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On a complex analytic manifold there exists the sheaf D X of differential operators with holomorphic coefficients. This is a coherent sheaf of rings, but in contrast to the sheaf O X it is non-commutative which leads to more involved ring-theoretic problems as compared with O-module theory. Analytic D-module theory is devoted to the study of sheaves of D X -modules, left or right. In most applications there occur coherent modules which locally are quotients of free D X -modules of finite rank. Therefore coherent D X -modules correspond to systems of differential equations with analytic coefficients. Of course, these systems are over-determined in general.
KeywordsFinite Rank Coherent Sheaf Coherent Module Fuchsian System Regular Singularity
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