Regular singular differential equations and free proalgebraic groups

Throughout this article we work over the field $\mathbb{C}$ of complex numbers. We use the term “algebraic group” in lieu of “affine group scheme of finite type over $\mathbb{C}$”. Similarly, a “proalgebraic group” is an “affine group scheme over $\mathbb{C}$”. By a closed subgroup of a proalgebraic group, we mean a closed subgroup scheme. Following [Mil17, Def. 5.5] a morphism $G\to H$ of proalgebraic groups is called a quotient map if it is faithfully flat. We use $G\twoheadrightarrow H$ to indicate quotient maps.

We begin by recalling the definition of free proalgebraic groups from [Wib20]. Let $\Gamma$ be a proalgebraic group and let $X$ be a set. A map $\varphi\colon X\to\Gamma(\mathbb{C})$ converges to $1$ if almost all elements of $X$ map to $1$ in any algebraic quotient of $\Gamma$, i.e., for every algebraic group $G$ and every quotient map $\phi\colon\Gamma\twoheadrightarrow G$, all but finitely many elements of $X$ map to $1$ under $X\xrightarrow{\varphi}\Gamma(\mathbb{C})\xrightarrow{\phi_{\mathbb{C}}}G(\mathbb{C})$.

The following definition is the special case of [Wib20, Def. 2.18], where $\mathcal{C}$ is the formation of all algebraic groups and $R=k=\mathbb{C}$.

As the pair $(\iota,\Gamma(X))$ is unique up to a unique isomorphism, we will usually speak of the free proalgebraic group $\Gamma(X)$ on $X$.

The map $\iota\colon X\to\Gamma(X)(\mathbb{C})$ is injective. In fact, the induced map $F(X)\to\Gamma(X)(\mathbb{C})$ from the (abstract) free group $F(X)$ on $X$ to $\Gamma(X)(\mathbb{C})$ is injective ([Wib21, Lem. 1.6]). We will therefore in the sequel identify $X$ with a subset of $\Gamma(X)(\mathbb{C})$ via $\iota$.

For a proalgebraic group $G$ and a subset $X$ of $G(\mathbb{C})$, we denote the smallest closed subgroup $H$ of $G$ such that $X\subseteq H(\mathbb{C})$ with $\langle X\rangle$. In other words, $\langle X\rangle$ is the closed subgroup of $G$ generated by $X$. By [Wib20, Thm. 2.17] we have $\Gamma(X)=\langle X\rangle$.

We next explain why the proof of Douady’s theorem (as presented in [Dou64] or [Sza09, Sec. 3.4]) does not have a direct differential analog. This proof uses a result ([Dou64, Prop. 1] or [Sza09, Lem. 3.4.11]) attributed to Serre by Douady, stating that, any set of $n$ elements that topologically generates a free profinite group of rank $n$ is a basis. Equivalently, a surjective endomorphism of a free profinite group of finite rank is an isomorphism. The proof of this result uses a counting argument and therefore does not apply in our context, where finite groups are replaced by algebraic groups. In fact, as shown in the following example, Serre’s result does not hold in our context.

We will need the notion of proalgebraic completion of an abstract group. See e.g., [BLMM02]. Note that the proalgebraic completion is sometimes also referred to as the proalgebraic hull (e.g., in [Sau16]) or as the Hochschild-Mostow group (in honor of [HM57]).

For $X$ a finite (!) set and $F(X)$ the (abstract) free group on $X$, it follows from the universal properties that $F(X)^{\operatorname{alg}}\simeq\Gamma(X)$. The proalgebraic completion $F^{\operatorname{alg}}$ of $F$ can be constructed as the fundamental group of the neutral tannakian category of all finite dimensional $\mathbb{C}$-linear representations of $F$.

Introductions to this topic can be found in [Mag94, vdPS03],[CH11] and [Sau16]. We recall the basic definitions and results, introducing our notation for the subsequent sections along the way.

We fix a differential field $K$ with derivation $\delta\colon K\to K$. We assume that the field of constants $K^{\delta}=\{a\in K|\ \delta(a)=a\}$ of $K$ is the field $\mathbb{C}$ of complex numbers. We are mainly interested in the case when $K=\mathbb{C}(x)$ is the rational function field in one variable $x$ and $\delta=\frac{d}{dx}$. We consider a family $\mathcal{F}=(\delta(y)=A_{i}y)_{i\in I}$ of linear differential equations indexed by some set $I$, where $A_{i}\in K^{n_{i}\times n_{i}}$ is a square matrix for every $i\in I$.

