Some Studies On Koszul-Type Objects And "Minimal" Horseshoe Lemma

The Horseshoe Lemma plays an important role in Homological Algebra, which provides a method to construct new project resolutions from the given ones. It is noted that using minimal projective resolutions is more convenient than using the ordinary ones in computation. Unfortunately, some examples show that the "Minimal" Horse-shoe Lemma is not to be hold in general. In algebra and ring theory, the extension method is a very important way to construct new algebras or rings from the given ones. The main aim of the paper is to find the conditions for the "Minimal" Horseshoe Lemma to be hold in the graded case. Then we discuss the one-point extensions ofλ-Koszul algebras.In the first chapter we describe the research background and list the main theorems of this article.The second chapter is the core of this paper, and we obtain some sufficient and necessary conditions, i.e. Koszul-type modules preserve kernels of epimorphisms if and only if the "Minimal" Horseshoe Lemma to be hold. Then, some simple applica-tions of the "Minimal" Horseshoe Lemma are given.In the third chapter the equivalent conditions that the extension algebras becomeλ-Koszul algebras are given.