On the Degree in Categories of Complexes of Fixed Size

We consider $$\Lambda $$Λ an artin algebra and $$n \ge 2$$n≥2. We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander–Reiten component of $${{\mathbf {C_n}}(\mathrm{proj}\, \Lambda )}$$Cn(projΛ) with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in $${\mathbf {C_n}}(\mathrm{proj}\, \Lambda )$$Cn(projΛ) belong to such a category. For a finite dimensional hereditary algebra H over an algebraically closed field, we determine when an irreducible morphism has finite left (or right) degree and we give a characterization, depending on the degrees of certain irreducible morphisms, under which $${\mathbf {C_n}}(\mathrm{proj} \,H)$$Cn(projH) is of finite type.