Abstract :
Let (R, image, k) be a commutative noetherian local ring, n ≥ 2 an integer, X a 2n + 1 × 2n + 1 alternating matrix with entries from image, Y a 1 × 2n + 1 matrix with entries from image, I the ideal I = I1(YX), and A the quotient ring R/I. Assume that the grade of I is at least 2n. (In this case, I is a perfect ideal of grade equal to 2n and I is minimally generated by 2n + 1 elements.) We prove that the minimal resolution of A by free R-modules is a DGΓ-algebra. Furthermore, we identify the algebra TorR·(A, k) and prove that, if R is regular and char k = 0 or (n + 2)/2 ≤ char k, then the Poincaré series PMA(z) = Σ∞i = 0 dimk TorAi(M, k)zi is a rational function for every finitely generated A-module M. As a consequence, we deduce that if the projective dimension of M is infinite, then, eventually, the betti numbers of M form an increasing sequence with strong exponential growth.
