Allow me to reconstruct what is written here, in order for me to present the question. Let $(\mathscr{M},\mathscr{O}_{\mathscr{M}})$ be a smooth manifold, and let $\mathscr{M\times M}$ be the Cartesian product of $\mathscr{M}$ with itself. Let $\Delta$ denote the diagonal mapping, and let $\mathscr{I}$ be the sheaf of germs of smooth functions on $\mathscr{M×M}$ which vanish on $\mathrm{Im}(\Delta)$. Then the quotient sheaf $\mathscr{I}/\mathscr{I}^2$ consists of equivalence classes of functions which vanish on $\mathrm{Im}(\Delta)$ modulo higher order terms. The pullback of this sheaf to $\mathscr{M}$ is the cotangent sheaf, $\Delta^{*}(\mathscr{I}/\mathscr{I}^2)=\Gamma(\mathrm{T}^{*}(\mathscr{M}))$. $\Delta^{*}(\mathscr{I}/\mathscr{I}^2)$ is obviously a sheaf of $\mathscr{O_M}$ modules.
Is $\Delta^{*}(\mathscr{I}/\mathscr{I}^2)$ a quasi-coherent sheaf of $\mathscr{O_M}$ modules? Is it a coherent sheaf of $\mathscr{O_M}$ modules?
If $M$ is a smooth manifold of dimension $n$ the Jacobian matrices of the coordinate transformation induced by the local trivializations of $M$ yield smooth maps between open subsets of $\mathbb{R}^{2n}$. These smooth functions are the transition functions of a rank $n$ vector bundle on $M$. This is the tangent bundle $T_M$ of $M$. The tangent sheaf $\mathcal{T}_X$ is nothing but the sheaf of sections of $T_M$. Therefore it is a rank $n$ locally free sheaf of $\mathcal{O}_M$-modules. The cotangent sheaf $\Omega_{M}$ is the dual of $\mathcal{T}_M$. Therefore it is locally free and in particular coherent.
For algebraic schemes, by Theorem $8.15$ in
R. Hartshorne, "Algebraic Geometry",
one has that an irreducible separated $n$-dimensional scheme $X$ of finite type over an algebraically closed field is smooth if and only if $\Omega_{X}$ is locally free of rank $n$.