On the Properties of the Ker-Coker-Sequence in a Semiabelian Category
by
Ya.A. Kopylov and V.I. Kuz'minov
Sobolev Institue of Mathematics, Novosibirsk

The class of semiabelian categories (introduced by Raikov) contains all abelian categories as well as many important non-abelian categories of the functional analysis and topological algebra (such as those of Banach spaces and topological abelian groups). We study the Ker-Coker-sequence

Ker\alpha@ > \epsilon >> Ker\beta@ > \zeta >> Ker\gamma@ > \delta >> Coker\alpha@ > \tau >> Coker\beta@ > \theta >> Coker\gamma\tag1

associated with the commutative diagram

\CD @. A0 @ > j0 >> B0 @ > \psi0 >> C0 @ >> > 0 @. @V\alphaVV @V\betaVV @V\gammaVV @. 0 @ >> > A1 @ > j1 >> B1 @ > \psi1 >> C1 @. \endCD \tag2

A morphism \mu in an additive category with kernels and cokernels is called strict (we write it as \mu in O_c) if, in its canonical factorization \mu = im\mu[`(\mu)] coim \mu, [`(\mu)] is both a monomorphism and an epimorphism.

The main results are as follows:

Theorem 1. If in (2) \beta in O_c then \delta in O_c in (1) and sequence (1) is exact in Ker\gamma and Coker\alpha.

Theorem 2. If in (2) j in O_c then sequence (1) is exact in Ker\beta and \epsilon in O_c; if \psi₁ in O_c then it is exact in Coker\beta and \theta in O_c.

Theorem 3. If in (2) \alpha in O_c then in (1) \zeta in O_c and the sequence is exact in Ker\beta and Ker\gamma; if \gamma in O_c then \tau in O_c and sequence (1) is exact in Coker\beta and Coker\alpha.

The research was supported by the INTAS (Grant 96-0712) and the RFBR (Grant 97-01-00846).

Date received: February 8, 2000