The Koszul property for spaces of quadrics of codimension three

Abstract

A b s t r ac t . In this paper we prove that, if k is an algebraically closed field of characteristic different from 2, almost all quadratic standard graded k-algebras R such that dim k R 2 = 3 are Koszul. More precisely, up to graded k-algebra homomorphisms and trivial fiber extensions, we find out that only two (or three, when the characteristic of k is 3) algebras of this kind are non-Koszul.Moreover, we show that there exist nontrivial quadratic standard graded kalgebras with dim k R 1 = 4, dim k R 2 = 3 that are Koszul but do not admit a Gröbner basis of quadrics even after a change of coordinates, thus settling in the negative a question asked by Conca. I n t ro d u c t i o nA (commutative) standard graded algebra R over a field k is a quotient of a polynomial ring over k in a finite number of variables by a homogeneous ideal I not containing any linear form. We say that R is Koszul when the minimal graded free resolution of k as an R-module is linear: for a survey about Koszulness in the commutative setting, see [7]. It is well-known that any Koszul algebra is quadratic, i.e. its defining ideal I is generated by quadrics. Note that, if R is a trivial fiber extension, i.e. it contains a linear form ∈ R 1 such that R 1 = 0, then R is Koszul if and only if R/ is Koszul.For the rest of this introduction, let R be a quadratic standard graded k-algebra, where k is an algebraically closed field of characteristic different from two. Backelin proved in [1] that, if dim k R 2 = 2, then R is Koszul (this actually holds for any field k). Conca then proved in [5] that, if dim k R 2 = 3 and R is Artinian, then R is Koszul. The main problem we are addressing in this paper is to find out what happens when, in the latter case, we drop the Artinian assumption. We will prove in Theorem 3.1 that, up to trivial fiber extension, the only non-Koszul algebras live in embedding dimension three. More precisely, these non-Koszul algebras in three variables are the two (or three, when char k = 3) objects exhibited by Backelin and Fröberg in [2]. As a byproduct, we shall also obtain a list of all possible Hilbert series when dim k R 2 = 3, see Theorem 3.3.Our results are in agreement with the numerical data presented by Roos in the characteristic 0 four-variable case, see [15].