Burchʹs inequality and the depth of the blow up rings of an ideal

Let image be a local noetherian ring with infinite residue field 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. Burchʹs inequality says that l(I)+inf{depth A/In, 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/In, 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).