Free summands of conormal modules and central elements in homotopy Lie algebras of local rings (Q2706564)

Considering a noetherian local ring \(R\) with maximal ideal \({\mathfrak m}\) and residue field \(k\), \(\text{Tor}^R(k,k)\) is a commutative algebra but not \(\text{Ext}_R(k,k)\). The last one is the universal enveloping algebra of \(\pi(R)\), the homotopy Lie algebra of \(R\). This paper considers the centre \(\zeta(R)\) of \(\pi(R)\).NEWLINENEWLINENEWLINEThe main theorem says that, for \((Q,{\mathfrak n},k)\) a local ring and \(I\) an ideal in \(Q\) and \(R=Q/I\), if there is an ideal \(J\) such that \(I^2\subseteq J\subseteq I\) and the \(R\)-module \(I/J\) is free, then NEWLINE\[NEWLINE\text{rank}_k\zeta^2(R)\geq\text{rank}_R(I/J)-\text{rank}_k(I/J+I\cap {\mathfrak n}^2).NEWLINE\]NEWLINE As a corollary one has the following result. If \((Q,{\mathfrak n})\to(R,{\mathfrak m})\) is a surjective local homomorphism with kernel \(I\), such that \(I\subset{\mathfrak n}^2\) and the conormal module \(I/I^2\) has a free summand of rank \(n\), the degree 2 central subspace of \(\pi(R)\) has dimension \(\geq n\).