I believe a statement in this Ian fletcher notes is Iano fletcher paper on Weighted projective spaces is wrong. For lemma 5.5 he states that the Proj$S(a_0,...,a_n)$ is isomorphic to Proj$S(qa_0,...,qa_n)$. He states this is because the graded rings $S(a_0,...,a_n)$ and $S(qa_0,...,qa_n)$ are isomorphic.
This is clearly wrong, as we will not be able to define a graded isomorphism that preserve grading.
Please could I have some clarification and why has no one spotted this clear error.