0 B Note that -j n the second of these two sequences an extra term appears, corresponding to the additional freedom ¢ ~ ¢ + c, which leaves ~A'B A' ~ ~A'B + vA' ¢, where c is any invariant under the transformation W B s complex constant(4).
Zl '" l e Z. Thus the space Hl(X,o w) effectively conta in s H1(X, O). and is strictly larger than H1(X,O) if the map to Z II l is not simply to the zero elemen t. ;ma~e The ;n the fi r"st Z is always zero, but the image in the second Z is the value Of the t';;w'gr> . (This much follows. for' example, from examination of n ,3, ) From this we see that the Hlp,O·) description only works if the 3. e. :e)'O : 2 d n, where f is homogeneous of degree zero, implies vanishing charge. ] to H (X,O( . 4» .