dim(R/a(M)) < d iff R/p is unmixed and the generic formal fiber of R/p is Cohen-Macaulay for all p in Supp_R(M) with dim(R/p)=d.
Local cohomology: an algebraic introduction with geometric applications
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Cohen-Macaulayness of formal fibers and dimension of local cohomology modules
dim(R/a(M)) < d iff R/p is unmixed and the generic formal fiber of R/p is Cohen-Macaulay for all p in Supp_R(M) with dim(R/p)=d.