The Bongartz’s Theorem on the Homotopy Category

Abstract

Let $$\Lambda $$ be an Artin algebra and $$M\in $$ mod $$\Lambda $$ . It is well-known that the Bongartz’s Theorem plays an important role in the representation theory of Artin algebras. In this paper, we investigate the relative version of the Bongartz’s Theorem in the bounded homotopy category $$K^{b}(\text {add}M)$$ and construct a Bongartz’s completion for an $$\text {add}M$$ -relative presilting complex in $$K^{b}(\text {add}M)$$ such that it becomes an $$\text {add}M$$ -relative silting complex. Finally, we give some applications.