Situation 75.24.1. 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.
75.24 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.
Lemma 75.24.2. In Situation 75.24.1. 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
E_0 is perfect and K_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0}), or
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 75.9.3. Condition (2) of this lemma follows immediately from Mayer-Vietoris for hom in the derived category, see Lemma 75.10.4. 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 75.4.2 and use the translation of properties explained in Lemmas 75.13.2, 75.13.3, and 75.13.5. \square
Lemma 75.24.3. In Situation 75.24.1 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 75.9.3. First, we observe that we already know the functor is fully faithful by Lemma 75.24.2. 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 75.10.8 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 75.4.2 and use the translation of Lemma 75.13.5. \square
Comments (0)