The paper proves that astab(I)=1 for every graded ideal in dimension two with vstab(I) arbitrary, and constructs ideals realizing every pair (astab(I), vstab(I))=(a,b) in higher dimension.
Brodmann, Asymptotic stability of Ass(M/I nM ), Proc
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Comparison of stability indices of powers of graded ideals
The paper proves that astab(I)=1 for every graded ideal in dimension two with vstab(I) arbitrary, and constructs ideals realizing every pair (astab(I), vstab(I))=(a,b) in higher dimension.