On Monomial Golod Ideals

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On Monomial Golod Ideals Hailong Dao1 · Alessandro De Stefani2 Received: 20 May 2019 / Revised: 12 March 2020 / Accepted: 19 May 2020 / © The Author(s) 2020

Abstract We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod. Keywords Golod rings · Product of ideals · Koszul homology · Koszul cycles Mathematics Subject Classification (2010) Primary 13D02 · Secondary 05E40

1 Introduction Let k be a field, and let Q = k[x1 , . . . , xn ] be a polynomial ring on n variables over k, with deg(xi ) = 1 for all i. We denote by m = (x1 , . . . , xn ) the homogeneous maximal ideal of Q. Let I ⊆ m2 be a homogeneous ideal and R = Q/I . Serre proved a coefficient-wise inequality of formal power series for the Poincare series of R: PkR (t) :=

 i≥0

i dimk TorR i (k, k)t 

1−



(1 + t)n

i>0 dimk

Q

Tori (k, R)t i+1

.

When equality happens, the ring R (and the ideal I ) is called Golod. The notion is defined and studied extensively in the local setting, but in this paper we shall restrict ourselves to the graded situation. Golod rings and ideals have attracted increasing attention recently (see [6, 8, 9, 12, 15]), but they remain mysterious even when n = 3. For instance, we do not know if the product of any two homogeneous ideals in Q = k[x, y, z] is Golod. Another reason for the increasing interest is their connection to moment-angle complexes (for example, see [7, 10, 14]).

 Alessandro De Stefani

[email protected] Hailong Dao [email protected] 1

Department of Mathematics, The University of Kansas, Lawrence, KS 66045, USA

2

Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35, 16146 Genova, Italy

H. Dao, A. De Stefani

It was asked by Welker whether it is always the case that the product of two proper homogeneous ideals is Golod (for example, see [16, Problem 6.18]) but a counter-example, even for monomial ideals, was constructed by the second author in [8]. In this work, we provide a concrete characterization of Golod monomial ideals in three variables, and use it to show that the product of any two proper monomial ideals in Q = k[x, y, z] is Golod. The following is our first main result: Theorem 1.1 Let Q = k[x, y, z] and I ⊆ m2 be a monomial ideal. Then, I is Golod if and only if the following conditions hold: (1) [I : x1 ] · [I : (x2 , x3 )] ⊆ I for all permutations {x1 , x2 , x3 } of {x, y, z}. (2) [I : x1 ] · [I : x2 ] ⊆ x3 [I : (x1 , x2 )] + I for all permutations {x1 , x2 , x3 } of {x, y, z}. We point out that rings with embedding codepth at most three have been studied extensively. For instance, their Koszul homology has been completely classified up to isomorphism (see [1, 3–5, 18]). To obtain Theorem 1.1, we first list in Proposition 2.1 a set of necessary conditions for Golodness for general ideals in any number of variables, that are easy to check and are of independent interest. They can be used to pr

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