Lemma 36.13.10. Let X be an affine scheme. Let T \subset X be a closed subset such that X \setminus T is quasi-compact. Let U \subset X be a quasi-compact open. For every perfect object F of D(\mathcal{O}_ U) supported on T \cap U the object F \oplus F[1] is the restriction of a perfect object E of D(\mathcal{O}_ X) supported in T.
Proof. Say T = V(g_1, \ldots , g_ s). After replacing g_ j by a power we may assume multiplication by g_ j is zero on F, see Lemma 36.13.9. Choose E as in Lemma 36.13.8. Note that g_ j : E \to E restricts to zero on U. Choose a distinguished triangle
E \xrightarrow {g_1} E \to C_1 \to E[1]
By Derived Categories, Lemma 13.4.11 the object C_1 restricts to F \oplus F[1] \oplus F[1] \oplus F[2] on U. Moreover, g_1 : C_1 \to C_1 has square zero by Derived Categories, Lemma 13.4.5. Namely, the diagram
\xymatrix{ E \ar[r] \ar[d]_0 & C_1 \ar[d]_{g_1} \ar[r] & E[1] \ar[d]_0 \\ E \ar[r] & C_1 \ar[r] & E[1] }
is commutative since the compositions E \xrightarrow {g_1} E \to C_1 and C_1 \to E[1] \xrightarrow {g_1} E[1] are zero. Continuing, setting C_{i + 1} equal to the cone of the map g_ i : C_ i \to C_ i we obtain a perfect complex C_ s on X supported on T whose restriction to U gives
F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s]
Choose morphisms of perfect complexes \beta : C' \to C_ s and \gamma : C' \to C_ s as in Lemma 36.13.7 such that \beta |_ U is an isomorphism and such that \gamma |_ U \circ \beta |_ U^{-1} is the morphism
F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s] \to F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s]
which is the identity on all summands except for F where it is zero. By Lemma 36.13.7 we also have C' = C_ s \otimes ^\mathbf {L} I for some perfect complex I on X. Hence the nullity of g_ j^2\text{id}_{C_ s} implies the same thing for C'. Thus C' is supported on T as well. Then \text{Cone}(\gamma ) is a solution. \square
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