Fuchsian modules. (Moduli fuchsiani.)(Italian)Zbl 0166.04301

Let $$k$$ be a field, $$D$$ a $$k$$-linear space of derivations of $$k$$ into $$k$$, $$M$$ a vector space over $$k$$. The author calls $$M$$ a $$D$$-module if an extension of the operations of $$D$$ to $$M$$ is defined such that $$\partial(m n) = \partial m + \partial n$$, $$(\partial x m) = (\partial x) m + x \partial m$$, $$m, n \in M$$, $$x\in k$$, $$\partial \in D$$. By the dimension of the $$D$$-module $$M$$ is meant the dimension of $$M$$ as a vector space. In particular, if $$k$$ is an ordinary differential field with derivation $$\partial$$, $$L = 0$$ is a linear homogeneous differential equation with coefficients in $$k$$, and $$M$$ is generated by a solution $$u$$ of $$L$$ and the derivatives of $$u$$, then $$M$$ is a $$D$$-module, $$D = k \partial$$, of dimension not exceeding the order of $$L$$. If $$L$$ is Fuchsian, then $$M$$ is a Fuchsian $$D$$-module in the sense defined by the author and stated below. The tensor product of $$D$$-modules is a $$D$$-module with derivations defined in the obvious way. The author shows that the isomorphism classes of one-dimensional $$D$$-modules form a group $$\Delta_k$$ under the multiplication induced by the tensor product. He determines the structure of this group explicitly.
If $$M$$ is any finite-dimensional $$D$$-module one can associate with it a uniquely determined element $$\delta(M)$$ of $$\Delta_k$$. Define $$(\dim, \delta) M = (\dim M, \delta(M)) \in Z\times\Delta_k$$. A ring $$R$$ is defined on $$Z\times\Delta_k$$ by $(z, \alpha) + (z', \alpha') = (z + z', \alpha + \alpha');\ (z, \alpha) (z', \alpha') = (zz', z\alpha' + z'\alpha).$ Then there is a homomorphism $$\varphi$$ of the Grothendieck ring $$K$$ of the category of finite-dimensional $$D$$-modules onto $$R$$ such that if $$M$$ is a finite-dimensional $$D$$-module, and $$\gamma(M)$$ the corresponding member of $$K$$, then $$\varphi\gamma(M) = (\dim, \delta) M$$.
Now let $$k = k_0\ll t\gg$$, where $$k_0$$ is a field of characteristic 0, and let $$D$$ be the one-dimensional space of continuous derivations of $$k$$ trivial on $$k_0$$. Let $$A = k_0 [[t]]$$, and let $$P$$ be the maximal ideal of $$A$$. Let $$D_0 = \{\partial\in D); \partial P \subseteq P\}$$. In particular $$\partial_t \in D_0$$, where $$\partial_t$$ is the continuous derivation such that $$\partial_tt = t$$. A $$D$$-module $$M$$ is called (locally) Fuchsian if for every $$m\in M$$ the minimal $$A$$ submodule containing $$m$$ and closed under $$D_0$$ is finitely generated. The isomorphism classes of one-dimensional Fuchsian modules form a subgroup $$\Delta_k^0$$ of $$\Delta_k$$ canonically isomorphic to $$k_0^+/Z$$, where $$k_0^+$$ is the additive group of $$k_0$$. Every finite-dimensional Fuchsian module is a direct sum of Fuchsian modules which are indecomposable (as $$D$$-modules).
If $$k_0$$ is algebraically closed, the structure of finite-dimensional Fuchsian modules is then completely determined by the following results: Every indecomposable, finite-dimensional, Fuchsian module $$M$$ contains a unique one-dimensional Fuchsian submodule $$M^*$$ [not in general in $$\delta(M)]$$. If $$\xi\in k_0^+/Z$$, $$a\in Z$$, $$a\ge 1$$, then the module $$M = M^\xi \otimes M^{(a)}$$ is an indecomposable Fuchsian module, where $$M^\xi$$ is the one-dimensional Fuchsian module of class $$\xi$$, and $$M^{(a)} = \sum_{i=0}^{a-1} k m_i$$, $$\partial_tm_i = m_{i-1}$$, $$m_{-1} = 0$$. Furthermore $$M^* \cong M^\xi$$, $$\dim M = a$$; and if $$N$$ is indecomposable and Fuchsian $$N^* \cong M^*$$, and $$\dim N = a$$, then $$N \cong M$$.
Lastly, let $$k$$ be a field of algebraic functions of one variable over an algebraically closed field $$k_0$$ of characteristic 0, and let $$D$$ be the one-dimensional space of derivations of $$k$$ trivial on $$k_0$$. Let $$V$$ be the set of discrete valuations of $$k$$ trivial on $$k_0$$, and for $$v\in V$$ let $$k_v$$ be the completion of $$k$$ under $$v$$, and let $$D_v = D\otimes k_v$$. A $$D$$-module $$M$$ is called (globally) Fuchsian if for each $$v\in V$$ the $$D_v$$-module $$M_v = k_v \otimes M$$ is (locally) Fuchsian. It is shown that the isomorphism classes of one-dimensional Fuchsian modules form a subgroup of $$\Delta_k$$, which is a homomorphic image of the group of differentials of $$k$$ with poles of order at most 1.
Classical results on Fuchsian differential equations are generalized by the following theorem: Let $$M$$ be a Fuchsian $$D$$-module of finite dimension $$a$$. There is a finite subset $$S$$ of $$V$$ such that if $$v\notin S$$, then $$M_v \cong \otimes_1^a k_v$$. For $$v_i\in S$$, let $$M_{v_i} \cong M^{\xi_{ij}} \otimes M^{(a_{ij})}$$, $$\xi_{ij}\in k_0^+/Z$$, $$a_{ij}\in Z$$, be the representation of $$M_{v_i}$$ as a direct sum of indecomposable $$D_v$$-modules. Then $$\sum_{i,j} a_{ij}\xi_{ij} = 0$$.

MSC:

 13-XX Commutative algebra
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References:

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