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Suppose $i: X \hookrightarrow \mathbb{P}^n_k$ is a projective scheme, over some field $k$ (I'm not sure if algebraically closed is necessary here). So $X$ comes together with an ideal sheaf and a short exact sequence $$ 0 \rightarrow \mathcal{I}_X \rightarrow \mathcal{O}_{\mathbb{P}^n} \rightarrow i_* \mathcal{O}_X \rightarrow 0.$$ Topologically $X$ consists of finitely many irreducible components. I would like to know if it is always possible to write $X$ as a scheme-theoretic union of irreducible subschemes $X_i \hookrightarrow X$, i.e. $\mathcal{I}_X = \bigcap_i \mathcal{I}_{X_i}$.

I think this works if $X$ is reduced, then one can just take the irreducible components together with the reduced structure as $X_i$, and $\mathcal{I}_X = \bigcap_i \mathcal{I}_{X_i}$, because the intersection of radical ideals is again radical, and two radical ideals with the same vanishing locus are equal, by Hilbert's Nullstellensatz.

Does this have anything to do with primary decomposition? Wikipedia states that there is some scheme theoretic interpretation of primary decomposition, but suppose $\mathcal{I}_X$ is the intersection of non-primary ideals, then why should it be possible to write it as the intersection of primary ideals?

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  • $\begingroup$ It's a theorem that in a noetherian ring every ideal has a primary decomposition (c.f., Atiyah-MacDonald, Thm 7.13), which deals with the case of affine schemes. I think there is a similar result for graded rings in Eisenbud, which would mean the same is true for projective schemes. $\endgroup$ – André 3000 Jan 29 at 17:54
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    $\begingroup$ Yeah I'm still trying to figure out primary ideals. Is it true that an ideal $\mathfrak{a} \subset A$ is primary if and only if $\text{Spec}(A/\mathfrak{a})$ is irreducible? $\endgroup$ – red_trumpet Jan 29 at 17:57
  • $\begingroup$ Yes, I think that follows from this question: math.stackexchange.com/q/541943 $\endgroup$ – André 3000 Jan 29 at 18:46
  • $\begingroup$ I don't think this is true anymore :D See the discussion here, and the counterexample here. $\endgroup$ – red_trumpet Jan 29 at 18:49
  • $\begingroup$ But...you're not using the definition of primary, just a strictly weaker property. $\endgroup$ – André 3000 Jan 29 at 19:02

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