Let \(R\) be a discrete valuation ring with quotient field \(K\), and let \(X=\Spec R\).
To give an \(\OO_X\)-module is equivalent to giving an \(R\)-module \(M\), a \(K\)-vector space \(L\), and a homomorphism \(\rho:M\otimes_R K\to L\).
That \(\OO_X\)-module is quasi-coherent if and only if \(\rho\) is an isomorphism.
Solution
Choose a uniformizer \(\pi\in R\) and write \(\mathfrak m=(\pi)\). Let \(x\in X\) correspond to \(\mathfrak m\), and let \(\eta\in X\) correspond to \((0)\). All homomorphisms \(\rho:M\otimes_R K\to L\) are taken to be \(K\)-linear.
1. The nonempty open sets of \(X\) are \(X\) and \(\{\eta\}\), with section rings \(R\) and \(K\), respectively.
Every nonzero element of \(R\) is \(u\pi^n\) for a unit \(u\) and an integer \(n\ge0\). A nonzero prime ideal therefore contains \(\pi\), and must equal the maximal ideal \(\mathfrak m\). Thus \(X\) has precisely the points \(x,\eta\). The distinguished open \(D(\pi)\) is \(\{\eta\}\), and its ring of sections is \(R[\pi^{-1}]=K\). Any open set containing \(x\) contains a distinguished neighborhood \(D(a)\) with \(a\notin\mathfrak m\). Such an \(a\) is a unit, so \(D(a)=X\). Hence \(X\) is the only open neighborhood of \(x\). The restriction map on the structure sheaf is the inclusion \(R\hookrightarrow K\).
2. The correspondence in part (a) is an equivalence, including morphisms.
Given an \(\OO_X\)-module \(\mcf\), set \[ M=\mcf(X),\qquad L=\mcf(\{\eta\}). \] The restriction map \(r:M\to L\) is \(R\)-linear for the inclusion \(R\hookrightarrow K\). It extends uniquely to a \(K\)-linear map \[ \rho:M\otimes_R K\longrightarrow L, \qquad\rho(m\otimes a)=a\,r(m). \]
Conversely, from \((M,L,\rho)\) define the sections on \(X\), \(\{\eta\}\), and \(\varnothing\) to be \(M\), \(L\), and \(0\), with restriction \[ r(m)=\rho(m\otimes1). \] Use the given \(R\)-module and \(K\)-module structures on the two nonempty opens. The \(K\)-linearity of \(\rho\) gives the required compatibility of \(r\) with \(R\hookrightarrow K\). Every open cover of either nonempty open set contains that open set itself, so locality and gluing hold: a compatible family is uniquely determined by its section on that member of the cover. The empty-set sheaf condition holds because its module of sections is zero. Thus these data define an \(\OO_X\)-module.
A morphism from \((M,L,\rho)\) to \((M',L',\rho')\) is a pair consisting of an \(R\)-linear map \(u:M\to M'\) and a \(K\)-linear map \(v:L\to L'\) such that \[ v\circ\rho=\rho'\circ(u\otimes\operatorname{id}_K). \] This equation is equivalent to compatibility with the restriction maps, so such pairs are precisely sheaf-module morphisms. Both constructions preserve composition and identities. They are inverse on section modules and restriction maps, proving the equivalence.
3. The sheaf corresponding to \((M,L,\rho)\) is quasi-coherent exactly when \(\rho\) is an isomorphism.
The sheaf \(\widetilde M\) associated to \(M\) has sections \(M\) on \(X\) and \(M\otimes_R K\) on \(D(\pi)=\{\eta\}\), with restriction \(m\mapsto m\otimes1\) [@Har10a, Proposition II.5.1]. It corresponds to \((M,M\otimes_R K,\operatorname{id}_{M\otimes_R K})\). If \(\rho\) is an isomorphism, the pair \((\operatorname{id}_M,\rho)\) therefore gives an isomorphism \(\widetilde M\to\mcf\), proving that \(\mcf\) is quasi-coherent.
Conversely, if \(\mcf\) is quasi-coherent, its defining affine-local description supplies an affine neighborhood \(U\) of \(x\) and a module \(N\) with \(\mcf|_U\cong\widetilde N\) [@Har10a, Chapter II, §5]. Step 1 forces \(U=X\). The resulting isomorphism gives an \(R\)-linear isomorphism \(u:N\to M\) and a \(K\)-linear isomorphism \(v:N\otimes_R K\to L\). By step 2 they satisfy \(v=\rho\circ(u\otimes\operatorname{id}_K)\). Hence \(\rho=v\circ(u\otimes\operatorname{id}_K)^{-1}\) is an isomorphism.
4. Q.E.D.
Step 2 proves part (a), and step 3 proves part (b).