I will call a (nonzero) ring primary if every zero divisor is nilpotent. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call primary if for every non-empty open set $U$, the ring $\Gamma(U, \mathcal{O})$ is primary.

I will say that a scheme $X$ has a primary decomposition if there exist finitely many primary closed subschemes $Y_i$ such that the morphism $\phi \colon Y = \coprod_i Y_i \to X$ of schemes is surjective and, moreover, for every open $U \subset X$, the induced map $\Gamma(U, \mathcal{O}_X) \to \Gamma(\phi^{-1}(U), \mathcal{O}_Y)$ is injective. (Thus, a section is determined by its restrictions to the $Y_i$.) If I have not made any errors in setting up these definitions, then it is a standard theorem that every Noetherian affine scheme has a primary decomposition. However, these primary decompositions, even if we require them to be minimal, are not unique or canonical. Thus, if they can be glued together for a global construction, this is not immediately obvious.

Do primary decompositions exist for Noetherian schemes in general? If not, are there "reasonable" hypotheses (other than affineness) under which they do exist?

  • $\begingroup$ Charles, you want the concept of "associated point" of a scheme and its good behavior in the (locally) noetherian case, so you then take the $Y_i$ to be the schematic closures of the local rings at the associated points. So by bizarre coincidence, I sort of answered this one yesterday (do search on "prime cycle"): see EGA IV$_2$, 3.1ff. $\endgroup$ – Boyarsky Jul 7 '10 at 2:20
  • $\begingroup$ Are you sure? What you are describing sounds uniquely defined, which primary decomposition is not, even in the affine case. $\endgroup$ – Charles Staats Jul 7 '10 at 3:34
  • $\begingroup$ Charles, if you follow my reference suggestion you'll quickly come upon EGA IV$_2$, 3.2.5 & 3.2.6, which are precisely a useful notion of primary decomposition for coherent sheaves $\mathcal{F}$ on locally noetherian schemes $X$ (recovering my suggestion for the structure sheaf). Beware of two typos: in 3.2.5 the quotients $\mathcal{F}_ {\alpha}$ of $\mathcal{F}$ should be required to be quasi-coherent (equivalently, coherent), and more importantly in 3.2.6 at the end $\kappa(x)$ should be $\mathcal{O}_ {X,x}$. Hopefully my preceding comment clarifies things, but seriously, just read EGA. $\endgroup$ – Boyarsky Jul 7 '10 at 4:06
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    $\begingroup$ One last thing: you should unravel my suggestion (= special case of EGA definition, applied to the structure sheaf) in the affine case to see that it is not primary decomposition, but rather a different kind of expression for $(0)$ as an intersection of ideals with prime radical. It satisfies your desired injectivity, surjectivity, and irreducibility properties, which is ultimately what matters. Your definition of primary scheme has problems because you allow empty or non-affine $U$, but after you read the EGA discussion you'll agree that Grothendieck's way is the right one...as usual. :) $\endgroup$ – Boyarsky Jul 7 '10 at 4:13
  • $\begingroup$ Okay, if it's not actually supposed to be primary decomposition in the affine case, then I can believe it. Thanks! $\endgroup$ – Charles Staats Jul 7 '10 at 14:53

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