Definitions
Λ is a complete Noetherian local ring with residue field ''k'', and ''C'' is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitely generated and Artinian) with residue field ''k''. A small extension in ''C'' is a morphism ''Y''→''Z'' in ''C'' that is surjective with kernel a 1-dimensional vector space over ''k''. A functor is called representable if it is of the form ''h''''X'' where ''h''''X''(''Y'')=hom(''X'',''Y'') for some ''X'', and is called pro-representable if it is of the form ''Y''→lim hom(''X''''i'',''Y'') for a filtered direct limit over ''i'' in some filtered ordered set. A morphism of functors ''F''→''G'' from ''C'' to sets is called smooth if whenever ''Y''→''Z'' is an epimorphism of ''C'', the map from ''F''(''Y'') to ''F''(''Z'')×''G''(''Z'')''G''(''Y'') is surjective. This definition is closely related to the notion of a formally smooth morphism of schemes. If in addition the map between the tangent spaces of ''F'' and ''G'' is an isomorphism, then ''F'' is called a hull of ''G''.Grothendieck's theorem
showed that a functor from the category ''C'' ofSchlessinger's representation theorem
One difficulty in applying Grothendieck's theorem is that it can be hard to check that a functor preserves all pullbacks. Schlessinger showed that it is sufficient to check that the functor preserves pullbacks of a special form, which is often easier to check. Schlessinger's theorem also gives conditions under which the functor has a hull, even if it is not representable. Schessinger's theorem gives conditions for a set-valued functor ''F'' on ''C'' to be representable by a complete local Λ-algebra ''R'' with maximal ideal ''m'' such that ''R''/''m''''n'' is in ''C'' for all ''n''. Schlessinger's theorem states that a functor from ''C'' to sets with ''F''(''k'') a 1-element set is representable by a complete Noetherian local algebra if it has the following properties, and has a hull if it has the first three properties: *H1: The map ''F''(''Y''×''X''''Z'')→''F''(''Y'')×''F''(''X'')''F''(''Z'') is surjective whenever ''Z''→''X'' is a small extension in ''C'' and ''Y''→''X'' is some morphism in ''C''. *H2: The map in H1 is a bijection whenever ''Z''→''X'' is the small extension ''k'' 'x''(''x''2)→''k''. *H3: The tangent space of ''F'' is a finite-dimensional vector space over ''k''. *H4: The map in H1 is a bijection whenever ''Y''=''Z'' is a small extension of ''X'' and the maps from ''Y'' and ''Z'' to ''X'' are the same.See also
*References
* *{{Citation , last1=Schlessinger , first1=Michael , title=Functors of Artin rings , jstor=1994967 , mr=0217093 , year=1968 , journal= Transactions of the American Mathematical Society , issn=0002-9947 , volume=130 , pages=208–222 , doi=10.2307/1994967, doi-access=free Theorems in algebraic geometry