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I couldn't find any info on what set of compact objects generates the following subcategory:

Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group scheme over $k$. Then let $\mathrm{stab}(k[G])$ be a stable model category consiting of finitely generated $k[G]$-modules (modulo the projectives). It is a subcategory of $\mathrm{Stab}(k[G])$ - the stable model category of all $k[G]$-modules (modulo the projectives). For $\mathrm{Stab}(k[G])$, it is known that compact objects are precisely finitely-generated modules and the simple modules generate $\mathrm{Stab}(k[G])$.

More generally, let $\mathcal{K}$ be a compactly generated stable model category with a set $\mathcal{C}$ of compact objects and a set $\mathcal{G}$ of compact generators. What are the compact generators of the subcategory of $\mathcal{K}$ spanned by objects in $\mathcal{C}$? I realize that this may be unknown, so I'd appreciate any particular examples (I'm more interested in cases where the orignal stable model category has a set of compact generators rather than a single one).

For the definitions I use, I refer to Schwede-Shipley. Of course, substituting the term "stable model category" for "triangulated category" across my question makes no difference.

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    $\begingroup$ I'm familiar with the terms "compact objects" and "set of compact generators" for a triangulated category with arbitrary coproducts, and I'm pretty sure the paper of Schwede and Shipley that you link to only uses the terms in this context. Could you be more explicit about your definitions in the case of a category like $\mathrm{stab}(k[G])$ which doesn't have infinite coproducts? $\endgroup$
    – Jeremy Rickard
    Commented May 19, 2019 at 10:19

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