The $K$-subalgebra $R$ of $L$ generated by all of entries of all $Y_{i}$’s and $\frac{1}{\det(Y_{i})}$’s is a $K$-$\delta$-subalgebra of $L$ and called a Picard-Vessiot ring for $\mathcal{F}$.

For a given family $\mathcal{F}$, a Picard-Vessiot extension exists and is unique up to a $K$-$\delta$-isomorphism. The differential Galois group $G(L/K)$ of the Picard-Vessiot extension $L/K$, or of the family $\mathcal{F}$, is the functor $T\rightsquigarrow\operatorname{Aut}(R\otimes_{\mathbb{C}}T/K\otimes_{\mathbb{C}}T)$, from the category of $\mathbb{C}$-algebras to the category groups, where $T$ is considered as a constant differential ring and the automorphisms are required to commute with the derivation. The functor $G(L/K)$ can be represented by a $\mathbb{C}$-algebra, i.e., $G(L/K)$ is a proalgebraic group.

A Picard-Vessiot extension $L/K$ is of finite type if it is the Picard-Vessiot extension for a single differential equation. This is the case if and only of $G(L/K)$ is algebraic.

Since $L$ is the field of fractions of $R$, any $g\in G(L/K)(\mathbb{C})$ extends uniquely to a $K$-$\delta$-automorphism of $L$. For a closed subgroup $H$ of $G=G(L/K)$ we set

An alternative definition of the differential Galois group of a family of linear differential equations can be given via the tannakian formalism ([DM82, Del90]). As a first step, one has to define the “category of differential equations”. This is formalized through the notion of a differential module. A differential module over $K$ is a finite dimensional $K$-vector space $M$ equipped with an additive map $\partial\colon M\to M$ such that $\partial(am)=\delta(a)m+a\partial(m)$ for all $a\in K$ and $m\in M$. A morphism of differential modules over $K$ is a $K$-linear map that commutes with $\partial$.

To a linear differential equation $\delta(y)=Ay$ with $A\in K^{n\times n}$, one associates the differential module $M_{A}$ by setting $M_{A}=K^{n}$ and $\partial(\xi)=\delta(\xi)-A\xi$ for all $\xi\in K^{n}$. Conversely, if $(M,\partial)$ is a differential module with basis $\underline{e}=(e_{1},\ldots,e_{n})$, we can write $\partial(\underline{e})=\underline{e}(-A)$ for a unique matrix $A=A_{M,\underline{e}}\in K^{n\times n}$. Then $\partial(\underline{e}\xi)=\underline{e}\delta(\xi)+\partial(\underline{e})\xi=\underline{e}(\delta(\xi)-A\xi)$ so that $M$ is isomorphic to $M_{A}$. Via the choice of the basis $\underline{e}$, we can thus associate to $M$ the differential equation $\delta(y)=Ay$. A different choice of basis leads to a Gauge equivalent differential equation.

For a differential module $(M,\partial)$ over $K$ one sets $M^{\partial}=\{m\in M|\ \partial(m)=0\}$. This is a $\mathbb{C}$/̄subspace of $M$. If $\underline{e}$ is a $K$-basis of $M$ and $A\in K^{n\times n}$ is such that $\partial(\underline{e}\xi)=\underline{e}(\delta(\xi)-A\xi)$ for all $\xi\in K^{n}$, then $M^{\partial}$ can be identified with the $\mathbb{C}$-space of all solutions of $\delta(y)=Ay$ in $K^{n}$. More generally, if $L$ is a differential field extension of $K$, then $M\otimes_{K}L$ is a differential module over $L$ and $M\otimes_{K}L$ identifies with the space of all solutions of $\delta(y)=Ay$ in $L^{n}$.

The category of all differential modules over $K$ is a neutral Tannakian category over $\mathbb{C}$. If $\mathcal{F}=(\delta(y)=A_{i}y)_{i\in I}$ is a family of differential equations over $K$ and $\mathcal{M}=(M_{A_{i}})_{i\in I}$ is the corresponding family of differential modules, then the differential Galois group of $\mathcal{F}$ is isomorphic to the fundamental group of the neutral Tannakian category $\langle\langle\mathcal{M}\rangle\rangle$ generated by $\mathcal{M}$.

