The situation.
I am currently reading Daniel Perrin's Algebraic geometry : an introduction and I am at the point where he defines the sheaf of regular functions on an affine algebraic set $V$ defined over an algebraically closed field $k$. It was shown that it suffices to define it on the principal open subsets $D(f)$ for $f \in \Gamma(V)$ where $\Gamma(V)$ is the algebra of global functions on $V$. One must then check that the restriction and gluing operations behave well on these open subsets. Restriction is not that hard. Now for gluing ...
So we take $D(f) = \bigcup_{i=1}^r D(f_i)$, sections $s_i \in D(f_i)$ and we want to show that there exists $s \in \Gamma(V)_f$ such that $s_{|U_i} = s_i$. When $V$ is irreducible, ie $\Gamma(V)$ is an integral domain, everything goes well.
But now let $V = \bigcup_{\ell = 1}^s V_\ell$ where $s \geqslant 2$ and the $V_\ell$ are the irreducible components of $V$. We can write $s_i = \frac{a_i}{f_i^n}$ and the fact that the $s_i$'s agree on the intersections $D(f_i) \cap D(f_j)$ means that there exists some $N$ such that for all $i, j$ :
\begin{equation} \label{rel} f_i^Nf_j^N(a_if_j^n - a_jf_i^n) = 0 \end{equation}
This holds on $D(f_i) \cap D(f_j)$. We can take the same $N$ because there is only a finite number of $f_i$. We are looking for $a \in \Gamma(V)$ and $m \in \mathbb{N}$ such that for all $i$ :
$$\frac{a}{f^m} = \frac{a_i}{f_i^n}$$
on $D(f_i)$, ie $f_i^{N'}(a f_i^n - a_i f^m) = 0$ for some $m,N'$. We know that $f \in V(f_1^{n+N}, \dots, f_r^{n+N})$ therefore (Nullstellensatz) we can write $f^m = \sum_{j = 1}^r b_j f_j^{n+N}$ for some $m \in \mathbb{N}$ and $b_j \in \Gamma(V)$.
Then $f_i^N a_i f^m = \sum_{j = 1}^r a_i b_j f_i^N f_j^{n+N} = \sum_{j = 1}^r a_j b_jf_j^Nf_i^{n+N} = f_i^{N}f_i^na$ where $a = \sum_{j=1}^r a_j b_j f_j^N$.
This is true because $a_i b_j f_i^N f_j^{n+N} = a_j b_jf_j^Nf_i^{n+N}$ holds on $D(f_i) \cap D(f_j)$ and outside of it, both terms are zero if one of $n$ or $N$ is positive which we can assume.
My questions.
Is my proof correct ? At first I wanted to ask you for advice about how to do it but as I wrote, I got ideas haha.
Is it possible to make use of the fact that we have done the irreducible case before ? We still have the relation on $D(f_i) \cap D(f_j)$ and by density, it holds on all the irreducible components of $V$ that intersect it. By the irreducible case, this gives functions $a_\ell$ on $V_\ell$ such that $s_\ell := \frac{a_{\ell}}{f_{|V_\ell}^m} \in \Gamma(V_\ell)_{f_{|V_\ell}}$ satisfies $s_{{\ell}_{V_\ell \cap D(f_i)}} = s_{{i}_{|V_\ell \cap D(f_i)}}$. To make it work, I suppose we then have to show that the $s_\ell$'s agree on the intersections of the $V_\ell$'s and that the function $s$ we define by glueing the $s_\ell$'s is indeed in $\Gamma(V)_f$.
Thank you for your time ! :)