Almost Split Sequences and Approximations

Abstract

Abstract. Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. First, we give new characterizations of an almost split sequence in A, which yields some necessary and sufficient conditions for A to have an almost split sequence with prescribed end terms. Then, we study when an almost split sequence in A induces an almost split sequence in an exact subcategory C of A. In case A has almost split sequences and C is Ext-finite and Krull-Schmidt, we obtain a necessary and sufficient condition for C to have almost split sequences. Finally, we show some applications of these results.

Almost Split Sequences for Complexes via Relative Homology

“…It follows from Proposition 2.14 that every element of P(ξ)(W, W ) is nilpotent, and so P(ξ)(W, W ) ⊆ rad(End C (W )), where P(ξ)(W, W ) is the two sided ideal consisting all endomorphisms which factor through C-E projective complexes. Hence, we have rad(End C/P(ξ) (W )) = rad(End C (W ))/P(ξ)(W, W ) (also see [17,Lemma 2.2]). …”

“…Also, we note that if ζ is a non-zero extension in ξxt 1 C (W, U ), then there exists always an R-linear form ϕ : ξxt 1 C (W, U ) → E such that ϕ(ζ) = 0. The following lemma is initiated by a result of Gabriel and Roiter [10, (9.3)], which is explicitly stated in [17,Prop. 3.1].…”

“…starting) in a non-C-E projective (resp. non-C-E injective) complex, which are inspired by [17,Ths. 3.2,3.3].…”

“…Proof. We will only prove the equivalence of assertions (1) and (2) by using an idea of the proof for [17,Th. 3.3].…”

“…The theory of almost split sequences developed further, for example, some results for subcategories of modΛ [5,14,18], the corresponding almost split exact triangles were introduced by Happel [11,12], and studied extensively [7,13,15,19]. The existence of almost split sequences was studied in more general abelian categories [16,17], and in monomorphism categories [21]. In [19], by letting the Nakayama functor act degree-wise, Salarian and Vahed defined a translation τ in the category C(modΛ) of complexes of finitely generated left Λ-modules.…”