Let $A$ be a commutative unital ring. Let $M$ be a finite flat $A$-module. We can present $M$ as the cokernel $$A^{\oplus J}\xrightarrow{\varphi} A^n\to M\to0$$ and there exists $a_{ij}$ for $1\leq i\leq n$ and $j\in J$ such that $\varphi(e_j)=\sum_{i=1}^na_{ij}e_i$. Consider the ideal generated by minors of $(a_{ij})$ $$\operatorname{Fit}_r(M)=\langle (n-r)\times(n-r)\textrm{-minors of }(a_{ij})\rangle. $$
Are $\operatorname{Fit}_r(M)$ flat over $A$?
