affineness

Affineness is a broadly used term in geometry and algebraic geometry describing a situation in which global algebraic data determine the object and its maps, and where many problems reduce to working with rings of functions.

In affine geometry, an affine space is a geometric structure modeled on a vector space, but without a distinguished origin. Points can be added to and subtracted to form displacement vectors, and lines, planes, and higher-dimensional affine subspaces are defined combinatorially by incidences. Affine transformations preserve parallelism and are of the form x ↦ Ax + b, with A a linear map and b a fixed vector. The study of affine geometry emphasizes incidence relations, straight lines, and affine combinations, contrasted with purely linear structures that require a fixed origin.

In algebraic geometry, affineness refers to objects that can be described completely by rings of functions. An affine scheme is a space X that is isomorphic to Spec(R) for some ring R; equivalently, X can be covered by a single affine open. An affine variety (over a field k) is a subset of affine space defined by polynomial equations and, in modern language, corresponds to the coordinate ring k[V] = k[x1, ..., xn]/I(V). A key feature is that global regular functions on an affine object encode rich structural information: for affine schemes, higher cohomology of quasi-coherent sheaves vanishes, and many problems reduce to commutative algebra of R or k[V].

A morphism f: X → Y is called affine if the preimage of every affine open in Y is affine; in particular, affine morphisms preserve the affineness property under base change. Examples include the projection from a product with an affine scheme, or a finite type map from an affine X.

Affineness contrasts with non-affine objects such as general projective varieties, which typically cannot be described solely by a global ring of functions. The concept provides a unifying framework across geometry and algebra for understanding when global algebraic data suffices to control structure and maps.