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Burch's inequality and the depth of the blow up rings of an ideal

✍ Scribed by Teresa Cortadellas; Santiago Zarzuela

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
187 KB
Volume
157
Category
Article
ISSN
0022-4049

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✦ Synopsis

Let (A; m) be a local noetherian ring with inÿnite residue ÿeld and I an ideal of A. Consider RA(I ) and GA(I ), respectively, the Rees algebra and the associated graded ring of I , and denote by l(I ) the analytic spread of I . Burch's inequality says that l(I )+inf {depth A=I n ; n ≥ 1} ≤ dim(A), and it is well known that equality holds if GA(I ) is Cohen-Macaulay. Thus, in that case one can compute the depth of the associated graded ring of I as depth GA(I )=l(I )+inf {depth A=I n ; n ≥ 1}.

We study when such an equality is also valid when GA(I ) is not necessarily Cohen-Macaulay, and we obtain positive results for ideals with analytic deviation less or equal than one and reduction number at most two. In those cases we may also give the value of depth RA(I ).

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