Enlargements of complexes of fixed size

Let $A$ be an artin algebra. The aim of this work is to describe the enlargements of an indecomposable complex in ${\mathbf{C}}_n(projA)$, and to study the irreducible morphisms between them. Precisely, we prove that any indecomposable complex in ${\mathbf{C}}_{[0,n]}(projA)$ or in ${\mathbf{C}}_{n+1}(projA)$ for $n$ a positive integer is a shift or an enlargement of an indecomposable complex in ${\mathbf{C}}_n(projA)$. We also describe the entrances of the irreducible morphisms in ${\mathbf{C}}_{[0,n]}(projA)$ between enlargements of an indecomposable complex $X$ in ${\mathbf{C}}_n(projA)$.