In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets , where each is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and compact. As with Noetherian rings, the concept is named after Emmy Noether.
It can be shown that, in a locally Noetherian scheme, if is an open affine subset, then A is a Noetherian ring. In particular, is a Noetherian scheme if and only if A is a Noetherian ring. Let X be a locally Noetherian scheme. Then the local rings are Noetherian rings.
A Noetherian scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring.
is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and...
no infinite chains. Noetherianscheme, a scheme in algebraic geometry that admits a finite covering by open spectra of Noetherian rings. Artinian ring...
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied...
deduce theorems of interest for usual schemes. A locally Noetherianscheme is a locally Noetherian formal scheme in the canonical way: the formal completion...
regular scheme is a locally Noetherianscheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite...
over A {\displaystyle A} . When X {\displaystyle X} is a locally Noetherianscheme, F {\displaystyle {\mathcal {F}}} is coherent if and only if it is...
Spec(R), the prime spectrum of R, is a Noetherian topological space. More generally, a Noetherianscheme is a Noetherian topological space. The converse does...
the Quot scheme is a scheme parametrizing sheaves on a projective scheme. More specifically, if X is a projective scheme over a Noetherianscheme S and if...
quasi-separated schemes factors as an open immersion followed by a proper morphism. Proper morphisms between locally noetherianschemes preserve coherent...
locally Noetherian The Ai are Noetherian rings. If in addition a finite number of such affine spectra covers X, the scheme is called noetherian. While...
Gorenstein scheme is a locally Noetherianscheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a...
group scheme over a field need not be commutative, however; for example, any finite group scheme is complete. A group scheme G over a noetherianscheme S...
Noether identities NoetherianNoetherian group Noetherian module Noetherian ring Noetherian space Noetherian induction Noetherianscheme "Noether boys",...
all schemes. Not all schemes have resolutions of their singularities: Grothendieck & Dieudonné (1965, section 7.9) showed that if a locally Noetherian scheme...
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called...
subschemes of projective space in the following sense: For any locally Noetherianscheme S, the set of S-valued points Hom ( S , H i l b ( n ) ) {\displaystyle...
{\displaystyle i>n} . This is especially useful for X {\displaystyle X} a Noetherianscheme (for example, a variety over a field) and F {\displaystyle {\mathcal...
on a scheme is flat or free. They are due to Alexander Grothendieck. Generic flatness states that if Y is an integral locally noetherianscheme, u : X...
divisorial schemes is quite large: it includes affine schemes, separated regular (noetherian) schemes and subschemes of a divisorial scheme (such as projective...
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