On the cohomology sequence in a semiabelian category
by
Ya. A. Kopylov
Sobolev Institute of Mathematics, Novosibirsk, Russia
Coauthors: V.I.Kuz'minov (Sobolev Institute of Mathematics)

The class of semiabelian categories, introduced by Raikov in 1969, contains all abelian categories as well as many important nonabelian categories of functional analysis and topological algebra (basic examples are given by the categories of Banach spaces and topological abelian groups). We study exactness of the cohomology sequence

... ¶n-1 ®   Hn(A) xn ®   Hn(B) wn ®   Hn(C) ¶n ®   Hn+1(A) xn+1 ®   ...(*)

corresponding to a short exact sequence

0 ® A j ®   B y ®   C ® 0

of cochain complexes A=(Aⁿ, aⁿ)_{n Î Z}, B=(Bⁿ, bⁿ)_{n Î Z}, and C=(Cⁿ, gⁿ)_{n Î Z} the morphisms of whose lines are strict in each dimension. We prove

Theorem. a) If aⁿ is strict then (*) is exact in Hⁿ(B) and Hⁿ(C), and also wⁿ is strict.

b) If bⁿ is strict then (*) is exact in Hⁿ(C) and Hⁿ(A), and also ¶ⁿ is strict.

c) If gⁿ is strict then (*) is exact in Hⁿ⁺¹(A) and Hⁿ⁺¹(B), and also xⁿ⁺¹ is strict.

Date received: February 6, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-17.