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Relative Completions and the Cohomology of Linear Groups Over Local Rings
Author(s) -
Knudson Kevin P.
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610701002836
Subject(s) - unipotent , mathematics , abelian group , nilpotent , group (periodic table) , homomorphism , nilpotent group , pure mathematics , central series , inverse , cohomology , section (typography) , combinatorics , limit (mathematics) , algebra over a field , geometry , physics , computer science , mathematical analysis , quantum mechanics , operating system
For a discrete group G there are two well known completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group U, defined over Q, together with a homomorphism ψ : G → U that is universal among maps from G into prounipotent Q‐groups. To construct U, it suffices for us to consider the case where G is nilpotent; the general case is handled by taking the inverse limit of the Malcev completions of the G /Γ r G , where Γ • G denotes the lower central series of G . If G is abelian, then U = G ⊗ Q. We review this construction in Section 2.