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algebraic space

In algebraic geometry one can glue affine schemes in various topologies; this way one obtains various kinds of locally affine ringed spaces. For example, schemes locally affine in Zariski topology?. Étale topology? is finer than Zariski, hence the category of locally affine (ringed) spaces in étale topology is larger than the category of schemes. Algebraic spaces make a category which is includes the subcategory of all schemes and is close to the category of locally affine spaces in étale topology, namely it consists of those ringed spaces which may be obtained as a quotient of a scheme $S$ by an equivalence relation $R\subset S\times S$ which is a closed subscheme, and whose projections $p_1,p_2:R\to S$ are étale morphisms of schemes.

The standard reference is the monograph

• D. Knutson, Algebraic spaces, LNM 203, Springer 1971.

• Some related MO questions: why-is-this-not-an-algebraic-space, Can an algebraic space fail to have a unviersal map to a scheme?…