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Let A be a regular local ring and let F = {F n } n ∈ z be a filtration of ideals in A such that R(F) = ○+ n>0 F n is a Noetherian ring with dimR(F) = dimA + 1. Let G(F) = ○+ n>0 F n /F n+1 and let a(G(F)) be the a-invariant of G(F). Then the theorem says that F 1 is a principal ideal and F n = F n 1 for all n ∈ Z if and only if G(F) is a Gorenstein ring and a(G(F)) = -1. Hence a(G(F)) ≤ -2, if G(F) is a Gorenstein ring, but the ideal F 1 is not principal.

The $a$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings