We determine the universal deformation ring, in the sense of Mazur, of a re
sidual representation <(rho)over bar> :G(K) --> GL(2)(k), where k is a fini
te field of characteristic p and K is a local field of residue characterist
ic p. As one might hope for, but is not proven in the global case, the defo
rmation ring is a complete intersection, flat over W(k), with the exact num
ber of equations given by the dimension of H-2(G(K), ad(<(rho)over bar>)).
We then go on to determine the ordinary locus inside the deformation space
and, using ideas of Mazur, apply this to compare the universal and the univ
ersal ordinary deformation spaces. Provided that the universal ring for ord
inary deformations with fixed determinant is finite flat over W(k), as was
shown in many cases by Diamond, Fujiwara, Taylor-Wiles and Wiles, we show t
hat the corresponding universal deformation ring - with no restriction of o
rdinariness or fixed determinant - is a complete intersection, finite flat
over W(k) of the dimension conjectured by Mazur, provided that the restrict
ion of det(<(rho)over bar>) to the inertia subgroup is different from the i
nverse cyclotomic character.