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Charles Staats
Charles Staats
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I will call a (nonzero) ring primary if every zero divisor is nilpotent, i.e., if the nilradical is prime. (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?

I will call a (nonzero) ring primary if every zero divisor is nilpotent, i.e., if the nilradical is prime. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call primary if for every 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?

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?

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Charles Staats
Charles Staats
  • 7.3k
  • 5
  • 67
  • 86

Primary decomposition for non-affine schemes

I will call a (nonzero) ring primary if every zero divisor is nilpotent, i.e., if the nilradical is prime. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call primary if for every 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?