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I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.

Let $k$ be a field and $R$ a $k$-algebra. The stable module category StMod(R) is the category of R-modules where we've modded out by all morphisms that factor through a projective. This triangulated category is useful in representation theory (see Happel's book) and for the study of Tate Cohomology (see work of Benson, Carlson, Rickard, Iyengar, Christensen, Minac, Beligiannis, Gillespie, Bravo, Stovicek, Krause, Pevtsova). The stable module category is also the homotopy category of a stable model structure on $R$-mod (shown originally by Pirashvili then by Hovey in his book). This is a monoidal model category when $R = k[G]$ for a finite group $G$ (see section 9 of Hovey's "Cotorsion pairs, model category structures, and representation theory), and this is the setting I am interested in.

Right now the only examples of operad-algebras in this category I can think of are:

  • Algebras over $Ass$ are associative $R$-algebras
  • Algebras over $Com$ are commutative $R$-algebras
  • Algebras over $Lie$ should work here and be $R$-algebras that additionally have a Lie structure.

My reason is that, motivated by applications in Top and Ch(k), I recently did some model categorical work on the interplay between localization and colored operads. I've realized the model structure on $R$-mod from above also satisfies all my hypotheses, so there might be applications of my work in this setting. Obviously, I need to read up first on what has been done and what sorts of questions people care about, hence the reference request. The first two examples above seem intrinsically interesting, but I don't know if their homotopy theory has ever been studied elsewhere. I know almost nothing about the third example. References where these three examples or others have been studied in StMod(R) would help me figure out what is currently known and what sorts of questions are the right questions to work on in this area.

Thanks!

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  • $\begingroup$ I forgot some names of people I know to have worked in this area. Greenlees has a great paper on this subject: Tate Cohomology in Axiomatic Stable Homotopy theory. I think Greg Stevenson also wrote a paper about the thick subcategories in this setting. Also, this MO question appears to contain related references: mathoverflow.net/questions/12782/… $\endgroup$ – David White Oct 8 '15 at 12:12
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The answer to this question is now yes. I asked this question while working on the following paper with Donald Yau, about preservation of operad-algebra structure under right Bousfield localization: http://arxiv.org/abs/1512.07570

After getting no answers here (as well as Dan Christensen sharing that he'd never heard of this being done), we went ahead and did it in Section 12, i.e. proved the stable module category satisfied our conditions and got results about preservation of operad algebras. If anyone comes across this thread later and thinks of another paper that looked at operads in the stable module category, please let me know so I can cite that paper.

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