462 ANDREW WILES itself is of type D.Again this is a slight abuse of terminology as we are really condering the xteio ofcand ot bt do this without further mention if the context makes it clear.(The analysis of this section actually applies to any characteristic zero lifting of po but in all our applications we will be in the more restrictive context we have described here.) With these hypotheses there is a unique local homomorphism R of O-algebras which takes the universal deformation to (the class of)pf.Let =ker:Rp.LetK be the field of fractions of and let U) with the Galois action taken from PfA.Similarly,let V=AdpfK/O (K/O)4 with the adjoint representation so that ≌W;⊕K/O where Wf has Galois action via Sym2pf detp and the action on the second factor is trivial.Then if po is ordinary the filtration of U under the Adp action of Dp induces one on Wr which we write 0c. Often to simplify the notation we will drop the index f from Wi,V etc.There is also a filtration on Wam={ker A":W- -Wf}given by Win =Wannwi (compatible with our previous description for n=1).Likewise we write Vn for{kerλn:V -+V. We now explain h to extend the definition of Hp to give meaning to H(Qs/Q,Vam)and H(Qs/Q,V)and these are O/a"and O-modules,re- spectively.In the case where po is ordinary the definitions are the same with oeplacingV and orreplacing One checks easily that (1.7) H(Qx/Q,Vin)H(Qs/Q,V)n, where as usual the subscript "denotes the kernel of multiplication by A". This just uses the divisibility of H(Qs/Q,V)and H(Qp,W/Wo)in the strict case.In the Selmer case one checks that for m>n the kernel of H(Qpnr,Van/Won)H(Qnr,Vam/Wom) has only the zero element fixed under Gal(Q/Qp)and the ord case is similar. Checking conditions at EM is done with similar arguments.In the Selmer and strict cases we make analogous definitions with Wan in place of Van and W in place of V and the analogue of(1.7)still holds. We now conider the case whereo is flat (but not ordinry).We claim first that there is a natural map of O-modules (1.8) H'(Qp,）一Ext6D,iCm,Un） for each m >n where the extensions are of O-modules with local Galois action.To describe this suppose thatV).Then we can asso- ciate to a a representation pa:Gal(Qp/Qp)-GL2(One])(where One]= 462 ANDREW WILES itself is of type 27.Again this is a slight abuse of terminology as we are really considering the extension of scalars pf,x 18 0 and not pf,x itself, but we will "f,~ do this without further mention if the context makes it clear. (The analysis of this section actually applies to any characteristic zero lifting of po but in all our applications we will be in the more restrictive context we have described here.) With these hypotheses there is a unique local homomorphism Rv + 0 of 0-algebras which takes the universal deformation to (the class of) pf,~. Let pv = ker : Rv + 0. Let K be the field of fractions of 0and let Uf = (~10)~ with the Galois action taken from pf,~. Similarly, let Vf = AdpfIx8" K/O = (~/0)*with the adjoint representation so that Vf = Wf @ K/0 where Wf has Galois action via sym2 pf,~ 8 det and the action on the second factor is trivial. Then if po is ordinary the filtration of Uf under the Adp action of Dp induces one on Wf which we write 0 c w~O c ~f c Wf. Often to simplify the notation we will drop the index f from Wj, Vf etc. There is also a filtration on Wxn = {ker An: Wf -Wf) given by Win = Wxn n wi (compatible with our previous description for n = 1). Likewise we write Vxn for {ker An: Vf -Vf). We now explain how to extend the definition of H&to give meaning to H&(Qc/Q, VAn) and H&(Qc/Q, V) and these are O/An and 0-modules, re￾spectively. In the case where po is ordinary the definitions are the same with VAn or V replacing Vx and O/An or K/0 replacing k. One checks easily that as 0-modules (1.7) H&(QclQ,Vxn) = H&(QC/Q,Vxn, where as usual the subscript An denotes the kernel of multiplication by An. This just uses the divisibility of HO(Qc/Q, V) and H'(Q*, w/wO) in the strict case. In the Selmer case one checks that for m > n the kernel of HI (QF~ , VA~/W;~) + H1(&inrlVA~/W;~) has only the zero element fixed under Gal(QpUnr/Qp) and the ord case is similar. Checking conditions at q E M is done with similar arguments. In the Selmer and strict cases we make analogous definitions with Wxn in place of Vxn and W in place of V and the analogue of (1.7) still holds. We now consider the case where po is flat (but not ordinary). We claim first that there is a natural map of 0-modules (1.8) H' (4p. Vxn ) + EX~~~D~~ (~xm, Uxn ) for each m 2 n where the extensions are of 0-modules with local Galois action. To describe this suppose that a E H1(Qp, VAn). Then we can asso￾ciate to a a representation pa: Gal(Qp/Qp) + GL2(0,[.c]) (where On[&] =