I am trying to do exercise III. A, 2. of Daniel Perrin's Algebraic geometry : an introduction and I'm not that sure about how to go about question a. I've looked for some insight on the internet but I didn't find much so I was hoping you could help me ! Here it is :
Let $V$ be an affine variety such that its coordinate ring $\Gamma(V)$ is a UFD.
a) Let $f_1, \dots, f_n \in \Gamma(V)$ be non zero elements and $h$ their gcd. Show that $D(f_1) \cup \dots \cup D(f_n) \subseteq D(h)$. Show that the restriction homomorphism
$$r : \Gamma(D(h), \mathcal{O}_V) \rightarrow \Gamma(D(f_1) \cup \dots \cup D(f_n), \mathcal{O}_V) $$
is an isomorphism.
Here's what I did :
I showed the inclusion. Now to show that the restriction is an isomorphism, I tried showing that it is injective and surjective. In the following, we'll write $f_i = \tilde{f_i}h$.
When $n = 2$ :
Injectivity. Suppose $\frac{g}{h^m} \in \Gamma(D(h), \mathcal{O}_V) $ is such that its restriction to $D(f_1) \cup D(f_2)$ is $0$. Then it is $0$ on $D(f_1)$ which is a non empty open subset of the irreducible space $D(h)$ (as an open subset of $V$ which must be irreducible since its coordinate ring is a UFD). Hence $D(f_1)$ is dense in $D(h)$ and $\frac{g}{h^m}$ is $0$ on $D(h)$.
Surjectivity. Let $P$ be an element of $\Gamma(D(f_1) \cup D(f_2), \mathcal{O}_V)$. Then $P$ can be written as an irreducible fraction $\frac{g_1}{f_1^a}$ (resp. $\frac{g_2}{f_2^b}$ ) on $D(f_1)$ (resp. $D(f_2)$). Therefore on the (non empty) intersection, we have
$$P = \frac{g_1}{f_1^a} = \frac{g_2}{f_2^b} $$
(with $a \geqslant b$ for example) and thus $\tilde{f_2}^b g_1 = h^{a-b}\tilde{f_1}^ag_2$. So $\tilde{f_1}^a$ divides $\tilde{f_2}^bg_1 $ which implies $a = b = 0$ given that neither $\tilde{f_2}$ nor $g_1$ share a common factor with $\tilde{f_1}$. We then have that $P$ was in fact the restriction to $D(f_1) \cup D(f_2)$ of a regular function $g = g_1 = g_2$ on $V$ and therefore of one on $D(h)$.
My questions are :
Is it correct ? I can usually tell when I am but I am always second guessing when it comes to algebraic geometry and what I did for the case $n = 2$ seems to be a bit stronger than what the exercice asks.
If it is, then how to go about general $n$ ? The exercise hints that one can do it inductively. The proof for injectivity would be the same. How about surjectivity ?
If $f_1, \dots, f_n, f_{n+1} \in \Gamma(V)$ are non zero and $h$ is their gcd. Denote by $h_1$ the gcd of the first $n$ $f_i$ and $U_i = D(f_i)$. Then by induction hypothesis we have that
$$\Gamma(D(h_1), \mathcal{O}_V) \rightarrow \Gamma(U_1 \cup \dots \cup U_n, \mathcal{O}_V) $$
is an isomorphism. By the case $n = 2$ and $h = h_1 \land f_{n+1}$ :
$$\Gamma(D(h), \mathcal{O}_V) \rightarrow \Gamma(D(h_1) \cup U_{n+1}, \mathcal{O}_V) $$
is also an isomorphism.Therefore it suffices to show that
$$\Gamma(D(h_1) \cup U_{n+1}, \mathcal{O}_V) \rightarrow \Gamma(U_1 \cup \dots \cup U_n \cup U_{n+1}, \mathcal{O}_V)$$
is an isomorphism. Injectivity seems to work the same way than before. For surjectivity, if $P \in \Gamma(U_1 \cup \dots \cup U_n \cup U_{n+1}, \mathcal{O}_V)$ then the restriction of $P$ to $\bigcup_{i=1}^n U_i$ is the restriction to this open subset of a function $Q \in \Gamma(D(h_1),\mathcal{O}_V)$. By sheaf property, the function $P$ is the restriction of an element in $\Gamma(D(h_1) \cup U_{n+1}, \mathcal{O}_V)$. Is this good ?
I apologize for the lengthy post and asking mostly for confirmation (and advice if what I've written is nonsense :) ).
Thank you for your time !