Situation 36.29.1. Let $S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i$ be a limit of a directed system of schemes with affine transition morphisms $f_{i'i} : S_{i'} \to S_ i$. We assume that $S_ i$ is quasi-compact and quasi-separated for all $i \in I$. We denote $f_ i : S \to S_ i$ the projection. We also fix an element $0 \in I$.
36.29 Limits and derived categories
In this section we collect some results about the derived category of a scheme which is the limit of an inverse system of schemes. More precisely, we will work in the following setting.
Lemma 36.29.2. In Situation 36.29.1. Let $E_0$ and $K_0$ be objects of $D(\mathcal{O}_{S_0})$. 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 is an isomorphism if either
$E_0$ is perfect and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{S_0})$, or
$E_0$ is pseudo-coherent and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{S_0})$ has finite tor dimension.
\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{S_ i})}(E_ i, K_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ S)}(E, K) \]
Proof. For every open $U_0 \subset S_0$ 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 $U = f_0^{-1}(U_0)$ and $U_ i = f_{i0}^{-1}(U_0)$. We will prove $P$ holds for all quasi-compact opens $U_0$ by the induction principle of Cohomology of Schemes, Lemma 30.4.1. Condition (2) of this lemma follows immediately from Mayer-Vietoris for hom in the derived category, see Cohomology, Lemma 20.33.3. Thus it suffices to prove the lemma when $S_0$ is affine.
Assume $S_0$ is affine. Say $S_0 = \mathop{\mathrm{Spec}}(A_0)$, $S_ i = \mathop{\mathrm{Spec}}(A_ i)$, and $S = \mathop{\mathrm{Spec}}(A)$. We will use Lemma 36.3.5 without further mention.
In case (1) the object $E_0^\bullet $ corresponds to a finite complex of finite projective $A_0$-modules, see Lemma 36.10.7. We may represent the object $K_0$ by a K-flat complex $K_0^\bullet $ of $A_0$-modules. In this situation we are trying to prove
\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(A_ i)}(E_0^\bullet \otimes _{A_0} A_ i, K_0^\bullet \otimes _{A_0} A_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(A)}(E_0^\bullet \otimes _{A_0} A, K_0^\bullet \otimes _{A_0} A) \]
Because $E_0^\bullet $ is a bounded above complex of projective modules we can rewrite this as
\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{K(A_0)}(E_0^\bullet , K_0^\bullet \otimes _{A_0} A_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{K(A_0)}(E_0^\bullet , K_0^\bullet \otimes _{A_0} A) \]
Since there are only a finite number of nonzero modules $E_0^ n$ and since these are all finitely presented modules, this map is an isomorphism.
In case (2) the object $E_0$ corresponds to a bounded above complex $E_0^\bullet $ of finite free $A_0$-modules, see Lemma 36.10.2. We may represent $K_0$ by a finite complex $K_0^\bullet $ of flat $A_0$-modules, see Lemma 36.10.4 and More on Algebra, Lemma 15.66.3. In particular $K_0^\bullet $ is K-flat and we can argue as before to arrive at the map
\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{K(A_0)}(E_0^\bullet , K_0^\bullet \otimes _{A_0} A_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{K(A_0)}(E_0^\bullet , K_0^\bullet \otimes _{A_0} A) \]
It is clear that this map is an isomorphism (only a finite number of terms are involved since $K_0^\bullet $ is bounded). $\square$
Lemma 36.29.3. In Situation 36.29.1 the category of perfect objects of $D(\mathcal{O}_ S)$ is the colimit of the categories of perfect objects of $D(\mathcal{O}_{S_ i})$.
Proof. For every open $U_0 \subset S_0$ 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 $U = f_0^{-1}(U_0)$ and $U_ i = f_{i0}^{-1}(U_0)$. We will prove $P$ holds for all quasi-compact opens $U_0$ by the induction principle of Cohomology of Schemes, Lemma 30.4.1. First, we observe that we already know the functor is fully faithful by Lemma 36.29.2. Thus it suffices to prove essential surjectivity.
We first check condition (2) of the induction principle. Thus suppose that we have $S_0 = U_0 \cup V_0$ and that $P$ holds for $U_0$, $V_0$, and $U_0 \cap V_0$. Let $E$ be a perfect object of $D(\mathcal{O}_ S)$. 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
\[ a : E_{U, i} \to (Rf_{i, *}E)|_{U_ i} \quad \text{and}\quad b : E_{V, i} \to (Rf_{i, *}E)|_{V_ i} \]
the maps adjoint to the isomorphisms $Lf_ i^*E_{U, i} \to E|_ U$ and $Lf_ i^*E_{V, i} \to E|_ V$. By fully faithfulness, after increasing $i$, we can find an isomorphism $c : E_{U, i}|_{U_ i \cap V_ i} \to E_{V, i}|_{U_ i \cap V_ i}$ which pulls back to the identifications
\[ Lf_ i^*E_{U, i}|_{U \cap V} \to E|_{U \cap V} \to Lf_ i^*E_{V, i}|_{U \cap V}. \]
Apply Cohomology, Lemma 20.45.1 to get an object $E_ i$ on $S_ i$ and a map $d : E_ i \to Rf_{i, *}E$ 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 $Lf_ i^*E_ i \to E$.
Finally, we check condition (1) of the induction principle, in other words, we check the lemma holds when $S_0$ is affine. Say $S_0 = \mathop{\mathrm{Spec}}(A_0)$, $S_ i = \mathop{\mathrm{Spec}}(A_ i)$, and $S = \mathop{\mathrm{Spec}}(A)$. Using Lemmas 36.3.5 and 36.10.7 we see that we have to show that
\[ D_{perf}(A) = \mathop{\mathrm{colim}}\nolimits D_{perf}(A_ i) \]
This is clear from the fact that perfect complexes over rings are given by finite complexes of finite projective (hence finitely presented) modules. See More on Algebra, Lemma 15.74.17 for details. $\square$
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