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. Noetherian scheme, a scheme in algebraic geometry that admits a finite covering by open spectra of Noetherian rings. Artinian ring...

deduce theorems of interest for usual schemes. A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion...

are the Noetherian schemes, in which the coordinate rings are Noetherian rings. Formally, a scheme is a ringed space covered by affine schemes. An affine...

affine spectra covers X, the scheme is called noetherian. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse...

Spec(R), the prime spectrum of R, is a Noetherian topological space. More generally, a Noetherian scheme is a Noetherian topological space. The converse does...

over A {\displaystyle A} . When X {\displaystyle X} is a locally Noetherian scheme, F {\displaystyle {\mathcal {F}}} is coherent if and only if it is...

isomorphic to the Riemann sphere CP1. Let X be an integral locally Noetherian scheme. A prime divisor or irreducible divisor on X is an integral closed...

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...

algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any Noetherian scheme X {\displaystyle X} . If we look at the...

locally Noetherian scheme then any morphism from X to any scheme is quasi-separated, and in particular X is a quasi-separated scheme. Any separated scheme or...

quasi-separated schemes factors as an open immersion followed by a proper morphism. Proper morphisms between locally noetherian schemes preserve coherent...

group scheme over a field need not be commutative, however; for example, any finite group scheme is complete. A group scheme G over a noetherian scheme S...

the Quot scheme is a scheme parametrizing sheaves on a projective scheme. More specifically, if X is a projective scheme over a Noetherian scheme S and if...

sheaves F on a complex manifold X (resp. quasi-coherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see (Serre 1956) (resp. (Serre...

regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite...

See also Generalized Cohen–Macaulay ring. We say that a locally Noetherian scheme X {\displaystyle X} is Cohen–Macaulay if at each point x ∈ X {\displaystyle...

divisorial schemes is quite large: it includes affine schemes, separated regular (noetherian) schemes and subschemes of a divisorial scheme (such as projective...

all schemes. Not all schemes have resolutions of their singularities: Grothendieck & Dieudonné (1965, section 7.9) showed that if a locally Noetherian scheme...

to points. One version for schemes states the following: (EGA, III.4.3.1) Let X be a scheme, S a locally noetherian scheme and f : X → S {\displaystyle...

subschemes of projective space in the following sense: For any locally Noetherian scheme S, the set of S-valued points Hom ⁡ ( S , H i l b ( n ) ) {\displaystyle...

corollary to 1. above, if f is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image R p f ∗ F {\displaystyle...

multiplication Div(X), the group of Weil divisors on an integral locally Noetherian scheme X span and div, HTML tags that implement generic elements div, a C...

semi-simple abelian category. The category of coherent sheaves on a Noetherian scheme is semi-simple if and only if X {\displaystyle X} is a finite disjoint...

on a scheme is flat or free. They are due to Alexander Grothendieck. Generic flatness states that if Y is an integral locally noetherian scheme, u : X...

In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of Z / ℓ n {\displaystyle \mathbb {Z} /\ell ^{n}} -modules...

statements about coherent sheaves on Noetherian schemes. Dévissage is an adaptation of a certain kind of Noetherian induction. It has many applications...

Gorenstein scheme is a locally Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a...

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...

Skolem–Noether theorem Noetherian Noetherian group Noetherian induction Noetherian module Noetherian ring Noetherian scheme Noetherian topological space "Noether...