# Primary decomposition for noetherian schemes

I'm trying to understand the natural scheme theoretic structure for the irreducible components of a noetherian scheme $$X$$.

The idea would be that if $$\text{Ass}(\mathcal{O}_X)=\{x_1,\dots x_n\}$$ then we can put a scheme structure on $$X_i=\overline{\{x_i\}}$$ such that we have $$X=X_1\cup\dots \cup X_n$$ where this is a scheme theoretic union, i.e, $$X$$ is the colimit of the $$\bigcap_{i\in I} X_i$$ when $$I$$ goes through the subset of $$\{1,\dots,n\}$$. For example, for $$n=2$$ this is the cartesian product:

$$\begin{array}{ccc}X_1\cup X_2&\leftarrow& X_1\\\uparrow & &\uparrow\\X_2&\leftarrow& X_1\times_X X_2\end{array}$$

For $$X$$ affine I think I figure it out. The idea is to first prove:

Lemma: $$\text{Spec}(A/I\cap J)$$ is the scheme theoretic union of $$\text{Spec}(A/I)$$ with $$\text{Spec}(A/I)$$.

I think this can be done as follows

Sketch of prove: First you prove it in the category of affine schemes, this reduces to manipulations of rings that are not so difficult. In order to prove it for the category of schemes, given a scheme $$X$$ you need to prove the existence and uniqueness of a morphism $$\text{Spec}(A/I\cap J)\rightarrow X$$.

The idea for this is to take an affine cover $$X=\cup U_i$$ and take preimages of $$U_i$$ along the projections to reduce to the affine case. If $$U_i=\text{Spec} B_i$$, then you have that $$\text{Spec }(A/I\cap J)\otimes B_i$$ is the colimit of $$\text{Spec }(A/I)\otimes B_i$$ with $$\text{Spec }(A/J)\otimes B_i$$ along $$\text{Spec}(A/I\cap J)\otimes B_i$$ (in the category of affine schemes). So there are unique morphisms from $$\text{Spec }(A/I\cap J)\otimes B_i$$ to $$U_i$$ that glue to a unique morphism to $$X$$.

Using induction you can pass from two to finitely many ideals and then you get:

If $$A$$ is a noetherian ring and $$(0)=q_1\cap\dots \cap q_n$$ is a primary decomposition then $$\text{Spec}(A)=\text{Spec}(A/q_1)\cup \dots \cup \text{Spec}(A/q_n)$$

Now I want to globalize this. Is easy to generalize the lemma to

If $$\mathcal{I}_1\dots \mathcal{I}_n$$ are quasi-coherent ideals of a scheme $$X$$ then $$V(\mathcal{I}_1) \cup \dots \cup V(\mathcal{I}_n) = V(\cap \mathcal{I}_i)$$

Because one can prove $$X_1\cup \dots \cup X_n = X \iff (X_1\cap U) \cup \dots \cup (X_1\cap U) = X\cap U \; \forall \, U \subseteq X \ \text{affine}$$

But I can't construct the analogue of the primary decomposition. So my question is

Given $$X$$ a noetherian scheme. Are there quasicoherent ideals $$\mathcal{I}_1,\dots,\mathcal{I}_n$$ such that for every affine open set $$U\subseteq X$$ with $$U=\text{Spec}(A)$$ one have that $$\mathcal{I}_1(U)\cap \dots\cap\mathcal{I}_n(U)=(0)$$ is a minimal (if one exclude the terms $$\mathcal{I}_i(U)=A$$) primary decomposition in $$A$$?

If this exists we would have that $$X_i=V(\mathcal{I_i})$$ would be a nice scheme theoretic structure for the irreducible components $$X_i$$.

As an observation if $$X=\text{Spec}(A)$$ is affine then we can take $$\mathcal{I}_i=\tilde{q_i}$$ for a primary decomposition $$(0)=q_1\cap\dots \cap q_n$$ because primary decompositions behave well with localizations (see Atiyah-Macdonald Proposition 4.9).

It would be nice if the uniqueness theorems for primary decompositions work in this context as well.

• Have a look at EGA IV, §3: Cycles premiers associés et décomposition primaire. – abx Jan 30 '19 at 5:35