Restricted homological dimensions and Cohen-Macaulayness

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G-dimension over local homomorphisms. Applications to the Frobenius endomorphism (2003, background): “utative Noetherian ring, not necessarily local, and Wa homologically bounded complex of R-modules; we do not assume that H(W) is ﬁnite. The large restricted ﬂat dimension of Wover R, as introduced in [13], is the quantity Rfd R(W) = sup{sup(F⊗L W) | Fan R-module with fdR(F) ﬁnite} This number is ﬁni”
Global Gorenstein dimensions (2006, background): “es a partial aﬃrmative answer to each of the two following questions: 1. Foxby’s Conjecture: For any ring R, if Gfd R(M) is ﬁnite, then Gfd R(M) = Rfd R(M). Recall, the large restricted ﬂat dimension [12], Rfd R(M), of an R-module M, is deﬁned by: Rfd R(M) = sup{i ≥ 0|∃L ∈ F(R) : Tor i(L,M) 6= 0 } s”