Approximate liftings in local algebra and a theorem of Grothendieck

Abstract

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 O X -module, where X = Spec A - V ( a ) . Under suitable conditions on A and B and assuming that Ext X 2 ( F , F ) = 0 and E xt X 1 ( F , O X ) = 0 , it is shown in this article that the dual sheaf F v can be extended to a reflexive coherent O Y -module, where Y = Spec B - 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.