Abstract
Let K be a field and X, Y denote matrices such that the entries of X are either indeterminates over K or 0, not all zero, and the entries of Y are indeterminates over K which are different from those appearing in X. We consider the ideal \(I_{1}(XY)\), which is the ideal generated by the homogeneous polynomials of degree 2 given by the \(1\times 1\) minors of the matrix XY. We prove that d-sequences and regular sequences arise naturally as part of generators of \(I_{1}(XY)\) for some special cases. We use this information to calculate the equations defining the Rees algebra of \(I_{1}(XY)\).
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Herzog J, Certain Complexes Associated to a Sequence and a Matrix, Manuscr. Math. 12 (1974) 217–248
Huneke C and Ulrich B, Divisor Class Groups and Deformations, Amer. J. Math. 107(6) (1985) 1265–1303
Saha J, Sengupta I and Tripathi G, Transversal Intersection and Sum of Polynomial Ideals, arXiv:1611.04732v2 [math.AC] 2018 (to appear in J. Ramanujan Math. Soc.)
Saha J, Sengupta I and Tripathi G, Primary decomposition and normality of certain determinantal ideals, Proc. Indian Acad. Sci. (Math. Sci.) 129 (2019) 55
Saha J, Sengupta I and Tripathi G, Ideals of the form \(I_{1}(XY)\), J. Symbolic Computation 91 (2019) 17–29
Saha J, Sengupta I and Tripathi G, Quadrics defined by skew-symmetric matrices, Int. J. Algebra 11(8) (2017) 349–356
Swanson I and Huneke C, Integral Closure of Ideals, Rings, and Modules, LMS Lecture Note Series 336 (2006) (UK: Cambridge University Press)
Vasconcelos W, Integral Closure, Springer Monograph in Mathematics (2005)
Acknowledgements
The first author, JS is supported by the NBHM post-doctoral fellowship, Ref. No. 0204/3/2020/R &D-II/2459. The second author, IS is supported by the MATRICS research grant MTR/2018/000420, sponsored by the SERB, Government of India.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D S Nagaraj.
Rights and permissions
About this article
Cite this article
Saha, J., Sengupta, I. & Tripathi, G. A note on \({\varvec{d}}\)-sequences and regular sequences of quadrics. Proc Math Sci 132, 75 (2022). https://doi.org/10.1007/s12044-022-00719-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12044-022-00719-x