Ideals associated to two sequences and a matrix

Let \u_{1\times n}, \X_{n\times n}, and \v_{n\times 1} be matrices of indeterminates, \Adj \X be the classical adjoint of \X, and $H(n)$ be the ideal I_1(\u\X)+I_1(\X\v)+I_1(\v\u-\Adj \X). Vasconcelos has conjectured that $H(n)$ is a perfect Gorenstein ideal of grade $2n$. In this paper, we obtain the minimal free resolution of $H(n)$; and thereby establish Vasconcelos' conjecture

Socle degrees, Resolutions, and Frobenius powers

We first describe a situation in which every graded Betti number in the tail of the resolution of $\frac RJ$ may be read from the socle degrees of $\frac RJ$. Then we apply the above result to the ideals $J$ and $J^{[q]}$; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of $R/J^{[q]}$ are equal to the graded Betti numbers in the tail of a shift of the resolution of $R/J$.Comment: 19 page

Artinian Gorenstein algebras with linear resolutions

Fix a pair of positive integers d and n. We create a ring R and a complex G of R-modules with the following universal property. Let P be a polynomial ring in d variables over a field and let I be a grade d Gorenstein ideal in P which is generated by homogeneous forms of degree n. If the resolution of P/I by free P-modules is linear, then there exists a ring homomorphism from R to P such that P tensor G is a minimal homogeneous resolution of P/I by free P-modules. Our construction is coordinate free