connections on coherent sheaves

More details for the proof of Malgrange.

First we easily reduce to the local case. Fix a point $x$ and a minimal system of generators $v_1,\dots, v_n$ of $F_x$. We want to prove it is free. Let $m_x$ be the maximal ideal of $O_{X,x}$, let $$a_1v_1+\dots+a_nv_n=0, \quad a_i\in O_{X,x}$$ and let $r$ be the smallest non-negative integer such that $a_i\in m_x^r$ for all $i$ and $a_i\notin m_x^{r+1}$ for at least one $i$. Using the connection, we are going to find a new vanishing relation as above, but with smaller $r$.

So apply the connection to the above equality, we get $$\sum_i a_i\nabla(v_i)+\sum_i v_i\otimes da_i=0.$$ Note that $a_i\nabla(v_i)\in M\otimes \Omega_{X}^1$ and $da_i\in m_x^{r-1}\Omega_{X}^1\setminus m_x^r\Omega_{X}^1$ (here we use the characteristic $0$). As $\Omega_X^{1}$ is free, considering the above equality components by components, we see that there exist $b_i\in O_{X,x}$ such that $$\sum_i b_iv_i=0$$ at leat one $b_i\notin m_x^{r}$. So we get a new vanishing relation with smaller $r$. Repeating util we get $r=1$, so one of the coefficients is invertible and the system of generators is not minimal. Contradiction.

See https://mathoverflow.net/questions/81338/coherent-sheaf-with-connection-is-locally-free