IPM - Institute for Research in Fundamental Sciences

“Rashid Zaare-Nahandi”

Tel: +98 241 4249023

Fax: +98 241 4152222

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IPM Positions

Non Resident Researcher (non-resident), School of Mathematics
(2002 - 2003 )

Past IPM Positions

Associate Researcher (non-resident), School of Mathematics
(2000 - 2001)
Student Researcher (non-resident), School of Mathematics
(1997 - 2000)

Non IPM Affiliations

Assistant Professor of IASBS, Zanjan

Research Interests

Algebric Geometry, Commutative Algebra

Research Activities

Let I be a an ideal of the polynomial ring S=k[X₁,?,X_n] generated by square-free monomials. A simplicial complex D_I can be associated to the ring S/I. In this manner S/I is called Stanley-Reisner ring and some of its invariants as Hilbert series and h-vector and Betti numbers, and primary decomposition of the ideal I can be deduced by combinatorial computations in D_I.

Let X be a matrix of linear forms in the ring S. Let I_t(X) be the ideal generated by all t-minors of X. Free complexes as Eagon-Northcott complex is associated to the quotient ring over ideal of maximal minors of X.

Stanley-Reisner rings and determinantal rings are widely studied by mathematicians. In this project, we aim to make a connection between these two subjects. First we determine Stanley-Reisner rings which can be regarded as a determinantal ring, and vise versa.

For example ideal of t-minors of a pluri-circulant matrix after a linear change of coordinates, is a monomial ideal and has a simplicial complex. And, the Stanley-Reisner ideal generated by < ₁-chain monomials of degree t, is a determinantal ideal, were, we say a monomial X_i₁?X_{i_s} is a < ₁-chain monomial if i_j+1 < i_j+1 for all 1 ? j ? s.

References

Bruns, W.; Herzog, J. Cohen-Macaulay Rings, Cambridge University Press: Cambridge, 1993; 403 pp.

[]

Conca, A., Straightening law and powers of determinantal ideals of Hankel matrices, Adv. Math., 138, no. 2 (1998), pp 263-292.

[]

Eagon, J. and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London, 269 (1962), pp 188-204.

Zaare-Nahandi, R. On the ideals of minors of pluri-circulant matrices. Comm. Algebra. to appear.

Present Research Project at IPM

Relations between Stanley-Reisner and determinantal rings

Related Papers

1. Rash. Zaare-Nahandi and Rah. Zaare-Nahandi
The minimal free resolution of a class of square-free monomial ideals
J. Pure Appl. Algebra 189 (2004), 263-278 [abstract]

2. Rah. Zaare-Nahandi and Rash. Zaare-Nahandi
Gr·· obner basis and free resolution of the ideal of 2-minors of A 2×n matrix of linear forms
Comm. Algebra 28 (2000), 4433-4453 [abstract]

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“Rashid Zaare-Nahandi”

IPM Positions

Past IPM Positions

Non IPM Affiliations

Research Interests

Research Activities

References

Present Research Project at IPM

Related Papers

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IPM Positions
Non Resident Researcher (non-resident), School of Mathematics (2002 - 2003 )
Past IPM Positions
Associate Researcher (non-resident), School of Mathematics (2000 - 2001) Student Researcher (non-resident), School of Mathematics (1997 - 2000)
Non IPM Affiliations
Assistant Professor of IASBS, Zanjan
Research Interests
Algebric Geometry, Commutative Algebra
Research Activities
Let I be a an ideal of the polynomial ring S=k[X₁,?,X_n] generated by square-free monomials. A simplicial complex D_I can be associated to the ring S/I. In this manner S/I is called Stanley-Reisner ring and some of its invariants as Hilbert series and h-vector and Betti numbers, and primary decomposition of the ideal I can be deduced by combinatorial computations in D_I. Let X be a matrix of linear forms in the ring S. Let I_t(X) be the ideal generated by all t-minors of X. Free complexes as Eagon-Northcott complex is associated to the quotient ring over ideal of maximal minors of X. Stanley-Reisner rings and determinantal rings are widely studied by mathematicians. In this project, we aim to make a connection between these two subjects. First we determine Stanley-Reisner rings which can be regarded as a determinantal ring, and vise versa. For example ideal of t-minors of a pluri-circulant matrix after a linear change of coordinates, is a monomial ideal and has a simplicial complex. And, the Stanley-Reisner ideal generated by < ₁-chain monomials of degree t, is a determinantal ideal, were, we say a monomial X_i₁?X_{i_s} is a < ₁-chain monomial if i_j+1 < i_j+1 for all 1 ? j ? s. References Bruns, W.; Herzog, J. Cohen-Macaulay Rings, Cambridge University Press: Cambridge, 1993; 403 pp. [] Conca, A., Straightening law and powers of determinantal ideals of Hankel matrices, Adv. Math., 138, no. 2 (1998), pp 263-292. [] Eagon, J. and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London, 269 (1962), pp 188-204. Zaare-Nahandi, R. On the ideals of minors of pluri-circulant matrices. Comm. Algebra. to appear.
Present Research Project at IPM
Relations between Stanley-Reisner and determinantal rings
Related Papers


1.	Rash. Zaare-Nahandi and Rah. Zaare-Nahandi The minimal free resolution of a class of square-free monomial ideals J. Pure Appl. Algebra 189 (2004), 263-278 [abstract]


2.	Rah. Zaare-Nahandi and Rash. Zaare-Nahandi Gr·· obner basis and free resolution of the ideal of 2-minors of A 2×n matrix of linear forms Comm. Algebra 28 (2000), 4433-4453 [abstract]