High Quality Content by WIKIPEDIA articles! In representation theory, the stable module category is a category in which projectives are "factored out." Let R be a ring. For two modules M and N, define underline{mathrm{Hom}}(M,N) to be the set of R-linear maps from M to N modulo the relation that f~g if f-g factors through a projective module. The stable module category is defined by setting the objects to be the R-modules, and the morphisms are the equivalence classes underline{mathrm{Hom}}(M,N). Given a module M, let P be a projective module with a surjection p colon P to M. Then set ?(M) to be the kernel of p. Suppose we are given a morphism f colon M to N and a surjection q colon Q to N where Q is projective. Then one can lift f to a map P to Q which maps ?(M) into ?(N). This gives a well-defined functor ? from the stable module category to itself.

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