I am familiarizing myself with D-modules and in the book "D-modules, perverse sheaves and representation theory" by Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki I found a statement that seems off to me.
Lemma 1.5.17. Let $i : X → Y$ be a closed embedding of algebraic complex varieties. Set $U = Y\setminus X$ and denote by $j : U → X$ the complementary open embedding. Then for any $K ∈ D^b(\mathcal{O}_U)$ we have $\mathcal{O}_X \otimes^L_{i^{−1}\mathcal{O}_Y} i^{−1}Rj_∗K = 0$.
Of course, $\mathcal{O}_X \otimes^L_{i^{−1}\mathcal{O}_Y} i^{−1}Rj_∗K $ is just $Li^*Rj_*K$. But doesn't taking $Y=\mathbb{A}^2$ provide a counterexample?
Take $X=\{0\}$. Then $j_*$ is exact, and, basically by the Hartog's lemma, $j_*(\mathcal{O}_U)=\mathcal{O}_Y$, and hence $i^*\mathcal{O}_Y$ is nonzero?
Their proof is this:
Proof. We have $$i_∗(\mathcal{O}_X \otimes^L_{i^{−1}\mathcal{O}_Y} i^{−1}Rj_∗K) = i_∗\mathcal{O}_X \otimes^L_{\mathcal{O}_Y} Rj_∗K = Rj_∗(j^{−1}i_∗\mathcal{O}_X \otimes^L_{\mathcal{O}_U}K) = 0.$$
They refer to Hartshorne's "Residues and Duality", Proposition II.5.6, for the "projection formula". However, the projection formula in Hartshorne is seemingly different. Hartshorne states
$$Rf_*(F\otimes^L_{\mathcal{O}_X}Lf^*G)\cong (Rf_*F)\otimes^L_{\mathcal{O}_Y}G)$$ under some natural hypotheses on the involved morphism, schemes and sheaves. Basically, I know two versions of projection formula: one that involves $f_*$ and $f^{-1}$ for sheaves of modules over a fixed ring, and one that involves $f_*$ and $f^*$, sheaves of $\mathcal{O}$-modules and a change of the sheaf over which $\otimes$ is taken. This proof somehow uses something in-between.
Is this proof wrong? If not, then what's wrong with my supposed counterexample?