My major concerns can be concluded in 2 questions:
Why consideration only at closed point is sufficient for the proof?
By the end of the proof of (ii)$\Rightarrow$(i), it mentioned the use of the theorem: $$A\text{ is Cohen-Macaulay }\Rightarrow A_{\mathfrak{p}}\text{ is Cohen-Macaulay for any prime ideal }\mathfrak{p}$$ My question is: how can we conclude the scheme itself as Cohen-Macaulay since we have only proved the Cohen-Macaulay property is satisfied at closed point?
Thank you in advance for your reply.
The Cohen-Macaulay locus of a finite type scheme over a field is open. In this case, this means that the non-CM locus of this scheme is a closed subset containing no closed points, thus the non-CM locus is empty, or the entire scheme is CM.
The general idea at work here is that considering closed points is enough to check open/closed conditions for schemes where the closed points are dense in every closed subset (Jacobson schemes). This covers every scheme of finite type over a field (aka varieties), for instance.