Editing Commutative algebra (section)

==Connections with algebraic geometry==
Commutative algebra (in the form of [[polynomial ring]]s and their quotients, used in the definition of [[algebraic varieties]]) has always been a part of [[algebraic geometry]]. However, in the late 1950s, algebraic varieties were subsumed into [[Alexander Grothendieck]]'s concept of a [[scheme (mathematics)|scheme]]. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the [[duality (category theory)|duality]] between the category of affine algebraic varieties over a field ''k'', and the category of finitely generated reduced ''k''-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the sense of [[Grothendieck topology]]. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the [[étale topology]], and the two flat Grothendieck topologies: fppf and fpqc. Nowadays some other examples have become prominent, including the [[Nisnevich topology]]. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions, leading to Artin stacks and, even finer, [[Deligne–Mumford stack]]s, both often called algebraic stacks.