Free Lie subalgebras of the cohomology of local rings

Abstract

A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra Ext A ( k , k ) {\operatorname {Ext} _A}(k,\,k) to be a free module over the universal envelope of a free graded Lie subalgebra. It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure. For all rings with edim ⁡ A − depth ⁡ A ⩽ 3 \operatorname {edim} A - \operatorname {depth} A \leqslant 3 , and for Gorenstein rings with edim ⁡ A − depth ⁡ A = 4 \operatorname {edim} A - \operatorname {depth} A = 4 , the following dichotomy is proved: Either A A is a complete intersection, or Ext A ( k , k ) {\operatorname {Ext} _A}(k,\,k) contains a nonabelian free graded Lie subalgebra.