Example

(6) If $M$ is a graded A-module and $M(\lambda_0)$ is a graded A-module obtained by shifting (no. 2, Example 3), the $\mathbf{Z}$-linear mapping of $M(\lambda_0)$ into $M$ which coincides with the canonical injection on each $M_{\lambda + \lambda_0}$ is a graded homomorphism of degree $\lambda_0$ (which is bijective when $\Delta$ is a group).

(7) If $a$ is a homogeneous element of degree $\delta$ belonging to the centre of $A$, the homothety $x \mapsto ax$ of any graded A-module $M$ is a graded homomorphism of degree $\delta$.

Remark (3) A graded A-module $M$ is called a graded free A-module if there exists a basis $(m_i)_{i \in I}$ of $M$ consisting of homogeneous elements. Suppose it is and $\Delta$ is a commutative group; let $\lambda_i$ be the degree of $m_i$ and consider for each $i$ the shifted A-module $A(-\lambda_i)$ (no. 2, Example 3); if $e_i$ denotes the element 1 of $A$ considered as an element of degree $\lambda_i$ in $A(-\lambda_i)$, the A-linear mapping $u : \bigoplus_{i \in I} A(-\lambda_i) \to M$ such that $u(e_i) = m_i$ for all $i$, is a graded A-module isomorphism.

Assuming always that $\Delta$ is a commutative group, now let $N$ be a graded A-module, $(n_i)_{i \in I}$ a system of homogeneous generators of $N$ and suppose that $n_i$ is of degree $\mu_i$. Then the A-linear mapping $v : \bigoplus_{i \in I} A(-\mu_i) \to N$ such that $u(e_i) = n_i$ for all $i$ is a surjective graded A-module homomorphism of degree 0. If $N$ is a finitely generated graded A-module, there is always a finite system of homogeneous generators of $N$ and hence there is a surjective homomorphism of the above type with $I$ finite.