Reviewing Connection
Let \(\pi: X\to S\) be an \(S\)-scheme and \(\mathcal{E}\) a quasi-coherent \(\mathcal{O}_X\)-module. An \(S\)-connection on \(\mathcal{E}\) is a \(\pi^{-1}\mathcal{O}_S\)-linear map
\[\nabla : \mathcal{E}\to \Omega^1_{X/S} \otimes_{\mathcal{O}_S}\mathcal{E}\]
Such that for \(f\in \mathcal{O}_X, e\in \mathcal{E}\), we have
\[ \nabla(fe) = df \cdot e + f\cdot \nabla e \]
Choosing frame sections to (locally) trivialize connection
For locally free sheaves, locally one can choose an isomorphism \(\varphi : \mathcal{O}^n\to \mathcal{E}\) to trivialize it. This is the same as choosing \(n\) sections \(e_1,\dots,e_n\) locally so that they form a basis \(\mathcal{E}= \bigoplus \mathcal{O}_X e_i\). It allows us to associate any section of \(\mathcal{E}\) into \(n\) coordinate functions \((f_i)\), and allows us to operate the connection on coordinate functions,
We obtain the coordinate representation of the connection
\[ \nabla' : \mathcal{O}^n \to \Omega^1_{X/S} \otimes_{\mathcal{O}_S} \mathcal{O}^n = (\Omega^1_{X/S})^n \]
which is explicitly
\[\begin{align*} \nabla' (f^1,\dots,f^n)^t &= \nabla (f^i e_i) \\ &= df^i \otimes e_i + f^i \nabla e_i \\ &= (df^i + f^j \omega^i_j) \otimes e_i\\ &= d(f^1,\dots,f^n)^t + \Lambda \cdot (f^1,\dots,f^n)^t, \end{align*}\]
where \(\Lambda = (\omega^j_i)\) is the matrix of 1-forms of the \(i\)-th component of \(\nabla e_j\).
Matrix Decomposition
So in general we have a decomposition for any locally free connection \(\nabla\),
\[\nabla' = d^{\oplus n} + \Lambda \in \mathrm{Hom}(\mathcal{O}_X^n, (\Omega^1_{X/S})^n).\]
Flat sections in coordinate
A flat section \(\nabla s = 0\) when viewed in coordinate says
\[ \nabla (s^i e_i) = (ds^i + s^j \omega^i_j) \otimes e_i = 0 \Leftrightarrow ds = -\Lambda s. \]
where \(s = (s^1,\dots,s^n)^t\) is the coordinate representation of \(s\) and \(ds + \Lambda s = 0\) is a system of linear differential equations.
Example
Consider the family of elliptic curves \(\pi: E\to S = \mathrm{Spec}(k[t])\) given by the Weierstrass equation
\[ y^2 = x^3 + tx + 1. \]
Choose \(\omega = \frac{dx}{y}, x\omega\) as a basis section of \(\mathcal{H}^1_{dR}(E/S)\) which is a sheaf on \(S\), then there is a \(k\)-linear Gauss-Manin connection defined with \(\mathcal{H}^1_{dR}(E/S)\) on the space \(S\to k\) by
\[\nabla : \mathcal{H}^1_{dR}(E/S) \to \Omega^1_{S/k} \otimes_{k} \mathcal{H}^1_{dR}(E/S)\]
given by (using \(2y\,\mathrm{d}y = 3x^2 \,\mathrm{d}x + t\,\mathrm{d}x + x\,\mathrm{d}t\))
\[\begin{align*} \nabla \omega &= d\frac{dx}{y} = -\frac{1}{y^2} dy\wedge dx \\ &= \frac{x}{2y^3} dx\wedge dt \\ &= (... \text{ performing reduction algorithms}) \\ &= \left(\frac{2t^2}{27+4t^3}\omega + \frac{-3t}{27+4t^3}x\omega + dg\right)\wedge dt. \end{align*}\]
Combining with another similar computation on \(\nabla(x\omega)\), we obtain the connection matrix
\[\Lambda = \frac{1}{\Delta} \begin{pmatrix} -2t^2 & 3t \\ 9 & 2t^2 \end{pmatrix}\,\mathrm{d}t\]
where \(\Delta = -4t^3-27\).
