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Archive for April, 2015

Compact objects

If R is a noncommutative ring, then Morita theory tells us that R cannot in general be recovered from its category \text{Mod}(R) of modules; that is, there can be a ring R', not isomorphic to R, such that \text{Mod}(R) \cong \text{Mod}(R'). This means, for example, that “free” is not a categorical property of modules, since it depends on a choice of ring R, or equivalently on a choice of forgetful functor.

It’s therefore something of a surprise that “finitely presented” is a categorical property of modules, and hence that it does not depend on a choice of ring R. The reason is that being finitely presented is equivalent to a categorical property called compactness.

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