A scheme is a space that locally looks like a particularly simple ringed space: an affine scheme. This can be formalised either within the category of locally ringed spaces or within the category of presheaves of sets on the category of affine schemes .
The notion of scheme originated in algebraic geometry where it is, since Grothendieck’s revolution of that subject, a central notion.
However, the idea that
A scheme is a ringed space that is locally isomorphic to an affine space.
is much more general and need not be restricted to a notion of affine spaces that are formal duals of rings. But then one talks about locally affine spaces.
Zoran: this gives wrong impression that Grothendieck school was working with ringed spaces and not functor of points. This is true for EGA but not for most of works of Grothendieck school. The things about generalization called locally affine space should go under locally affine space and under functor of points and not here. There is also redundancy in new text.
For instance a smooth manifold is a ringed space that is locally isomorphic to a “smooth affine space” , with its standard smooth structure.
The standard concept of scheme in algebraic geometry is therefore usefully understood as a special case of generalized schemes that naturally appear for instance also in differential geometry, in synthetic differential geometry and many other topics.
A scheme is a locally ringed space with an open cover (as locally ringed spaces), by affine schemes: the spectra of unital commutative rings.
A morphism of schemes is a morphism of the underlying ringed spaces, such that for each point the induced map of local rings
is local (in that it carries the maximal ideal to the maximal ideal). See functor of points.
Zoran: This is superfluous: the definition of morphisms in the category of LOCALLY ringed spaces already asks for this condition, one does not need to introduce that as late as when introducing schemes.
Jacob Lurie argues that underlying locale point of view is better than underlying topological space point of view, see schemes as locally affine structured (∞,1)-toposes.
The fundamental theorem on morphisms of schemes is a fully faithful functor from the category of schemes to the category of presheaves on which sends to the functor
This identifies schemes with those presheaves on CRing that
The standard reference for the functor-of-points approach to schemes is
In algebraic geometry this is a basic object of study, since the revolution of Grothendieck. There are generalizations like relative schemes (which are just objects in a slice category ), relative schemes in noncommutative algebraic geometry introduced by A. Rosenberg in terms of categories and covers defined using pairs of adjoint functors, the generalized schemes of Nikolai Durov, the algebraic stacks of Deligne-Mumford and Artin, the dg-schemes of Kapranov, the derived schemes of Jacob Lurie, the higher algebraic stacks of Toën–Vezzosi, almost schemes (Ofer Gabber and Lorenzo Ramero), formal schemes (Cartier–Grothendieck), locally affine spaces in the fpqc, fppf or étale topology (Grothendieck), algebraic spaces, etc.
See also wikipedia. EGA says prescheme, for what we call algebraic scheme, and says scheme for what we call separated scheme?.
A useful quick introduction that presents the concept in the light of its higher categorical generalizations is at the beginning of