TITLE: Relative Ext of Avramov-Martsinkovsky as a derived Hom
QUESTION [6 upvotes]: Avramov-Martsinkovsky (http://mathserver.neu.edu/~martsinkovsky/Relative.pdf) have defined an exotic version of Ext between two modules over (for simplicity) Gorenstein rings. The basic idea of their construction in the Gorenstein case is to treat "totally reflexive" modules (=modules of G-dimension zero) as the analogue of projective modules. One then takes suitable left resolutions by these modules to obtain this exotic Ext. 
I am wondering whether anyone had formulated a "relative derived category" where these Exts are the Hom groups in the category? As a concrete question:
Positselski has defined exotic versions of derived categories in https://arxiv.org/pdf/0905.2621.pdf. Are the derived categories studied in Section 3.9 related to the [AM] construction? 
Remark: I am aware of "stable module categories," which I gather play the role that I want with "relative Ext" replaced by "Tate cohomology."

REPLY [2 votes]: Yes, there is a Gorenstein derived category where the Ext groups relative to the Gorenstein projectives can be expressed as Homs in this Gorenstein derived categories.
See https://arxiv.org/pdf/1712.04587.pdf section 3 (especially corollary 3.3.4) for a nice overview on this and https://www.sciencedirect.com/science/article/pii/S0021869310000517 for the original research article.