ag.algebraic geometry - Dualizing sheaf of reducible variety?

With regards part 2.

Let's assume that you have two components $X_1$ and $X_2$ (or even unions of components) such that $X_1 cup X_2 = X$=. Let $I_1$ and $I_2$ denote the ideal sheaves of $X_1$ and $X_2$ in $X$.

Set $Z$ to be the scheme $X_1 cap X_2$, in other words, the ideal sheaf of $Z$ is $I_1 + I_2$.

It is easy to see you have a short exact sequence
$$0 to I_1 cap I_2 to I_1 oplus I_2 to (I_1 + I_2) to 0$$
where the third map sends $(a,b)$ to $a-b$.

The nine-lemma should imply that you have a short exact sequence

$$0 to O_X to O_{X_1} oplus O_{X_2} to O_Z to 0$$

If you Hom this sequence into the dualizing complex of $X$, you get a triangle
$$omega_Z^. to omega_{X_1}^. oplus omega_{X_2}^. to omega_{X}^. to omega_Z^.[1]$$

You can then take cohomology and, depending on how things intersect (and what you understand about the intersection), possibly answer your question.

If $X_1$ and $X_2$ are hypersurfaces with no common components (which should imply everything in sight is Cohen-Macualay) then these dualizing complexes are all just sheaves (with various shifts), and you just get a short exact sequence
$$0 to omega_{X_1} oplus omega_{X_2} to omega_{X} to omega_{Z} to 0$$

Technically speaking, I should also probably push all these sheaves forward onto $X$ via inclusion maps.

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