For background on regular singular differential equations and the Riemann-Hilbert correspondence see [Sau16, Part 3], [vdPS03, Chapters 5 and  6], [MS16, Part I] and [ABC20, Part III].

We first treat the local definitions. Consider a differential module $(M,\partial)$ over the field $\mathbb{C}((t))$ of formal Laurent series in $t$ equipped with the usual derivation $\delta=\frac{d}{dt}$. Let $\mathbb{C}[[t]]\subseteq\mathbb{C}((t))$ be the differential subring of formal power series. A $\mathbb{C}[[t]]$/̄lattice in $M$ is a $\mathbb{C}[[t]]$/̄submodule $N$ of $M$ such that there exists a $\mathbb{C}[[t]]$-basis of $N$ that is also a $\mathbb{C}((t))$-basis of $M$.

The differential module $M$ is called regular if there exists a $\mathbb{C}[[t]]$/̄lattice $N$ in $M$ such that $\partial(N)\subseteq N$. The differential module $M$ is called regular singular if there exists a $\mathbb{C}[[t]]$/̄lattice $N$ in $M$ such that $t\partial(N)\subseteq N$. So a regular differential module is regular singular.

We now consider the global picture. Let $(M,\partial)$ be differential module over $\mathbb{C}(x)$. Here, as throughout the paper, the rational function field $\mathbb{C}(x)$ is considered as a differential field via the derivation $\delta=\frac{d}{dx}$.

Let $\mathbb{P}^{1}(\mathbb{C})=\mathbb{C}\cup\{\infty\}$ denote the Riemann sphere. For every point $p\in\mathbb{P}^{1}(\mathbb{C})$ we have a “local” differential field $\mathbb{C}(x)_{p}=\mathbb{C}((t))$. For $p\in\mathbb{C}$ this is $\mathbb{C}(x)_{p}=\mathbb{C}((x-p))=\mathbb{C}((t))$, the field of formal Laurent series in $t=x-p$ with derivation $\frac{d}{dt}=\frac{d}{d(x-p)}$. For $p=\infty$ this is $\mathbb{C}(x)_{p}=\mathbb{C}((x^{-1}))=\mathbb{C}((t))$, the field of formal Laurent series in $t=x^{-1}$ with derivation $\frac{d}{dt}=\frac{d}{dx^{-1}}$.

Note that $(\mathbb{C}(x),\frac{d}{dx})$ is a differential subfield of $(\mathbb{C}((x-p)),\frac{d}{d(x-p)})$ for $p\in\mathbb{C}$. Thus, we obtain a differential module $M_{p}=M\otimes_{\mathbb{C}(x)}\mathbb{C}(x)_{p}$ over $\mathbb{C}(x)_{p}$ for every $p\in\mathbb{C}$. For $p=\infty$ it is not true that $(\mathbb{C}(x),\frac{d}{dx})$ is a differential subfield of $(\mathbb{C}((x^{-1})),\frac{d}{dx^{-1}})$. However, $(\mathbb{C}(x),-x^{2}\frac{d}{dx})$ is a differential subfield of $(\mathbb{C}((x^{-1})),\frac{d}{dx^{-1}})$. So we can base change the differential module $(M,{-x^{2}}\partial)$ over $(\mathbb{C}(x),-x^{2}\frac{d}{dx})$ to a differential module $M_{\infty}$ over $\mathbb{C}(x)_{\infty}$. This awkwardness at infinity is one of the reasons why some authors prefer to work with connections, rather than differential modules. This way one can avoid the a priori choice of a derivation on $\mathbb{C}(x)$. However, since the module $\Omega_{\mathbb{C}(x)/\mathbb{C}}$ of differentials of $\mathbb{C}(x)$ over $\mathbb{C}$ is one dimensional, these two approaches are equivalent.

A point $p\in\mathbb{P}^{1}(\mathbb{C})$ is a singularity of $M$ if the differential module $M_{p}$ over $\mathbb{C}((t))$ is not regular. A point $p\in\mathbb{P}^{1}(\mathbb{C})$ is regular singular for $M$ if the differential module $M_{p}$ over $\mathbb{C}((t))$ is regular singular. Finally, $M$ is called regular singular if every point $p\in\mathbb{P}^{1}(\mathbb{C})$ is regular singular for $M$.

