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
Moreover, the corresponding homomorphisms of cohomology groups
of these complexes are isomorphisms as follows .
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 , 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  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,....