Remarks on partially abelian exact categories

Abstract

The purpose of this short and elementary note is to identify some classes of exact categories introduced in L. Previdi's thesis. Among other things we show: (1) An exact category is partially abelian exact if and only if it is abelian. (2) An exact category satisfies the axioms AIC and AIC° if and only if it is quasi-abelian in the sense of J.-P. Schneiders. (3) An exact category satisfies AIC if and only if it is an additive category of the type considered by G. Laumon in his work on derived categories of filtered $\mathcal{D}$-modules. In all of the above classes all morphisms have kernels and coimages and the exact structure must be given by all kernel-cokernel pairs.

5 pages