Primary decomposition and normality of certain determinantal ideals

Abstract

In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree 2 polynomials and occur naturally from determinantal conditions. Normality is derived from these results.2010 Mathematics Subject Classification. Primary 13C40, 13P10.

Ideals of the form I1(XY)

“…, g n form a regular sequence as well; see Lemma 4.3 and Theorem 6.1. However, this Gröbner basis is too small in size to be of much help in applications like computing primary decomposition of I 1 (XY ) or computing Betti numbers of ideals of the form I 1 (XY ) + J, carried out in [15] and [16] respectively. This motivated us to look for a a different Gröbner basis for I; see Theorem 4.1.…”

“…We prove that g 1 , · · · , g n : g n+1 = g 1 , · · · , g n , ∆ ; where ∆ = det(X). This proof requires the fact that g 1 , · · · , g n , ∆ is a prime ideal, which has been proved in Theorem 5.4 in [15]. Step 3.…”

“…We need detailed information about the ideal g 1 , · · · , g n , ∆ , where ∆ = det(X). We need the fact that this ideal is a prime ideal, which has been proved in Theorem 5.4 in [15]. We also need a minimal free resolution for this ideal, which has been proved below in Lemma 6.10.…”

“…We will see that computing Betti numbers for I 1 (XY ) in the first two cases is not difficult, while the last two cases are not so straightforward. We will use some results from [15] and [16] which have some more deep consequences of the Gröbner basis computation carried out in this paper.…”

Quadrics defined by skew-symmetric matrices

“…Ideals of the form I 1 (X n Y n ) has been studied by [2] and they appear in some of our recent works; see [4], [5], [6], [7]. We described its Gröbner bases, primary decompositions and Betti numbers through computational techniques, mostly under the assumption that X n is either a generic or a generic symmetric matrix.…”