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We descibe some notions of duality for modules generalizing the dual of a vector space. We first discuss duality for free and projective modules, which is very siilar to the vector space case. Then we discuss duality for finite abelian groups, which is a sort of analog of Fourier analysis. As an application of duality we show the existence of many injective modules over a ring.
This video is part of a lecture course by Richard Borcherds from 2021.