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A finite locally free sheaf of ideals on a reduced scheme has rank at most one at every point. On a nonempty affine open subscheme where its rank is $r$, localize its inclusion into the structure sheaf at a minimal prime ideal. The resulting local ring is a field $K$, giving an injection $K^r\hookrightarrow K$, hence $r\le 1$. This works without a Noetherian assumption. The empty scheme is a vacuous exception to claims phrased as nonexistence of a sheaf of a prescribed rank.

For a locally free sheaf on an irreducible variety, the stalk inclusion identifies sections regular near $P$ inside its rational sections. The quotient records the obstruction to regularity at that point. This definition does not assume a nonsingular point or a discrete valuation parameter.

On an irreducible variety, every nonempty open subset is dense, so a germ determines a rational section. For a locally free sheaf, this map is injective: a local section which vanishes on a dense open has all its regular coefficient functions zero, hence is zero. Local freeness is important; a section of a torsion sheaf can vanish generically without having zero germ.

On an integral scheme, a quasi-coherent sheaf is torsion-free when its module on every affine chart is a torsion-free module over that chart's integral domain. It embeds into its generic fibre. Therefore a torsion-free sheaf with zero generic fibre is zero; locally free sheaves and their subsheaves are torsion-free.

For a fixed coherent sheaf on a smooth projective curve, one degree bound works for all divisors on an algebraic curve, rather than just powers of one ample divisor. Choose a point $P$. The complement of $P$ is affine, so the surjective transition maps on the finite-dimensional groups $H^1(C,\mathcal{F}(nP))$ eventually kill every class. The Euler characteristic of a coherent sheaf gives $h^0(\mathcal{O}(A))\ge \mathrm{deg}A+1-g$. Every divisor of sufficiently large degree is therefore linearly equivalent to $n_{0}P+E$ with $E$ effective. The point-supported quotient for adding $E$ transfers vanishing to that divisor. Torsion is first removed so that the relevant inclusions are injections of locally free sheaves.