For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|x|=3$, $|y|=4$, and $|z|=5$. But my question also applies to other graded $k$-algebras.
I have in front of me the nice book of Graham Leuschke and Roger Wiegand, open at Proposition 14.19, which looks wonderful. It gives me the list of indecomposable maximal Cohen-Macaulay $R$-modules in any characteristic, including two. So the singularity category of this ring has finite representation type.
My question is this. What I really want to do is to take the singularity category as a formal differential graded algebra. In other words, I want to start with the derived category whose objects are differential graded $R$-modules, where the degrees are cohomological (i.e., the differential increases degree), and whose arrows are degree preserving maps with the quasi-isomorphisms inverted. Then I want to restrict my attention to the modules whose cohomology with respect to the differential is finitely generated, and then finally I want to localise by getting rid of everything finitely built from the ring $R$. Can the indecomposables in this category be classified in a similar way? Are there general theorems about this?
Edit: (22 Jul 2023) Since nobody seems to have shown much interest in this question, perhaps I should add some context. The algebra $R$ above is the cohomology ring $H^*(M_{11},k)$ of the Mathieu sporadic group $M_{11}$ of order $7920$. It turns out that the cochains on the classifying space $C^*(BM_{11};k)$ give a formal differential graded algebra (this is a very rare phenomenon). In other words, it is quasi-isomorphic to its cohomology, which is $R$. So the category I'm interested in is the singularity category of $C^*(BM_{11};k)$. I'm trying to classify the indecomposables in this category as part of a larger programme whose intention is to understand as much as possible about such singularity categories.
