The theory of meromorphic connections is a modern viewpoint on local behaviour of a class of systems of ODE-s with meromorphic coefficients in a complex domain.
Consider the field of meromorphic functions in a neighborhood of with possible pole at and a finite dimensional -module . A meromorphic connection at is a -linear operator satisfying
In fact, it is customary, in modern literature to consider just a germ: the connections on two different neighborhoods agreeing on the intersection are identified. This way is isomorphic to the field of formal Laurant power series .
There is a natural tensor product on the category of -modules with meromorphic connections. Namely where
There is also an inner hom, namely is with a meromorphic connection
chapter 5, Theory of meromorphic connections, from R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser
P. Maisonobe, C. Sabbah, D-modules cohérents et holonomes, Hermann, Paris 1993.
D. Babbitt, V.S. Varadarajan, Deformations of nilpotent matrices over rings and reduction of analytic families of meromorphic differential equations, Mem. Amer. Math. Soc. 55 (325), iv+147, 1985; Local moduli for meromorphic differential equations, Astérisque 169-170 (1989), 1–217.
Pierre Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes in Math. 163, Springer-Verlag (1970)