The lifting of determinantal prime ideals
One main result (theorem 1.1) of this paper is inspired by the generic perfection of determinantal ideals in the sense of Hochster and Eagon. It shows that a perfect prime ideal \(I_r(A)\) generated by the maximal minors of an \(r\times s\) matrix \(A= (a_{ij})\) of elements \(a_{ij}\) in a noetherian ring \(R\), \(r\leq s\), has the equality between symbolic and ordinary powers if the grade of the ideals generated by the low order minors of the matrix is large. More precisely: If \({\mathfrak p}=I_r(A)\) is prime with \(\text{grade } {\mathfrak p} =s-r+1\) and if \(\text{grade} (I_t(A)) \geq(r-t+1) \cdot (s-r) +2\) for \(t=1, \dots, r-1\), then the form ring \(\text{gr}_{\mathfrak p} (R)\) of \({\mathfrak p}\) is a domain and \({\mathfrak p}^{(n)} ={\mathfrak p}^n\) for all \(n\geq 1\). For Cohen-Macaulay rings \(R\) this result was also proved by Eisenbud and Huneke but with other methods. The proof of theorem 1.1 is based on the possibility to transfer the property of \(\text{gr}_{\mathfrak p} (R)\) being a domain in (weakly) lifting prime ideals. The second main result is concerned with the question whether one can lift a homogeneous determinantal ideal to a determinantal prime ideal with nice properties by a regular sequence of linear forms. The statement (theorem 1.2 and 4.4) is as follows: Let \(R\) be a finitely generated graded algebra over a field which is a domain and let \(A\) be an \(r\times s\) matrix of homogeneous elements of \(R\) such that all 2-minors of \(A\) are homogeneous, \(r\leq s\). If \({\mathfrak p}=I_r(A)\) is a prime ideal of \(R\) with \(\text{grade} {\mathfrak p}= s-r+1\), then \({\mathfrak p}\) can be lifted by a regular sequence of linear forms to a prime ideal \(I\) generated by the maximal minors of a matrix of homogeneous elements of a finitely generated algebra \(S\) such that \(\text{gr}_I(S)\) is a domain and \(I^{(n)} =I^n\) for all \(n\geq 1\).
generic perfection of determinantal ideals
scientific article; zbMATH DE number 980724
scientific article