Lemma 36.14.5. Let X be a scheme. Let X = U \cup V be an open covering with U quasi-compact, V affine, and U \cap V quasi-compact. If approximation by perfect complexes holds on U, then approximation holds on X.
Proof. Let T \subset X be a closed subset with X \setminus T retro-compact in X. Let r_ U be the integer of Definition 36.14.2 adapted to the pair (U, T \cap U). Set T' = T \setminus U. Note that T' \subset V and that V \setminus T' = (X \setminus T) \cap U \cap V is quasi-compact by our assumption on T. Let r' be the number of affines needed to cover V \setminus T'. We claim that r = \max (r_ U, r') works for the pair (X, T).
To see this choose a triple (T, E, m) such that E is (m - r)-pseudo-coherent and H^ i(E) is supported on T for i \geq m - r. Let t be the largest integer such that H^ t(E)|_ U is nonzero. (Such an integer exists as U is quasi-compact and E|_ U is (m - r)-pseudo-coherent.) We will prove that E can be approximated by induction on t.
Base case: t \leq m - r'. This means that H^ i(E) is supported on T' for i \geq m - r'. Hence Lemma 36.14.4 guarantees the existence of an approximation P \to E|_ V of (T', E|_ V, m) on V. Applying Lemma 36.14.3 we see that (T', E, m) can be approximated. Such an approximation is also an approximation of (T, E, m).
Induction step. Choose an approximation P \to E|_ U of (T \cap U, E|_ U, m). This in particular gives a surjection H^ t(P) \to H^ t(E|_ U). By Lemma 36.13.10 we can choose a perfect object Q in D(\mathcal{O}_ V) supported on T \cap V and an isomorphism Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}. By Lemma 36.13.7 we can replace Q by Q \otimes ^\mathbf {L} I and assume that the map
Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V}
lifts to Q \to E|_ V. By Cohomology, Lemma 20.45.1 we find an morphism a : R \to E of D(\mathcal{O}_ X) such that a|_ U is isomorphic to P \oplus P[1] \to E|_ U and a|_ V isomorphic to Q \to E|_ V. Thus R is perfect and supported on T and the map H^ t(R) \to H^ t(E) is surjective on restriction to U. Choose a distinguished triangle
R \to E \to E' \to R[1]
Then E' is (m - r)-pseudo-coherent (Cohomology, Lemma 20.47.4), H^ i(E')|_ U = 0 for i \geq t, and H^ i(E') is supported on T for i \geq m - r. By induction we find an approximation R' \to E' of (T, E', m). Fit the composition R' \to E' \to R[1] into a distinguished triangle R \to R'' \to R' \to R[1] and extend the morphisms R' \to E' and R[1] \to R[1] into a morphism of distinguished triangles
\xymatrix{ R \ar[r] \ar[d] & R'' \ar[d] \ar[r] & R' \ar[d] \ar[r] & R[1] \ar[d] \\ R \ar[r] & E \ar[r] & E' \ar[r] & R[1] }
using TR3. Then R'' is a perfect complex (Cohomology, Lemma 20.49.7) supported on T. An easy diagram chase shows that R'' \to E is the desired approximation. \square
Comments (0)