Koszul property of diagonal subalgebras

Abstract

ABSTRACT. Let S = K[x 1 , . . . , x n ] be a polynomial ring over a field K and I a homogeneous ideal in S generated by a regular sequence f 1 , f 2 , . . . , f k of homogeneous forms of degree d. We study a generalization of a result of Conca, Herzog, Trung, and Valla [9] concerning Koszul property of the diagonal subalgebras associated to I. Each such subalgebra has the form K[(I e ) ed+c ], where c, e ∈ N. For k = 3, we extend [9, Corollary 6.10] by proving that K[(I e ) ed+c ] is Koszul as soon as c ≥ d2 and e > 0. We also extend [9, Corollary 6.10] in another direction by replacing the polynomial ring with a Koszul ring.

Diagonal subalgebras of residual intersections

“…. , x n ] of the same degree, explicit lower bounds are obtained in [5] and [13], such that for larger values of c and e, R(I) ∆ is Koszul. More generally, the authors in [6] also show that for given any standard bigraded k-algebra S, one has S ∆ is Koszul for c, e ≫ 0.…”