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    Noetherian rings and modules

    D-DEFNOETH

    (Noetherian ring)

    A ring \(A\) is Noetherian if it satisfies the ascending chain condition on ideals: any increasing chain \[ I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots \] of ideals of \(A\) eventually stabilises. Equivalently, every ideal of \(A\) is finitely generated. An \(A\)-module \(M\) is Noetherian if its submodules satisfy the same condition.

    (Noetherian scheme)

    A scheme \(X\) is locally Noetherian if it can be covered by open affine subsets \(\Spec A_i\) with each \(A_i\) a Noetherian ring. It is Noetherian if it is locally Noetherian and quasicompact, equivalently if it has a finite such cover. A scheme is locally Noetherian if and only if \(R\) is Noetherian for every open affine \(\Spec R \subseteq X\). [@Har10a, §II.3, Proposition II.3.2]

    By the Hilbert basis theorem, \(A\) Noetherian implies \(A[x]\) Noetherian; hence every finitely generated algebra over a field or over \(\ZZ\) is Noetherian. Quotients and localisations of Noetherian rings are Noetherian; subrings need not be, and infinitely generated polynomial rings such as \(k[x_1, x_2, \ldots]\) are not.

    Over a Noetherian ring, finitely generated modules are Noetherian and submodules of finitely generated modules are finitely generated; on a locally Noetherian scheme, a quasicoherent sheaf is coherent if and only if it is of finite type. A Noetherian local ring has finite Krull dimension, every ideal of a Noetherian ring has a primary decomposition, and for \(X\) projective over a Noetherian ring \(A\) and \(\mcf\) coherent, each \(H^i(X,\mcf)\) is a finitely generated \(A\)-module [@Har10a, Theorem III.5.2].

    What is a Noetherian ring, and what is the equivalent finiteness condition?
    Which operations preserve the Noetherian condition?
    Show that a scheme is locally Noetherian if and only if $R$ is Noetherian for every affine open $\Spec R \subseteq X$.
    If $(f_1, \ldots, f_n) = (1)$ in $A$ and every $A_{f_i}$ is Noetherian, show that $A$ is Noetherian.
    Give a scheme whose underlying space is Noetherian but which is not a Noetherian scheme.

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