Given a finitely presented $R$-module $M$ over a ring $R$ one can define, for every integer $k\geq 0$ the $k$-th Fitting ideal of $M$, for instance in this way, using exterior algebra. $\DeclareMathOperator{Fitt}{Fitt}$ $\DeclareMathOperator{rank}{rank}$
Actually, given any free presentation $$ \varphi: R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$ of $M$, we can describe $\Fitt_k(M)$ as the ideal of $R$ generated by the $(n-k)\times (n-k)$ minors of the matrix representation of $\varphi$.
Do you know if we can say something about the Krull dimension of $\Fitt_k(M)$? In particular I would like to show that (I think the claim is true, but I'm not sure) the codimension of $ \Fitt_k(M)$ in $R$ is at most $$ (k+1)(n-m+k+1) $$ Do you agree with me that this holds? If so, do you have any suggestion on how to prove it?
Any kind of help is appreciated!
A classical result of Eagon and Northcott says that $$\operatorname{height}(I_r(\varphi))\le (m-r+1)(n-r+1).$$ Since $\operatorname{Fitt}_k(M)=I_{n-k}(\varphi)$ we have $$\operatorname{height}(\operatorname{Fitt}_k(M))\le(k+1)(n-m+k+1).$$