Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

We consider a finitely generated graded module M over a standard graded commutative Noetherian ring R = d 0 R d and we study the local cohomology modules H i R + (M) with respect to the irrelevant ideal R + of R. We prove that the top nonvanishing local cohomology is tame, and the set of its minimal associated primes is finite. When M is Cohen-Macaulay and R 0 is local, we establish new formulas for the index of the top, respectively bottom, nonvanishing local cohomology. As a consequence, we obtain that the (S k )-loci of a Cohen-Macaulay R-module M, regarded as an R 0 -module, are open in Spec(R 0 ). Also, when dim(R 0 ) 2 and M is a Cohen-Macaulay R-module, we prove that H i R + (M) is tame, and its set of minimal associated primes is finite for all i.

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