Cyclic Vector Theorem
In the case \(S = \mathrm{Spec}(k[t])\) there is essentially only one tangent direction, so we can divide everything in \(\Omega^1_{S/k}\) by \(dt\) to obtain a derivation \(\partial_t: k[t]\to k[t]\).
In this special case a connection can be simplified into a differential operator on the \(k[t]\)-module \(\mathcal{E}= \mathcal{H}^1_{dR}(E/S)\), making it a differential module \((\mathcal{E}, D)\) over the differential ring \((k[t],\partial_t)\), or equivalently a \(D_S = k[t,\partial_t]\)-module.
The fact that its fibres / sections are finite dimensional vector space means, it has a cyclic vector by the cyclic vector theorem (over the differential ring \(k(t)\)), which will enable us to get a single differential equation describing all flat sections.
And as we’ve seen above, we will need to invert some functions (for example \(\Delta\)) and restrict to an open subset of \(S\) for these things to work.
Abstraction
Let’s consider now a two dimensional differential module \(M\) generated by \(e_1, e_2\) over the differential ring \(k(t)\), or equivalently a two dimensional \(k(t)[\partial_t]\)-module with connection matrix
\[ \Lambda = \left((D e_j)_i\right)_{1\le i,j\le 2} = \frac{1}{\Delta} \begin{pmatrix} -2t^2 & 3t \\ 9 & 2t^2 \end{pmatrix}. \]
Our goal will be trying to find a cyclic vector. By the dense generation result, most vectors should work, for example let’s try \(e_1\), for it to be cyclic it suffices that \(e_1\) and \(D e_1\) are linearly independent over \(k(t)\). We have
\[D e_1 = -\frac{2t^2}{\Delta} e_1 + \frac{9}{\Delta} e_2\]
and this is clearly true, so \(e_1\) is a cyclic vector. Now differentiate again we have
\[\begin{align*} D^2(e_1) &= \partial_t\left(-\frac{2t^2}{\Delta}\right)e_1 + \left(-\frac{2t^2}{\Delta}\right)D(e_1) + \partial_t\left(\frac{9}{\Delta}\right)e_2 + \left(\frac{9}{\Delta}\right)D(e_2) \\ \end{align*}\]
Simple Example
We consider the following simple example as an instructive case
\[\Lambda = \begin{pmatrix} 1 & 0 \\ t & 1 \\ \end{pmatrix}\]
i.e. we have
\[ De_1 = e_1 + t e_2, \quad De_2 = e_2. \]
Clearly \(e_1\) is a cyclic vector here, by computing \(D^2 e_1\) we have
\[ D^2 e_1 = e_1 + (1+2t)e_2. \]
This gives us a linear dependence relation
\[ (1+t)e_1 - (1+2t)De_1 + t D^2 e_1 = 0. \]
Now the generation map gives an coimage-image isomorphism
\[ k(t)[D] = D_S \to M, \quad Q \mapsto Q e_1 \]
\[ \Rightarrow M \cong D_S / D_S P, \quad P = (1+t) - (1+2t)D + tD^2. \]
But there is one thing that is unsatisfying, the quotient we obtained is of the form \(D_S/D_SP\), if we write it in terms of its canonical basis \(e_1,\dots, D^{n-1}e_1\), we have a left cyclic matrix
\[ \Lambda' = \begin{pmatrix} & & & -a_{0} \\ 1 & & & -a_{1} \\ & \ddots& & \vdots \\ & & 1 & -a_{n-1} \\ \end{pmatrix} \]
where \(a_i\) are the coefficients of (the monic version of) \(P\). Under this basis the equations for flat sections \((s_0,\dots,s_{n-1})\) are
\[ s_0' = a_0 s_{n-1}, \quad s_i' = s_{i-1} + a_i s_{n-1}, \quad i = 1,\dots,n-1, \]
which is not very clear. We would like to have a ‘right cyclic’ matrix that looks like the transpose of \(\Lambda'\),
\[\Lambda'' = \begin{pmatrix} & 1 & & \\ & & \ddots & \\ & & & 1 \\ -a_0 & -a_1 & \dots & -a_{n-1} \end{pmatrix}\]
for which the equations for flat sections \((s_0,\dots,s_{n-1})\) are simply
\[ s_{n-1}' +a_{n-1} s_{n-1} + \dots + a_{0} s_{0}=0,\quad s_i' = s_{i+1}, i = 0,\dots,n-2. \]