algebrailisation

Algebraisation (also spelled algebraization in American English) is a concept in algebraic geometry describing when a formal or analytic object can be realized as the completion of an algebraic object along a closed subscheme. In short, it asks whether a formal or infinitesimal piece can be globalized to an honest algebraic geometric object.

The basic setting involves formal schemes or formal families, often over a complete Noetherian local ring, and the question is whether there exists an algebraic scheme, morphism, or family whose formal completion along a subscheme matches the given object. Algebraisation connects local data, captured by formal geometry, with global algebraic models. The notion also appears in the study of coherent sheaves and morphisms, where one seeks to lift formal information to genuine algebraic data.

Key results include Grothendieck’s existence theorem, which relates coherent sheaves on a formal completion to compatible systems of coherent sheaves on thickenings, typically under properness hypotheses. This provides a way to algebraize certain formal sheaves and morphisms. Artin’s algebraization criterion gives conditions under which formal moduli problems are algebraizable by an algebraic space or scheme, often requiring finiteness and openness conditions on the deformation data. Together, these results form a framework for turning infinitesimal or formal information into legitimate algebraic objects.

Algebraisation interacts with related ideas such as formal GAGA and deformation theory, and it is central to questions about when local or formal structures arise from globally defined algebraic families. See also algebraization theorem, formal completion, and Grothendieck existence theorem.