If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$

Depends on

• Hilbert basis theorem: if $R$ is Noetherian then $R[x]$ is Noetherian
• Polynomial rings in finitely many commuting indeterminates by iteration

Used by

• A plane intersection with no common component is nonempty and zero-dimensional Corollary
• Every algebra of finite type over a Noetherian ring is a Noetherian ring Corollary
• Every algebra of finite type over a Noetherian ring is finitely presented Corollary
• A cusp family defeats the missing source hypothesis Counterexample
• Ramification points, branch points and unramifiedness Definition
• The intrinsic Zariski tangent space Definition
• The normalized Khovanov-Rozansky HOMFLYPT Euler series Definition
• A nonfree maximal Cohen--Macaulay module Example
• A parameter sequence regular in a hypersurface Example
• All twists on the projective line Example
• An upper jump of h0 in a flat projective family Example
• Fields and ℤ are Noetherian, and so are their polynomial rings in finitely many variables Example
• Localizing (x²,xy)=(x)∩(x,y)² keeps only the matching component Example
• The subalgebra k[x,xy,xy²,…] of k[x,y] is not Noetherian Example
• Two minimal decompositions of (x²,xy) share radicals but not the embedded component Example
• ℤ and k[x] are Noetherian but not Artinian Example
• False statement: in a Noetherian ring there is a single bound on the number of generators an ideal needs False statement
• A classical affine algebraic set has a unique finite irredundant decomposition Lemma
• An S-rational map defined after a faithfully flat smooth base change is defined Lemma
• Embedded flat deformations of a smooth hypersurface are deformations of its equation Lemma
• Finite local length exactly when no common local branch Lemma
• Finite twisted locally free resolutions on projective space Lemma
• Finite-type field extensions with zero Ω Lemma
• Generic freeness over a Noetherian domain Lemma
• High-degree section module is finite graded Lemma
• Over a Noetherian ring, an ideal filtration is stable exactly when its Rees module is finite, and the Rees algebra is Noetherian Lemma
• Standard smooth algebras are finitely presented and flat Lemma
• The blowup of a one-dimensional integral Noetherian scheme at a closed point is finite Lemma
• The Laurent polynomial ring is Noetherian and a unique factorisation domain Lemma
• The product of two projective lines is an integral smooth projective surface Lemma
• Which constructions preserve the Noetherian condition, and which do not Remark
• A finite graded module over a standard graded algebra has rational Hilbert series and eventual polynomial growth Theorem
• Completion of a Noetherian ring is Noetherian Theorem
• Degree of the coherent Hilbert polynomial Theorem
• Euler characteristic is a Hilbert polynomial Theorem
• localisation and polynomial extension of regular rings Theorem
• Serre vanishing for coherent sheaves and ample twists Theorem
• The dimension formula for affine domains Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• B. Totaro, Commutative Algebra (Michaelmas 2011), notes by Z. Norwood, §8 Corollary 8.4 (standard reference, not scraped)
• A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., (16.12) (standard reference, not scraped)