The quotient set of Deligne cocycles by Deligne coboundaries is the first Deligne

cohomology group [H.sup.[1].sub.D].

We let [Mathematical Expression Omitted] denote the image in [Mathematical Expression Omitted] of the subset of [Hom.sub.G](X.sub.S](-2), [U.sub.N,S]) consisting of those homomorphisms which induce isomorphisms between the respective Tate

cohomology groups of [X.sub.S](-2) and [U.sub.N,S].

Then the 2-th Hochschild

cohomology group of A with coefficients in X is defined by

Again, if w is not a coboundary, then there is no ample division algebra in C(G,[omega],R); if w is a coboundary, then the iso-classes of ample division algebras in C(G,[omega],R) with D = R or H are parametrized by the second

cohomology group [H.sup.2](G,[R.sup.*]).

We wish to construct equivariant Chern classes for Real bundles in equivariant

cohomology groups with integral coefficients and our main requirement is that one recovers the classical Chern classes by forgetting the [C.sub.2]-action.

The subgroup [[PHI].sub.y] [[pi], [delta]] of the invariant elements of A under the action [psi] was denoted [A.sup.Y] in [20] and shown to be the 0-dimensional

cohomology group [H.sup.0.sub.[phi]](Y, A).

The equation [[delta].sup.k+1][[delta].sup.k] = 0 implies that [Z.sup.k]([Alpha]) contains [B.sup.k]([Alpha]) and the [k.sup.th]

cohomology group [H.sup.k]([Alpha]) is defined to be the quotient space [Z.sup.k]([Alpha])/[B.sup.k]([Alpha]) for k [greater than or equal to] 1.

MacLane and Whitehead have proved in [MW] that equivalence classes of so called crossed modules are in bijection with the elements of the third

cohomology group [H.sup.3](G, A).

Let [[??].sup.r] (M; [Z.sub.2]), r < d, be the first nonzero reduced

cohomology group of M.

By the Riemann-Rock formula we see that X ( D, [2D.sub.|D])= -2 so that the last

cohomology group must have positive dimension.

Section 3 contains the proprieties of two-dimensional local field we need, such as, duality and the vanishing of the second

cohomology group. In section 4, we construct the generalized reciprocity map and study the Bloch-Ogus complex associated to X.

Basic forms are preserved by the exterior derivative and are used to define basic de-Rham

cohomology groups [H*.sub.B] (F).