Noetherian closed sets split into finitely many maximal irreducible pieces, uniquely; algebraically these are the minimal primes.Suggest a slogan
Every closed subset \(X\) of a Noetherian space is a finite union \[ X = X_1 \union \cdots \union X_r \] of irreducible closed subsets. If no \(X_i\) contains another, the \(X_i\) are unique, and are the irreducible components of \(X\).
Under the correspondence the components of \(V(J)\) are \(V(\mfp_i)\) for the minimal primes \(\mfp_i\) over \(J\), so decomposing a variety is finding minimal primes. A reducible hypersurface gives the basic computation: if \(f=\prod f_i^{e_i}\) with the \(f_i\) irreducible, then the components of \(V(f)\) are the \(V(f_i)\), while the multiplicities \(e_i\) are invisible to the topology. That invisibility is exactly what the scheme structure later restores.
Decompose a closed algebraic set into irreducibles.
In what sense is the decomposition into irreducible components unique?
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D-9DIKBIrreducible and Noetherian spacesuses
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P-AGHTHREECOMPDecomposing $V(x^2-yz,\ xz-x)$ into irreducible componentsuses