I am working through a book on linear algebraic groups, and without giving background and just at one single point the notion of a sheaf of a function is used. I never heard of that. I know what an algebraic variety is, and I know some commutative algebra, which is also mentioned in the book. But when I try to look up what is meant by this "sheaf" I am totally lost, and even the wikipedia article is quite long.
So, could anyone explain shortly for this specific situation why we need a sheaf in the following construction, and whats the intuition behind it?
The relevant section of the books is:
[...] projective $n$-space $\mathbb P^n$ may be defined as the set of equivalence classes of $k^{n+1} \setminus \{ (0,0,\ldots,0) \}$ modulo the diagonal action of $k^{\times}$ by multiplication. Taking common zeros of a collection of homogeneous polynomials in $k[T_0, T_1, \ldots, T_n]$ as closed sets defines a topology on $\mathbb P^n$. A projective variety is then a closed subset of $\mathbb P^n$ equipped with the induced topology.
The $k$-algebra of regular function on an affine variety here needs to be replaced by a sheaf of a function, as follows: First, for $X$ an irreducible affine variety and $x \in X$, let $I(x) \unlhd k[X]$ be the ideal of functions vanishing at $x$ and let $\mathcal O_x$ be the localization of $k[X]$ with respect to the prime ideal $I(x)$. Then setting $O_X = \bigcap_{x\in X} O_x$, for $U \subseteq X$ open, defines a sheaf of functions $\mathcal O_X$ on $X$. Clearly the fields of fractions of $\mathcal O_X(U)$ and $k[X]$ agree. More generally, if $X$ is reducible, say $X = X_1 \cup \ldots X_n$ with irreducible components $X_i$, then setting $$ O_X(U) := \{ f : U \to k \mid f_{|U\cap X_i} \in \mathcal O_{X_i}(U\cap X_i) \}, $$ for $U \subseteq X$ open, defines a sheaf on $X$. One can show that $O_X(X) = k[X]$, the usual $k$-algebra of regular functions on $X$.
I cannot make sense of all these construction. These construction are used in the book to show that linear algebraic groups admit quotients that are algebraic