By Fernando Q. Gouvea, Noriko Yui
The lawsuits of the 3rd convention of the Canadian quantity conception organization August 18-24, 1991
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Extra resources for Advances in number theory: the proceedings of the Third Conference of the Canadian Number Theory Association, August 18-24, 1991, the Queen's University at Kingston
Let A be a G-module and let g be a subgroup of G. Then A is always a gmodule and Ag is a G/g-module, provided g is normal in G. What are the relations among the cohomology groups H q (G/g, Ag ), H q (G, A) and H q (g, A) ? We first restrict our considerations to the case of positive dimension q ≥ 1. If g is normal in G, we associate with every q-cochain x : G/g × · · · × G/g −→ Ag a q-cochain y : G × · · · × G −→ A by defining y(σ1 , . . , σq ) = x(σ1 · g, . . , σq · g) . We call this y the inflation of x and denote it by y = inf x.
Free for private, non-commercial use. § 4. Inflation, Restriction and Corestriction 33 Along with inflation we obtain another cohomological map by associating with every q-cochain x : G × · · · × G −→ A its restriction y : g × · · · × g −→ A from G × · · · × G to g × · · · × g. We call this q-cochain y the restriction of x and denote it by y = res x. 2) Definition. Let A be a G-module and g a subgroup of G. The homomorphism resq : H q (G, A) −→ H q (g, A), q ≥ 1, induced by the restriction of the cochains of the G-module A to the group g is called restriction.
A particular class of such G-modules are the G-induced modules, which we will make use of in many of the proofs and definitions below. 9) Definition. A G-module A is called G-induced if it can be represented as a direct sum A= σD σ∈G with a subgroup D ⊆ A. In particular, the G-module ZZ[G] = σ∈G σ(ZZ·1) is G-induced, and it is clear that the G-induced modules are represented simply as the tensor products ZZ[G] ⊗ D with arbitrary abelian groups D. In fact, if we consider D as a trivial Gmodule, we have the G-isomorphism ZZ[G] ⊗ D = ZZσ ⊗ D = σ∈G ZZ(σ ⊗ D) = σ∈G σ(ZZ ⊗ D).
Advances in number theory: the proceedings of the Third Conference of the Canadian Number Theory Association, August 18-24, 1991, the Queen's University at Kingston by Fernando Q. Gouvea, Noriko Yui