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...

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...

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...

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...

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...

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...

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...

field (a point), Rqf∗(F ) is the same as Hq(F ). Suppose that X is a Noetherian scheme. An abelian étale sheaf F over X is called finite locally constant...

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

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...

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...