Locally free sheaves of finite rank

Let $X$ be a scheme and let $\mathcal{E}$ be a sheaf of ${\mathcal{O}}_X$-modules (Modules on a ringed space). For $r\ge 0$ write ${\mathcal{O}}_X^r$ for the sheaf $U\mapsto {\mathcal{O}}_X(U{)}^r$ with componentwise restriction, so that ${\mathcal{O}}_X^0=0$ and ${\mathcal{O}}_X^1={\mathcal{O}}_X$.

$\mathcal{E}$ is locally free of rank $r$ near $x\in X$ if there is an open neighbourhood $U\subseteq X$ of $x$ and an isomorphism of ${\mathcal{O}}_U$-modules $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r\mathrm{.}$ $\mathcal{E}$ is locally free of finite rank (or finite locally free) if every point $x\in X$ has such a neighbourhood with some rank $r=r(x)\ge 0$. It is then invertible if $r(x)=1$ for every $x$, a notion recorded separately on this page. A locally free sheaf is written with the rank function $r:X\to \mathbb{N}$ that its charts determine.

Well-definedness of the rank. Suppose $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ and $\mathcal{E}{\mathrm{\mid }}_V\cong {\mathcal{O}}_V^s$ with $x\in U\cap V$. Passing to stalks gives an isomorphism $({\mathcal{O}}_{X,x}{)}^r\cong ({\mathcal{O}}_{X,x}{)}^s$ of modules over the local ring ${\mathcal{O}}_{X,x}$, which is a nonzero commutative ring: for an affine open $\mathrm{Spec}A\ni x$ one has ${\mathcal{O}}_{X,x}\cong A_{{\mathfrak{p}}_x}$ with ${\mathfrak{p}}_x\in \mathrm{Spec}A$ (The stalk of the affine structure sheaf at a prime is the localization at that prime). A nonzero commutative ring has invariant basis number (Every nonzero commutative ring has invariant basis number for finite bases, Invariant basis number and the rank of a free module), so $r=s$. Hence $r(x)$ is independent of the chart, and on each chart $U$ the function $r$ is constant, so $r$ is locally constant on $X$; in particular it is constant on every connected component of $X$ and is a locally constant $\mathbb{N}$-valued function on $X$.

Immediate consequences. If $\mathcal{E}$ is locally free of rank $r$ on $X$, then $\mathcal{E}$ is quasi-coherent: on a chart $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r=\widetilde{A^r}$ with $A^r$ free, so the local affine-module condition of Quasi-coherent module on a scheme holds. A locally free sheaf of rank $0$ is the zero sheaf, since it vanishes on the members of an open cover; conversely the zero sheaf is locally free of rank $0$. The conditions are local on $X$ and invariant under isomorphism, and restriction to an open subscheme preserves local freeness with the same rank function. A finite locally free sheaf need not have globally constant rank: the rank may jump between connected components, and only the existence of a rank at each point is required.