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Transactions of the American Mathematical Society

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Cohen-Macaulay residual intersections and their Castelnuovo-Mumford regularity

Author: Seyed Hamid Hassanzadeh
Journal: Trans. Amer. Math. Soc. 364 (2012), 6371-6394
MSC (2010): Primary 13C40, 13D02, 13H10; Secondary 14C17
Published electronically: July 11, 2012
MathSciNet review: 2958940
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Abstract: In this article we study the structure of residual intersections via constructing a finite complex of not necessarily free modules. The complex provides information about an ideal which coincides with the residual intersection in the geometric case; and is closely related to it in general. A new success obtained through studying such a complex is to prove the Cohen-Macaulayness of residual intersections of a wide class of ideals. In particular, it is shown that in a Cohen-Macaulay local ring any geometric residual intersection of an ideal which satisfies the sliding depth condition is Cohen-Macaulay. This is an affirmative answer for one of the main open questions in the theory of residual intersections (Huneke and Ulrich, 1988, Question 5.7).

The complex that we come up with in this article suffices to obtain a bound for the Castelnuovo-Mumford regularity of a residual intersection in terms of the degrees of minimal generators. More precisely, in a positively graded Cohen-Macaulay *local ring $ R=\bigoplus _{n \geq 0} R_{n}$, if $ J=\mathfrak{a} :I$ is a ``geometric'' $ s$-residual intersection such that $ \operatorname {Ht} (I)=g>0$ and $ I$ satisfies a sliding depth condition, then $ \operatorname {reg}(R/J) \leq \operatorname {reg}( R) + \dim (R_0)+ \sigma ( \mathfrak{a}) -(s-g+1)\operatorname {indeg} (I/\mathfrak{a})-s$, where $ \sigma ( \mathfrak{a})$ is the sum of the degrees of elements of a minimal generating set of $ \mathfrak{a}$. It is also shown that the equality holds whenever $ I$ is a perfect ideal of height 2 and $ R_0$ is a field.

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Seyed Hamid Hassanzadeh
Affiliation: Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, 599 Taleghani Avenue, Tehran 15618, Iran – and – Institut de Mathematiques, Université Pierre et Marie Curie, 175 rue du Chevaleret, 75013, Paris, France
Address at time of publication: Departamento de Matemática, Universidade Federal de Pernambuco, Av. Jornalista Anibal Fernandes, sn, Cidade Universitária 50740-560, Recife, Pernambuco, Brazil

Keywords: Castelnuovo-Mumford regularity, generalized Koszul complex, $G_{s}$ condition, residual intersection, sliding depth condition
Received by editor(s): June 18, 2009
Received by editor(s) in revised form: December 31, 2010
Published electronically: July 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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