How can I find an injective resolution for the canonical line bundle on $\mathbb{P}^n$?

Question

I want to learn how to compute examples of dualizing complexes and it seems like the first step in this direction is learning how to construct an injective resolution for $\omega_{\mathbb{P}^n}$. This is because we can define the dualizing complex as $$ \omega_X^\bullet =\mathcal{RHom}_{\mathbb{P}^n}(\mathcal{O}_X,\omega_{\mathbb{P}^n}) $$ Are there constructive methods for doing this "by hand"? I am mainly interested in cases which are not cohen-macaulay, hence the shifted dualizing sheaf does not work. For example, consider the quasi-projective variety $$ X = \text{Proj}\left(\frac{\mathbb{C}[x,y,z,x]}{(x)(y,z)} \right) $$ which is a copy of $\mathbb{P}^2$ intersecting a copy of $\mathbb{P}^1$ at a point. What is it's dualizing complex?

Answer 1

The main propositions I used are III.6.5 and III.6.7 from Hartshorne's algebraic geometry book. Namely, given a resolution $\mathcal{L}_\bullet \to\mathcal{O}_X \to 0$, we have $$ \mathcal{RHom}_{\mathbb{P}^n}(\mathcal{O}_X,\omega_{\mathbb{P}^n}) = \mathcal{Hom}_{\mathbb{P}^n}(\mathcal{L}_\bullet, \omega_{\mathbb{P}^n}) $$ Then $\mathcal{L}_\bullet$ is given by the complex $$ \mathcal{O}_{\mathbb{P}^3}(-3) \xrightarrow{\begin{bmatrix} w \\ -y \end{bmatrix}} \mathcal{O}_{\mathbb{P}^3}(-2)\oplus \mathcal{O}_{\mathbb{P}^3}(-2) \xrightarrow{\begin{bmatrix} xy & xw \end{bmatrix}} \mathcal{O}_{\mathbb{P}^3} $$ Then using the second proposition, the dualizing complex is given by taking the dual complex of $\mathcal{L}_\bullet \otimes \mathcal{O}(4)$.