... thatevery irreducible separated curve is quasi-projective, and every nonsingular completesurface is projective, and so this is an example of minimum dimension. In Shafarevich 1994, VI .2. 3 there ... m mis m (because m ⊃ m m⇒ m ⊃ m) , and so the only maximal ideal inA /m mis m/ m m ; in particular, A /m mis a local ring. As s is not in m/ m m , it is a unitin A /m m,andsosa =0inA /m mimplies ... be arbitrarily given. Since the k-linear maps m/ m 2 → k have a similar description,the first isomorphism is now obvious.Second isomorphism. To give a k-algebra homomorphism A→ k isthesameastogive...