Thursday, 26 July 2012

triangulated categories - Is K(R-Mod) compactly generated when R is an artin algebra?

The answer is in general no - K(RtextmathrmMod) can fail to be well generated even when R is artinian. As you mention K(RtextmathrmMod) is compactly generated if R is of finite representation type. It turns out that the converse holds. This is a result of Jan Šťovíček which occurs as Proposition 2.6 in this paper. The precise result is:



Proposition Let R be a ring. The following are equivalent:



(i) K(RtextmathrmMod) is well generated;



(ii) K(RtextmathrmMod) is compactly generated;



(iii) R is left pure semisimple.



In particular, when R is artinian this occurs precisely when R has finite representation type.

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