I found the book a bit hard to read in places as assumptions on algebraic closedness are not always explicitly stated
A) At the start of Section 4 he writes from 4.8 (II) $k$ will be assumed to be algebraically closed. However it seems that this condition is not necessary for most of the rest of the chapter, am I correct that he only needs it in Chapter 4 in two more places, namely
1) for the remark on page 80 which directly invokes 4.8 II
2) for the second part of the proposition in 4.13 $$ k[V_f] = k[V][f^{-1}]$$ which is wrong over R e.g.
Or did I miss out other spots?
B) In Chapters 5 and 6 he does not write an explicit requirement that k is algebraically closed, however e.g. Proposition 6.3 the proof uses algebraically closed (as it should since the statement is wrong otherwise). He does write in 8.8 that "gave way in 3 to considerations over an arbitrary field $k$, promptly assumed to be algebraically closed." Or should I take his comment from Section 4 to mean from there onwards for the rest of the book
C) If it is the case that Ch5 and Ch6 assume algebraically closed how much remains valid over non-algebraically closed fields?