requires non-trivial generalization of topological spaces as "non-commutative spaces". For this reason there is some talk about non-commutative topology...

University. 4 September 2003. Retrieved 18 May 2023. Alain Connes, « Géométrie non-commutative », Université de tous les savoirs, 4, 175–190, Editions Odile...

The Éléments de géométrie algébrique (EGA; from French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a...

to be commutative when the diagram commutes for all objects A {\displaystyle A} and B {\displaystyle B} . The Kleisli category of a commutative monad...

Mathematics (N)". Maths History. Fontenelle, Bernard de (1727). Eléments de la géométrie de l'infini (in French). p. 3. Arithmetices principia: nova methodo (in...

This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic...

arbitrary commutative ring as a foundation for algebraic geometry. The theory took its definitive form in Grothendieck's Éléments de géométrie algébrique...

originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings...

Flatness was introduced by Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. A left module M over a ring R is flat if the following...

(5): 433–439, doi:10.1512/iumj.1974.23.23036 Connes, A. (1990), Géométrie non commutative, InterEditions, ISBN 2-7296-0284-4 Ferrara, S.; Mattiolia, G.;...

In algebraic geometry and commutative algebra, a ring homomorphism f : A → B {\displaystyle f:A\to B} is called formally smooth (from French: Formellement...

and C*-algebras are known as non-commutative measure theory and non-commutative topology, respectively. Non-commutative geometry is not merely a pursuit...

multiplication is non-commutative). This result is known as the Frobenius theorem. The octonions O, for which multiplication is neither commutative nor associative...

rule i 2 = − 1 {\displaystyle i^{2}=-1} along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative...

the structure of quasi-finite morphisms of schemes. Several results in commutative algebra imply the geometric forms of Zariski's main theorem, including:...

first of these comes from commutative algebra: subject to some finiteness conditions on f, it can be shown that there is a non-empty open subscheme Y ′...

of the subject. For the items in commutative algebra (the theory of commutative rings), see Glossary of commutative algebra. For ring-theoretic concepts...

notion of a complete intersection. Given a commutative ring R and an R-module M, an element r in R is called a non-zero-divisor on M if r m = 0 implies m...

incomplete, Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA). The collection Fondements de la Géometrie Algébrique (FGA),...

that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring...

Pontryagin duality taking finite commutative group schemes to finite commutative group schemes. Finite flat commutative group schemes over a perfect field...

above for Noetherian schemes: First, given a projective module E over a commutative ring A, an A-linear map u : E → A {\displaystyle u:E\to A} is called...

between (non-compact or non-Hausdorff) topological spaces need not be homeomorphisms. Negrepontis (1971, §1) shows that the functor from commutative C*-algebras...

of Éléments de géométrie algébrique? [1] scheme A scheme is a locally ringed space that is locally a prime spectrum of a commutative ring. Schubert 1...

Non-formal power series also generalize polynomials, but the multiplication of two power series may not converge. A polynomial f over a commutative ring...

groups. First there is Jean Dieudonne's theory of determinants over "non-commutative fields" (division rings). Artin describes GL(n, k) group structure...

duality Dévissage Affine scheme Scheme Éléments de géométrie algébrique Grothendieck's Séminaire de géométrie algébrique Fiber product of schemes Flat morphism...

advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge...

Grothendieck, J. Dieudonné, Eléments de géométrie algébrique, Ch. 0IV § 23, Publ. Math. IHÉS 20, (1964). H. Matsumura, Commutative algebra ISBN 0-8053-7026-9, chapter...

In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules...