A finite flat module over a Noetherian ring is finite projective

For every maximal ideal $\mathfrak{m}\subset R$, the localization $M_{\mathfrak{m}}$ is finite and flat over the Noetherian local ring $R_{\mathfrak{m}}$. By [L1], it is free.

By [L2], choose a finite presentation $R^m\stackrel{A}{\to }{R}^n\stackrel{\pi }{\to }M\to 0$, writing $x_i=\pi (e_i)$. Apply [L3] simultaneously to the finitely many relations given by the columns of $A$. It supplies elements $y_j\in M$ and coefficients $b_{ij}$ such that $x_i={\sum }_j{b}_{ij}{y}_j$ and every relation among the $x_i$ is also a relation among the corresponding coefficient columns. Choose lifts $z_j\in R^n$ of the $y_j$ and define $s_0(e_i)={\sum }_j{b}_{ij}{z}_j$. Then $\pi s_0=\pi$, while the relation condition says that $s_0$ kills $\mathrm{im}(A)=\ker (\pi )$. Hence $s_0$ descends to $s:M\to R^n$ with $\pi s=1_M$. Thus the presentation epimorphism splits, so [L4] makes $M$ projective. Since $M$ is finite, it is finite projective.

Thus every finite flat module over a Noetherian ring is finite projective.