Among the topics are genus change in inseparable extensions of functional fields, the homology of noetherian rings and local rings, the

cohomology groups of tori in infinite Galois extensions of number fields, an algorithm for determining the type of a singular fiber in an elliptic pencil, variation of the canonical height of a point depending on a parameter, the non-existence of certain Galois extensions of Q unramified outside two, and refining Gross' conjecture on the values of abelian L-functions.

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

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

Moreover, the corresponding homomorphisms of

cohomology groups of these complexes are isomorphisms as follows [9].

For the case of X being compact, the decomposition (2.1) of the

cohomology groups is proved by S.

The endomorphism algebra of a tilting module preserves many significant invariants, for example, the center of an algebra, the number of nonisomorphic simple modules, the Hochschild

cohomology groups, and Cartan determinants.

In [1], we studied Lie-Rinehart cohomology of singularities, and we gave an interpretation of these

cohomology groups in terms of integrable connections on modules of rank one defined on the given singularities.

The orbicycle index polynomial can be use to compute the even dimensions of the orbifold

cohomology groups for global orbifolds of the form [M.sup.n][/.sup.orb]G, where M is a compact smooth manifold, and G [subset] [S.sub.n].

It surveys several algebraic invariants: the fundamental group, singular and Cech homology groups, and a variety of

cohomology groups. Proceeding from the view of topology as a form of geometry, Wallace (professor emeritus, University of Pennsylvania) emphasizes geometrical motivations and interpretations.

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].

An important result by Johnson [10] is that a locally compact group G is amenable if and only if the group algebra [L.sup.1](G) is amenable as a Banach algebra; that is, the first

cohomology groups [H.sup,1]([L.sup.1](G),X*) vanishes for all Banach [L.sup.1](G)-bimodules X.

Here, a topological module V over a topological ring R is said to have vanishing U-cohomology, where U is a topological group which acts continuously and faithfully on V, if the

cohomology groups [H.sup.n](U, V) defined by continuous cochains are trivial for all n = 0,1,....