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Suppose that $X$ is a non-reduced scheme of finite type over a field, with multiple irreducible components $X_1,\ldots,X_n$, possibly intersecting each other. Is there a natural scheme structure on each $X_i$? I don't want the reduced induced structure: for example, if $X$ is generically non-reduced on $X_i$, I want my induced structure on $X_i$ to be non-reduced.

I am not sure if I should expect to have embedded points at $X_i \cap X$ or not. Possibly there is not a canonical choice for this structure.

tim p
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1 Answers1

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Yes. On affine charts, the scheme structure on $X_i = \mathrm{Spec}( R/P_i)$ is given by the (unique) primary ideal $Q_i$ with radical $P_i$ in a minimal primary decomposition of the zero ideal of $R$. This is called the "second uniqueness theorem" for primary decomposition. (Uniqueness ensures that the ideals $Q_i$ obtained from different charts patch to a unique scheme structure on $X_i$.)

It's worth noting that this theorem only exists for irreducible components, i.e. minimal prime ideals $P_i$, not embedded primes.

Jake Levinson
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