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
\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)
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.
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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