ON A SPECIAL CLASS OF EXACT SEQUENCES

Abstract

The notion of good exact sequence is introduced, which is a special class of exact sequences including split exact sequences as special examples. Several criteria characterizing good exact sequences are given. In particular, we prove that the exact sequence ξ: 0 → K → M → N → 0 is good if and only if the Horseshoe Lemma is true in the "minimal version" with respect to ξ. We also give a lot of nontrivial examples of good exact sequences via "Koszul" objects. Moreover, some applications of good exact sequences are also provided in the end.