##
**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 \(\sum_{i,j} a_{ij}\xi_{ij} = 0\).

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 \(\sum_{i,j} a_{ij}\xi_{ij} = 0\).

Reviewer: Richard M. Cohn (New Brunswick)

### MSC:

13-XX | Commutative algebra |

### Keywords:

structure of finite-dimensional Fuchsian modules; isomorphism classes of one-dimensional Fuchsian modules; Fuchsian differential equations; Fuchsian \(D\)-module
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\textit{Yu. I. Manin}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 19, 113--126 (1965; Zbl 0166.04301)

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