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Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of differential graded modules?

Moreover, if there is a well-defined notion, when are each of them finite? For example, does the Koszul complex $K^\bullet_R(M;f_1,\ldots,f_k)$ have computable K-dimensions?

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    $\begingroup$ I did not see at first that you are fixing one differential graded algebra. What does the letter "K" denote? Is that the name of your differential graded algebra? $\endgroup$
    – Jason Starr
    Commented Sep 3, 2017 at 23:33
  • $\begingroup$ K as in K-injective resolution. $\endgroup$
    – 54321user
    Commented Sep 3, 2017 at 23:36
  • $\begingroup$ For an Abelian category, such as the category of differential graded modules over a fixed differential graded algebra, there is a purely categorical notion of injective and projective objects. Is that the notion you are using? $\endgroup$
    – Jason Starr
    Commented Sep 3, 2017 at 23:42

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The answer is yes, this theory has been worked out. As a topologist, my favorite reference is Christensen-Hovey "Quillen model structures for relative homological algebra," which works in extreme generality (so covers your situation of interest). Hovey went on, with Lockridge, to write a series of papers about homological dimension. They are phrased in terms of ring spectra, but everything also works in the dg setting (and quite possibly was known before; I am not familiar with the history). See "Semisimple ring spectra" and "Homological dimensions of ring spectra"

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    $\begingroup$ is there a formula to compute the dimensions for the derived Koszul algebra mentioned in the question? $\endgroup$
    – Yosemite Sam
    Commented Sep 5, 2017 at 4:00

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