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

Abstract:

If $(Q,\mathfrak {n})\twoheadrightarrow (R,\mathfrak {m})$ is a surjective local homomorphism with kernel $I$, such that $I\subseteq \mathfrak {n}^2$ and the conormal module $I/I^2$ has a free summand of rank $n$, then the degree $2$ central subspace of the homotopy Lie algebra of $R$ has dimension greater than or equal to $n$. This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module.