Approximate liftings in local algebra and a theorem of Grothendieck

Let A and B denote local rings such that A=B/tB, where t is a regular nonunit, and let b denote an ideal in B such that the A-ideal a=b/(t) has codimension ⩾2. Let F be a reflexive OX-module, where X=SpecA-V(a). Under suitable conditions on A and B and assuming that ExtX2(F,F)=0 and ExtX1(F,OX)=0, it is shown in this article that the dual sheaf Fv can be extended to a reflexive coherent OY-module, where Y=SpecB-V(b). The infinitesimal procedure that leads to this sheaf extension makes use of the injective theory of sheaves. Applications to homomorphisms of divisor class groups come about as a consequence of this result, and a strong connection with Grothendieck's theorem on parafactoriality is drawn.