2 Mar 19:36 2008

### Minimal abelian subcategory

Tom Leinster <t.leinster <at> maths.gla.ac.uk>

2008-03-02 18:36:33 GMT

2008-03-02 18:36:33 GMT

My colleague Walter Mazorchuk has the following question. Being abelian is a*property*of a category, not extra structure. Given an abelian category A, it therefore makes sense to define a subcategory of A to be an ABELIAN SUBCATEGORY if, considered as a category in its own right, it is abelian. Note that a priori, the inclusion need not preserve sums, kernels etc. Now let R be a ring and M an R-module. Is there a minimal abelian subcategory of Mod-R containing M? If so, is there a canonical way to describe it? Any thoughts or pointers to the literature would be welcome. Feel free to assume hypotheses on R (it might be a finite-dimensional algebra etc), or to answer the question for full subcategories only. Thanks, Tom