In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition...

Noetherian. Noetherian relation, a binary relation that satisfies the ascending chain condition on its elements. Noetherian topological space, a topological space...

agree in a metric space, but may not be equivalent in other topological spaces. One such generalization is that a topological space is sequentially compact...

scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring. The...

mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper...

(quasi)compact, and if the ring in question is Noetherian then the space is a Noetherian topological space. However, these facts are counterintuitive: we...

locally Noetherian, but there are important constructions that lead to more general schemes. In any (not necessarily noetherian) topological space, every...

projective limits. One of the way to fix this is to consider Noetherian topological spaces; every open sets are compact so that the cokernel is a sheaf...

ring a Noetherian topological space. The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space are Noetherian...

space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that...

space, named for Oscar Zariski, has several different meanings: A topological space that is Noetherian (every open set is quasicompact) A topological...

define locally noetherian formal schemes. All rings will be assumed to be commutative and with unit. Let A be a (Noetherian) topological ring, that is...

noetherian. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false. For example, most schemes...

is a Hausdorff space. The theorem is a consequence of the Artin–Rees lemma together with Nakayama's lemma, and, as such, the "Noetherian" assumption is...

Sheaf cohomology groups Hi on a noetherian topological space vanish for i strictly greater than the dimension of the space. Thus the quantity, called the...

In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures...

sheaf of abelian groups F {\displaystyle {\mathcal {F}}} on a Noetherian topological space X {\displaystyle X} of dimension n < ∞ {\displaystyle n<\infty...

role of the finite-dimensional vector spaces in linear algebra. In particular, Noetherian rings (see also § Noetherian rings, below) can be defined as the...

a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than...

rings over a field are Noetherian is called Hilbert's basis theorem. Moreover, many ring constructions preserve the Noetherian property. In particular...

and a topological structure on each of the sets X, X2, X3, ... satisfying certain axioms. (N) Each of the Xn is a Noetherian topological space, of dimension...

have been studied in the categories of groups, rings, modules, and topological spaces. The terms "hopfian" and "cohopfian" have arisen since the 1960s,...

unique maximal ideal is a Noetherian local ring. The completion is a functorial operation: a continuous map f: R → S of topological rings gives rise to a...

Hausdorff space is also called separated, in that case, the 𝔞-adic topology is called separated. By Krull's intersection theorem, if R is a Noetherian ring...

chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over...

C*-algebras to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization of topological spaces as "non-commutative...

reformulated in purely topological terms, using the Zariski topology, for which the closed sets are the algebraic subsets: A topological space is irreducible...

mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets...

coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let R {\displaystyle R} be a Noetherian ring (for example, a field)...

algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring...