Proposition 3

Let A be a graded ring of type \Delta, M, N two graded A-modules of type \Delta and u: M \to N a graded A-homomorphism of degree \delta. Then: (i) Im(u) is a graded submodule of N. (ii) If \delta is a regular element of \Delta, Ker(u) is a graded submodule of M. (iii) If \delta = 0, the bijection M/Ker(u) \to Im(u) canonically associated with u is an isomorphism of graded modules.

Assertion (i) follows immediately from the definitions and Proposition 2(c). If x is an element of M such that u(x) = 0 and x = \sum_\lambda x_\lambda is its decomposition into homogeneous components (where x_\lambda is of degree \lambda), then

and u(x_\lambda) is of degree \lambda + \delta; if \delta is regular the relation \lambda + \delta = \mu + \delta implies \lambda = \mu, hence the u(x_\lambda) are the homogeneous components of u(x) and necessarily u(x_\lambda) = 0 for all \lambda \in \Delta, which proves (ii). The bijection v: M/Ker(u) \to Im(u) canonically associated with u is then a graded homomorphism of degree \delta, as follows from the definition of the quotient graduation; whence (iii) when \delta = 0.