A Note on Buchsbaum Rings and Localizations of Graded Domains

Source abstract

Let R = ⊕ i ≧0 R i be a graded integral domain, and let p ∈ Proj (R) be a homogeneous, relevant prime ideal. Let R (p) = { r / t | r ∈ R i , t ∈ R i \ p } be the geometric local ring at p and let R p = { r/t | r ∈ R , t ∈ R\p } be the arithmetic local ring at p . Under the mild restriction that there exists an element r 1 ∈ R 1 \ p , W. E. Kuan [ 2 ], Theorem 2, showed that r 1 is transcendental over R (p) and where S is the multiplicative system R\p. It is also demonstrated in [ 2 ] that R (p) is normal (regular) if and only if R p is normal (regular). By looking more closely at the relationship between R (p) and R (p) , we extend this result to Cohen-Macaulay (abbreviated C M.) and Gorenstein rings.