APPENDIX: LOCAL APPEARANCE OF BLOW UP Let us define a ring homomorphism φ as follows.

φ : A[X_{0}, …, X_{s}] →A[f_{0} · T, … f_{s} · T, (f_{j} · T)^{−1}]_{0}
∈  ∈
X_{i}  ↦ (f_{i} · T)(f_{j} · T)^{−1}

It is easy to see that φ is a surjective homomorphism.

p(x_{0}, x_{1}, … x_{s}) ∈ Ker(φ̄)

⟺ p (\frac{f_{0}T}{f_{j}T} , …, \frac{f_{s}T}{f_{j}T} ) = 0
⟺ ∑_{i_{0},i_{1},...i_{s}} p_{i_{0}i_{1}...i_{s}} (\frac{f_{0}T}{f_{j}T} )^{i_{0}} · · · · · (\frac{f_{s}T}{f_{j}T} )^{i_{s}} = 0
⟺ ∃N > 0 ∑_{i_{0},i_{1},...i_{s}} p_{i_{0}i_{1}...i_{s}} (f_{0}T)^{i_{0}} · · · · · (f_{s}T)^{i_{s}} (f_{j}T)^{N−(i_{0}+i_{1}+···+i_{s})} = 0
⟺ ∑_{i_{0},i_{1},...i_{s}} p_{i_{0}i_{1}...i_{s}}f_{0}^{i_{0}} · · · · · f_{s}^{i_{s}}f_{j}^{N−(i_{0}+i_{1}+···+i_{s})} = 0
⟺ p(f_{0}f_{j}^{−1}, … f_{s}f_{j}^{−1}) = 0 in A[f_{j}^{−1}].

We conclude:

Proposition 0.1. Let us denote by A[f_{0}f_{j}^{−1}, … f_{s}f_{j}^{−1}] the subalgebra of A[f_{j}^{−1}] generated by f_{0}f_{j}^{−1}, … f_{s}f_{j}^{−1}. (Note that this notation is ambiguous and should not be used without an explanation.) Then the ring homomorphism φ as above induces an algebra isomorphism

φ̄ : A[f_{0}f_{j}^{−1}, … f_{s}f_{j}^{−1}] ≅ A[f_{0}T, … f_{s}T, (f_{j}T)^{−1}]_{0}.