Consider a differential equation $\delta(y)=Ay$ with $A\in\mathbb{C}(x)^{n\times n}$. A point $p\in\mathbb{P}^{1}(\mathbb{C})$ is a singularity of $\delta(y)=Ay$ if it is a singularity of the associated differential module $M_{A}$. Note that this definition is at odds with the common terminology, referring to the poles of $A$ as the singularities of $\delta(y)=Ay$. If $p\in\mathbb{C}$ is a singularity of $\delta(y)=Ay$, then $p$ must be a pole of $A$. However, the converse is not true. A pole of $A$ that is not a singularity of $\delta(y)=Ay$ is sometimes called an apparent singularity. The differential equation $\delta(y)=Ay$ is called regular singular if the associated differential module $M_{A}$ is regular singular.

Fix a proper subset $X$ of $\mathbb{C}\subseteq\mathbb{P}^{1}(\mathbb{C})$. As $X$ is assumed to be a proper subset of $\mathbb{C}$, we can choose a “base point” $x_{0}\in\mathbb{C}$ with $x_{0}\notin X$. Let $\operatorname{RegSing}(\mathbb{C}(x),X)$ denote the category of all regular singular differential modules over $\mathbb{C}(x)$ with singularities contained in $X\cup\{\infty\}$. This is a Tannakian category over $\mathbb{C}$. A fibre functor $\omega_{X,x_{0}}\colon\operatorname{RegSing}(\mathbb{C}(x),X)\to\operatorname{Vec}_{\mathbb{C}}$, with values in the category $\operatorname{Vec}_{\mathbb{C}}$ of finite dimensional $\mathbb{C}$-vector spaces, is given by

where $\mathcal{M}_{x_{0}}$ is the differential field of germs of meromorphic functions at $x_{0}$. We denote with $\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$ the proalgebraic group of tensor automorphisms of $\omega_{X,x_{0}}$.

Now assume that $X$ is finite. Let $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})$ be the topological fundamental group of the Riemann sphere with the points $X\cup\{\infty\}$ removed, with base point $x_{0}$. Then the local solution space $(M\otimes_{\mathbb{C}(x)}\mathcal{M}_{x_{0}})^{\partial}$ is naturally equipped with the monodromy action of $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})$. We denote with $\operatorname{Rep}(\pi_{1}(\mathbb{C}\smallsetminus X,x_{0}))$ the category of finite dimensional $\mathbb{C}$-linear representations of $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})$. The following theorem is sometimes referred to as the (global) Riemann-Hilbert correspondence. The essential surjectivity of the functor in the theorem is also known as the solution of the (weak form of the) Riemann-Hilbert problem.

From Theorem 2.7 we immediately obtain:

We stress the fact that the morphism of groups $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})\to\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})(\mathbb{C})$ is canonical. If $X^{\prime}$ is a subset of the finite set $X\subseteq\mathbb{C}$, then $\operatorname{RegSing}(\mathbb{C}(x),X^{\prime})$ is a subcategory of $\operatorname{RegSing}(\mathbb{C}(x),X)$ and $\omega_{X^{\prime},x_{0}}$ is the restriction of $\omega_{X,x_{0}}$ to $\operatorname{RegSing}(\mathbb{C}(x),X^{\prime})$. We thus have a morphism $\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})\to\underline{\operatorname{Aut}}^{\otimes}(\omega_{X^{\prime},x_{0}})$ of proalgebraic groups. As $\mathbb{C}\smallsetminus X\subseteq\mathbb{C}\smallsetminus X^{\prime}$, we also have a morphism $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})\to\pi_{1}(\mathbb{C}\smallsetminus X^{\prime},x_{0})$ of groups. The diagram

commutes and so also the diagram

commutes.

The family of isomorphisms $\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})^{\operatorname{alg}}\simeq\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})$, one for every finite subset $X$ of $\mathbb{C}$ not containing $x_{0}$, can thus be seen as defining an isomorphism between two projective systems of proalgebraic groups. The projective limit on the right hand side is $\varprojlim_{X}\underline{\operatorname{Aut}}^{\otimes}(\omega_{X,x_{0}})=\underline{\operatorname{Aut}}^{\otimes}(\omega_{\mathbb{C}\smallsetminus\{x_{0}\},x_{0}})$, whereas the projective limit $\varprojlim_{X}\pi_{1}(\mathbb{C}\smallsetminus X,x_{0})^{\operatorname{alg}}$ on the left hand side is a projective limit of free proalgebraic groups. The following section provides the necessary tools to show that this limit itself is free.