Summary: EUKNAL OF ALGEBKA 134, 157-181 ( 1990)
omologicai Duality for Grosse
A group G is a duality group of dimension n if there exists a righr
G-module C and an element eETorzG( C. Z) such that the: mor
induced by cap product namely
is an isomorphism for each integer i and every left &module A (see[BE] ).
Becauseof the duality of G, the cohomological dimension of G is ~2and in
particular G is torsion free.
In addition to the above construction we also need to recall the
construction of the crossed product KTT (see [AR1 j ). In K;F. K is a
commutative ring, f is a group acting err it via a homomorphism
t : r-+ AutjK), and a is an element of the cohomology group H2(rF K")
(K* is the set of invertible elements of K viewed as a I'-module). The
crossed product K :r is isomorphic, as a left K module to the direct sum
ll 0EI Ku,. while the product is defined by the rule
where $ I-x I--+ K* is a 2-cocycle representing x and a(+;`) is the 11a)
action on 1:.
The crossedproduct KYI' is an associative K' algebra (K' is the subring
of K fixed by I?. Up to an isomorphism of algebras .K:S does not depend