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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups
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by Huy Tài Hà and Ngô Viêt Trung PDF
Trans. Amer. Math. Soc. 357 (2005), 3655-3672 Request permission


This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let $Y$ be the blow-up of a projective scheme $X = \operatorname {Proj} R$ along the ideal sheaf of $I \subset R$. It is known that there are embeddings $Y \cong \operatorname {Proj} k[(I^e)_c]$ for $c \ge d(I)e + 1$, where $d(I)$ denotes the maximal generating degree of $I$, and that there exists a Cohen-Macaulay ring of the form $k[(I^e)_c]$ (which gives an arithmetic Macaulayfication of $X$) if and only if $H^0(Y,\mathcal {O}_Y) = k$, $H^i(Y,\mathcal {O}_Y) = 0$ for $i = 1,..., \dim Y-1$, and $Y$ is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants $\varepsilon$ and $e_0$ such that $k[(I^e)_c]$ is Cohen-Macaulay for all $c > d(I)e + \varepsilon$ and $e > e_0$, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form $R[(I^e)_ct]$. If $R$ has negative $a^*$-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if $\pi _*\mathcal {O}_Y = \mathcal {O}_X$, $R^i\pi _*\mathcal {O}_Y = 0$ for $i > 0$, and $Y$ is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of $R[(I^e)_ct]$ for all $c > d(I)e + \varepsilon$ and $e> e_0$.
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Additional Information
  • Huy Tài Hà
  • Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65201
  • Address at time of publication: Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, Louisiana 70118
  • ORCID: 0000-0002-6002-3453
  • Email:,
  • Ngô Viêt Trung
  • Affiliation: Institute of Mathematics, 18 Hoang Quoc Viet, Hanoi, Vietnam
  • MR Author ID: 207806
  • Email:
  • Received by editor(s): January 10, 2004
  • Published electronically: January 21, 2005
  • Additional Notes: The second author was partially supported by the National Basic Research Program of Vietnam
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 357 (2005), 3655-3672
  • MSC (2000): Primary 14M05, 13H10, 13A30, 14E25
  • DOI:
  • MathSciNet review: 2146643