Koszul homology of Cohen-Macaulay rings with linear resolutions

Abstract. The first Koszul homology module of a Cohen-Macaulay ideal witha linear resolution is studied and some new examples of rings of minimal mul-tiplicity are presented. 1. Introduction Let R be a polynomial ring over a field k, and let / be a graded ideal ofR. The algebra S = R/I is Cohen-Macaulay (C-M for short) if the projectivedimension of S as an i?-module is equal to the height of /. The ideal / has ap-linear resolution if / is generated by forms of degree p and if all the mapsof its graded minimal resolution by free .R-modules have linear entries. We saythat S has a p-linear resolution if / does, and that / is C-M if S is.Examples of Cohen-Macaulay algebras with linear resolutions include ringsof minimal multiplicity [17], the coordinate ring of a variety defined by the submaximal minors of a generic symmetric matrix [13], the coordinate ring of a variety defined by the maximal minors of a generic matrix [3], and some facerings [5].Cohen-Macaulay rings with linear resolutions have been studied by Sally [ 17]for the case p — 2, and by Schenzel [ 18] for the general case; more generalrings with linear resolutions have been examined in [21, 9, 4].In this work we use some of the techniques introduced by Kustin, Miller, andUlrich [14], and by Vasconcelos [22] to study the Koszul homology of Cohen-Macaulay ideals with linear resolution.We now describe the contents of this paper. In §2 we consider a C-M ideal/ of height g with a p-linear resolution. If g = 2, Avramov and Herzog [ 1 ]have shown that the Koszul homology of / is Cohen-Macaulay. We are ableto prove that if / is generically a complete intersection satisfying g > 3 andp > 2 then the first Koszul homology module of / is not C-M.In §3 we present a somewhat different proof of a result due to Cavaliere,Rossi, and Valla [2] which characterizes C-M rings with linear resolution; such