Limits and derived categories

In this section we collect some results about the derived category of an algebraic space which is the limit of an inverse system of algebraic spaces. More precisely, we will work in the following setting.

Situation 73.24.1. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of algebraic spaces over $S$ with affine transition morphisms $f_{i'i} : X_{i'} \to X_ i$. We denote $f_ i : X \to X_ i$ the projection. We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$. We also choose an element $0 \in I$.

Lemma 73.24.2. In Situation 73.24.1. Let $E_0$ and $K_0$ be objects of $D(\mathcal{O}_{X_0})$. Set $E_ i = Lf_{i0}^*E_0$ and $K_ i = Lf_{i0}^*K_0$ for $i \geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ i})}(E_ i, K_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(E, K) \]

is an isomorphism if either

  1. $E_0$ is perfect and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$, or

  2. $E_0$ is pseudo-coherent and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$ has finite tor dimension.

Proof. For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that the canonical map

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{U_ i})}(E_ i|_{U_ i}, K_ i|_{U_ i}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(E|_ U, K|_ U) \]

is an isomorphism, where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for each $U_0$ by the induction principle of Lemma 73.9.3. Condition (2) of this lemma follows immediately from Mayer-Vietoris for hom in the derived category, see Lemma 73.10.4. Thus it suffices to prove the lemma when $X_0$ is affine.

If $X_0$ is affine, then the result follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.29.2. To see this use the equivalence of Lemma 73.4.2 and use the translation of properties explained in Lemmas 73.13.2, 73.13.3, and 73.13.5. $\square$

Lemma 73.24.3. In Situation 73.24.1 the category of perfect objects of $D(\mathcal{O}_ X)$ is the colimit of the categories of perfect objects of $D(\mathcal{O}_{X_ i})$.

Proof. For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that the functor

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} D_{perf}(\mathcal{O}_{U_ i}) \longrightarrow D_{perf}(\mathcal{O}_ U) \]

is an equivalence where ${}_{perf}$ indicates the full subcategory of perfect objects and where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for every $U_0$ by the induction principle of Lemma 73.9.3. First, we observe that we already know the functor is fully faithful by Lemma 73.24.2. Thus it suffices to prove essential surjectivity.

We first check condition (2) of the induction principle. Thus suppose that we have an elementary distinguished square $(U_0 \subset X_0, V_0 \to X_0)$ and that $P$ holds for $U_0$, $V_0$, and $U_0 \times _{X_0} V_0$. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$. We can find $i \geq 0$ and $E_{U, i}$ perfect on $U_ i$ and $E_{V, i}$ perfect on $V_ i$ whose pullback to $U$ and $V$ are isomorphic to $E|_ U$ and $E|_ V$. Denote

\[ a : E_{U, i} \to (R(X \to X_ i)_*E)|_{U_ i} \quad \text{and}\quad b : E_{V, i} \to (R(X \to X_ i)_*E)|_{V_ i} \]

the maps adjoint to the isomorphisms $L(U \to U_ i)^*E_{U, i} \to E|_ U$ and $L(V \to V_ i)^*E_{V, i} \to E|_ V$. By fully faithfulness, after increasing $i$, we can find an isomorphism $c : E_{U, i}|_{U_ i \times _{X_ i} V_ i} \to E_{V, i}|_{U_ i \times _{X_ i} V_ i}$ which pulls back to the identifications

\[ L(U \to U_ i)^*E_{U, i}|_{U \times _ X V} \to E|_{U \times _ X V} \to L(V \to V_ i)^*E_{V, i}|_{U \times _ X V}. \]

Apply Lemma 73.10.8 to get an object $E_ i$ on $X_ i$ and a map $d : E_ i \to R(X \to X_ i)_*E$ which restricts to the maps $a$ and $b$ over $U_ i$ and $V_ i$. Then it is clear that $E_ i$ is perfect and that $d$ is adjoint to an isomorphism $L(X \to X_ i)^*E_ i \to E$.

Finally, we check condition (1) of the induction principle, in other words, we check the lemma holds when $X_0$ is affine. This follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.29.3. To see this use the equivalence of Lemma 73.4.2 and use the translation of Lemma 73.13.5. $\square$


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