460 ANDREW WILES ring with residue field k,has the form,≈(g）with亓ramified then is of type (A). Globally,Proposition 1.1 says that if po is strict and if D=(Se,E,O,M) andD=(str,∑，O,M)then the natural map RD→Rp'isan orphism In each case the tangent space of Ro may be computed as in [Mal].Let A be a uniformizer for and let Uk2 be the representation space for po (The motivation for the subscript will become apparent later.)Let Vbe the representation space of Gal(Q/Q)on Adpo=Homk(U,UA)M2(k).Then there is an isomorphism of k-vector spaces(cf.the proof of Prop.1.2 below) (1.5) Homk(mp/(m品，k)≥H(Q/Q,) where H(Qs/Q,Va)is a subspace of H(Qx/Q,Va)which we now describe and mp is the maximal ideal of Rp.It consists of the cohomology classes which satisfy certain local restrictions at p and at the primes in M.We call mp/(m,A)the reduced cotangent space of RD. We begin with p.First we may write (since p 2),as k[Gal(Q/Q)]- modules, (1.6)V=Waek,where Wa {f E Homk(UA,U):tracef=0} (Sym2⑧det-l)po and k is the one-dimensional subspace of scalar multiplications.Then if po is ordinary the action of Dp on U induces a filtration of U and also on W and V.Suppose we write these,0C WWC Wa and 0CVCVC V.Thus U is defined by the requirement that Dp act on it via the character xI(cf.(1.2))and on U/U via x2.For Wa the filtrations are defined by W={f∈W:f(UR)C UR) w={U∈w:f=0onU} and the filtrations for V are obtained by replacing w by V.We note that these filtrations are often characterized by the action of Dp.Thus the action of Dp on We is viax/x on WA/Wg it is trivial and on W/W it is via X2/X1.These determine the filtration if either x1/x2 is not quadratic or polD is not semisimple.We define the k-vector spaces Vyrd={UeV以：f=0 in Hom(U/UR,U/U} Hse(Qp,Va)=ker{(Qp,Va)H(Qunr,Va/W)} Hdra(Qp,Va)=ker{H(Qp:Va)H(Qunr,Va/Vord)} Htr(Qp,Va)=ker{(Qp,Va)H(Qp:Wa/W)(Qunr,k)} 460 ANDREW WILES ring with residue field k, has the form (A T) with ?i ramified then T is of type (A). Globally, Proposition 1.1 says that if po is strict and if V = (Se, C, 0,M) and V"= (str, C, 0,M) then the natural map RD + RDl is an isomorphism. In each case the tangent space of RD may be computed as in [Mall. Let X be a uniformizer for 0 and let UA e k2 be the representation space for po. (The motivation for the subscript X will become apparent later.) Let VA be the representation space of Gal(Qc/Q) on Ad po = Hornk (UA, UA) 11 M2 (k). Then there is an isomorphism of k-vector spaces (cf. the proof of Prop. 1.2 below) where H&(Qc/Q, Vx) is a subspace of H1 (Qc/Q, VA) which we now describe and mz, is the maximal ideal of RD. It consists of the cohomology classes which satisfy certain local restrictions at p and at the primes in M. We call mD/(m&,A) the reduced cotangent space of RD. We begin with p. First we may write (since p # 2), as k[Gal(Qc/Q)]- modules, (1.6) Vx= Wx@k, where WA = {f EHomk(Ux,Ux):tracef =0} and k is the one-dimensional subspace of scalar multiplications. Then if po is ordinary the action of D, on Ux induces a filtration of Ux and also on WA and VA. Suppose we write these 0 c U: c UA, 0 c Wt c Wi c Wx and 0 c V: c V: c VA. Thus U: is defined by the requirement that D, act on it via the character ~1 (cf. (1.2)) and on Ux/U: via ~2. For Wx the filtrations are defined by and the filtrations for VA are obtained by replacing W by V. We note that these filtrations are often characterized by the action of D,. Thus the action of D, on W: is via ~11x2;on w~/w:it is trivial and on WA/W~it is via ~21x1. These determine the filtration if either x1/x2is not quadratic or polD, is not semisimple. We define the k-vector spaces