Regarding the Proof of Hartshorne III.7.11

191 Views Asked by At

enter image description here

I got two questions regarding the proof:

  1. Why we can just assume $A_\mathfrak{m}$ as $A$ if we pick a sufficiently small neighborhood of $x$? (the 7-th line)

  2. Why's the last isomorphism true?

Thank you very much in advance for answering my question.