Change of rings in deformation theory of modules

Abstract

Given a B B -module M M and any presentation B = A / J B=A/J , the obstruction theory of M M as a B B -module is determined by the usual obstruction class o A \mathrm {o}_{\! \scriptscriptstyle {A}}^{\scriptscriptstyle {}} for deforming M M as an A A -module and a new obstruction class o J \mathrm {o}_{\! \scriptscriptstyle {J}}^{\scriptscriptstyle {}} . These two classes give the tool for constructing two obstruction maps which depend on each other and which characterise the hull of the deformation functor. We obtain relations between the obstruction classes by studying a change of rings spectral sequence and by representing certain classes as elements in the Yoneda complex. Calculation of the deformation functor of M M as a B B -module, including the (generalised) Massey products, is thus possible within any A A -free 2 2 -presentation of M M .