The Stacks project

Proof. Let $V = X \setminus T$ and $W = U \cap V$. Note that $X = U \cup V$ is an open covering of $X$. Denote $j_ W : W \to V$ the open immersion. Let $E$ be an object of $D(\mathcal{O}_ X)$ whose cohomology sheaves are supported on $T$. By Lemma 20.32.4 we have $(Rj_*E|_ U)|_ V = Rj_{W, *}(E|_ W) = 0$ because $E|_ W = 0$ by our assumption. On the other hand, $Rj_*(E|_ U)|_ U = E|_ U$. Thus (1) is clear. Let $F$ be an object of $D(\mathcal{O}_ U)$ whose cohomology sheaves are supported on $T$. By Lemma 20.32.4 we have $(Rj_*F)|_ V = Rj_{W, *}(F|_ W) = 0$ because $F|_ W = 0$ by our assumption. We also have $(j_!F)|_ V = j_{W!}(F|_ W) = 0$ (the first equality is immediate from the definition of extension by zero). Since both $(Rj_*F)|_ U = F$ and $(j_!F)|_ U = F$ we see that (2) holds. $\square$