Sequentially Cohen-Macaulayness of bigraded modules

Ahad Rahimi

doi:10.1216/RMJ-2017-47-2-621

2017 Sequentially Cohen-Macaulayness of bigraded modules

Ahad Rahimi

Rocky Mountain J. Math. 47(2): 621-635 (2017). DOI: 10.1216/RMJ-2017-47-2-621

Abstract

Let $K$ be a field, $S=K[x_1,\ldots ,x_m, y_1,\ldots , y_n]$ a standard bigraded polynomial ring, and $M$ a finitely generated bigraded $S$-module. In this paper, we study the sequentially Cohen-Macaulayness of~$M$ with respect to $Q=(y_1,\ldots ,y_n)$. We characterize the sequentially Cohen-Macaulayness of $L\otimes _KN$ with respect to $Q$ as an $S$-~module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \ldots , x_m]$ and $K[y_1, \ldots , y_n]$, respectively. All hypersurface rings that are sequentially Cohen-Macaulay with respect to $Q$ are classified.

References

1.

W. Bruns and J. Herzog, Cohen-Macaulay rings, revised ed., Cambridge Stud. Adv. Math. {39}, Cambridge University Press, Cambridge, 1998. W. Bruns and J. Herzog, Cohen-Macaulay rings, revised ed., Cambridge Stud. Adv. Math. {39}, Cambridge University Press, Cambridge, 1998.

2.

M. Chardin, J.P. Jouanolou and A. Rahimi, The eventual stability of depth, associated primes and cohomology of a graded module, J. Commutative Algebra {5} (2013), 63–92. M. Chardin, J.P. Jouanolou and A. Rahimi, The eventual stability of depth, associated primes and cohomology of a graded module, J. Commutative Algebra {5} (2013), 63–92.

3.

N.T. Cuong and L.T. Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra {267} (2003), 156–177. N.T. Cuong and L.T. Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra {267} (2003), 156–177.

4.

S. Faridi, Monomial ideals via square-free monomial ideals, Lect. Notes Pure Appl. Math. {244} (2005), 85–114. S. Faridi, Monomial ideals via square-free monomial ideals, Lect. Notes Pure Appl. Math. {244} (2005), 85–114.

5.

J. Herzog and A. Rahimi, Local duality for bigraded modules, Illinois J. Math. {51} (2007), 137–150. J. Herzog and A. Rahimi, Local duality for bigraded modules, Illinois J. Math. {51} (2007), 137–150.

6.

J. Herzog and E. Sbarra, Sequentially Cohen-Macaulay modules and local cohomology, Tata Inst. Fund. Res. Stud. Math. {16} (2002), 327–340. J. Herzog and E. Sbarra, Sequentially Cohen-Macaulay modules and local cohomology, Tata Inst. Fund. Res. Stud. Math. {16} (2002), 327–340.

7.

M. Jahangiri and A. Rahimi, Relative Cohen-Macaulayness and relative unmixedness of bigraded modules, J. Commutative Algebra {4} (2012), 551–575. M. Jahangiri and A. Rahimi, Relative Cohen-Macaulayness and relative unmixedness of bigraded modules, J. Commutative Algebra {4} (2012), 551–575.

8.

L. Parsaei Majd and A. Rahimi, On the structure of sequentially Cohen-Macaulay bigraded modules, Czech. Math. J. {65} (2015), 1011–1022. L. Parsaei Majd and A. Rahimi, On the structure of sequentially Cohen-Macaulay bigraded modules, Czech. Math. J. {65} (2015), 1011–1022.

9.

H. Noormohamadi and A. Rahimi, Sequentially generalized Cohen-Macaulayness of bigraded modules, J. Alg. Appl. {15} (2016), 1650017–14. H. Noormohamadi and A. Rahimi, Sequentially generalized Cohen-Macaulayness of bigraded modules, J. Alg. Appl. {15} (2016), 1650017–14.

10.

A. Rahimi, Relative Cohen-Macaulayness of bigraded modules, J. Algebra {323} (2010), 1745–1757. A. Rahimi, Relative Cohen-Macaulayness of bigraded modules, J. Algebra {323} (2010), 1745–1757.

11.

H. Sabzrou, M. Tousi and S. Yassemi, Simplicial join via tensor product, Manuscr. Math. {126} (2008), 255–272. H. Sabzrou, M. Tousi and S. Yassemi, Simplicial join via tensor product, Manuscr. Math. {126} (2008), 255–272.

12.

P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Lect. Notes Pure Appl. Math. {206}, Dekker, New York, 1999. P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Lect. Notes Pure Appl. Math. {206}, Dekker, New York, 1999.

13.

R.P. Stanley, Combinatorics and commutative algebra, second ed., Birkhäuser, Berlin, 1996. R.P. Stanley, Combinatorics and commutative algebra, second ed., Birkhäuser, Berlin, 1996.

14.

M. Tousi and S. Yassemi, Sequentially Cohen-Macaulay modules under base change, Comm. Alg. {33} (2005), 3977–3987. M. Tousi and S. Yassemi, Sequentially Cohen-Macaulay modules under base change, Comm. Alg. {33} (2005), 3977–3987.

Citation Download Citation

Ahad Rahimi "Sequentially Cohen-Macaulayness of bigraded modules," Rocky Mountain Journal of Mathematics 47(2), 621-635, (2017). https://doi.org/10.1216/RMJ-2017-47-2-621

Published: 2017

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