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Winter 2021 On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups
Manoj Kummini, Shreedevi K. Masuti
J. Commut. Algebra 13(4): 505-522 (Winter 2021). DOI: 10.1216/jca.2021.13.505


Let (R,𝔪,𝕜) be a three-dimensional Cohen–Macaulay analytically unramified local ring and I an 𝔪-primary R-ideal. Write X=Proj(nIn¯tn). We prove some consequences of the vanishing of H2(X,𝒪X), whose length equals the constant term e¯3(I) of the normal Hilbert polynomial of I. Firstly, X is Cohen–Macaulay. Secondly, if the extended Rees ring A :=nIn¯tn is not Cohen–Macaulay, and either R is equicharacteristic or I¯=m, then e¯2(I)lengthRI2¯/II¯3; this estimate is proved using Boij–Söderberg theory of coherent sheaves on 𝕜2. The two results above are related to a conjecture of S. Itoh (1992). Thirdly, HE2(X,Im𝒪X)=0 for all integers m, where E is the exceptional divisor in X. Finally, if additionally R is regular and X is pseudorational, then the adjoint ideals In ˜, n1 satisfy In ˜=IIn1 ˜ for every n3. The last two results are related to conjectures of J. Lipman (1994).


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Manoj Kummini. Shreedevi K. Masuti. "On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups." J. Commut. Algebra 13 (4) 505 - 522, Winter 2021.


Received: 1 May 2018; Revised: 16 April 2019; Accepted: 15 May 2019; Published: Winter 2021
First available in Project Euclid: 18 January 2022

Digital Object Identifier: 10.1216/jca.2021.13.505

Primary: 13A30 , 13B22 , 13D45
Secondary: 13H10 , 13H15

Keywords: Adjoints , blow-up algebras , Cohomology , integral closure

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium


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Vol.13 • No. 4 • Winter 2021
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