Proposition 22

Let A (resp. B) be a graded ring of type A, M (resp. N) a graded module over the graded ring A (resp. B), S (resp. T) a multiplicative subset of A (resp. B) all of whose elements are homogeneous, $f : A \to B$ a homogeneous homomorphism of degree 0 such that $f(S) \subset T$ and $u : M \to N$ an A-linear mapping which is homogeneous of degree k. Then the homomorphism $f' : S^{-1}A \to T^{-1}B$ derived from f (no. 1, Proposition 2) is homogeneous of degree 0 and the $(S^{-1}A)$-linear mapping

derived from f and $u$ (no. 2, Proposition 5) is homogeneous of degree k.

This follows immediately from the equations $f'(a/s) = f(a)/f(s)$ and $u'(m/s) = u(m)/f(s)$.

Finally we note that, if E is a graded A-algebra and S a multiplicative subset of A consisting of homogeneous elements, $S^{-1}E$ with its $(S^{-1}A)$-algebra structure (no. 8) and the graduation $(S^{-1}E)_i$ is a graded $S^{-1}A$-algebra, as follows immediately from the definitions.