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Transactions of the American Mathematical Society

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Cohen-Macaulayness and canonical module of residual intersections


Authors: Marc Chardin, José Naéliton and Quang Hoa Tran
Journal: Trans. Amer. Math. Soc. 372 (2019), 1601-1630
MSC (2010): Primary 13C40, 14M06; Secondary 13D02, 13D40, 13H10, 14M10
DOI: https://doi.org/10.1090/tran/7607
Published electronically: May 9, 2019
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Abstract: We show the Cohen-Macaulayness and describe the canonical module of residual intersections $ J={\mathfrak{a}}\colon _R I$ in a Cohen-Macaulay local ring $ R$, under sliding depth type hypotheses. For this purpose, we construct and study, using a recent article of Hassanzadeh and the second author, a family of complexes that contains important information on a residual intersection and its canonical module. We also determine several invariants of residual intersections as the graded canonical module, the Hilbert series, the Castelnuovo-Mumford regularity and the type. Finally, whenever $ I$ is strongly Cohen-Macaulay, we show duality results for residual intersections that are closely connected to results by Eisenbud and Ulrich. It establishes some tight relations between the Hilbert series of some symmetric powers of $ I/{\mathfrak{a}}$. We also provide closed formulas for the types and for the Bass numbers of some symmetric powers of $ I/{\mathfrak{a}}.$


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Additional Information

Marc Chardin
Affiliation: Institut de Mathématiques de Jussieu, UPMC, 4 place Jussieu, 75005 Paris, France
Email: marc.chardin@imj-prg.fr

José Naéliton
Affiliation: Departamento de Matemática, CCEN, Campus I–sn–Cidade Universitária, Universidade Federal de Paraíba, 58051-090 João Pessoa, Brazil
Email: jnaeliton@yahoo.com.br

Quang Hoa Tran
Affiliation: University of Education, Hue University, 34 Le Loi Street, Hue City, Vietnam; and Institut de Mathématiques de Jussieu, UPMC, 4 place Jussieu, 75005 Paris, France
Email: tranquanghoa@hueuni.edu.vn

DOI: https://doi.org/10.1090/tran/7607
Keywords: Residual intersection, sliding depth, strongly Cohen--Macaulay, approximation complex, perfect pairing.
Received by editor(s): May 19, 2017
Received by editor(s) in revised form: March 23, 2018
Published electronically: May 9, 2019
Additional Notes: Part of this work was done while the second author was visiting the Université Pierre et Marie Curie, and he expresses his gratitude for the hospitality.
All authors were partially supported by the Math-AmSud program SYRAM, which gave them the opportunity to work together on this question.
Article copyright: © Copyright 2019 American Mathematical Society