Relating the Exterior Algebra of the Conormal Module to the Tor Algebra

Abstract

For any ideal $I$ (whose projective dimension need not be finite) in a local Noetherian ring $R$, we show that, if the conormal module $I/I^2$ has a free summand given by $F$, the natural map from $\bigwedge F$ to ${\text{Tor}}_{\ast }^R(R/I,R/I)$ splits as algebras. To achieve this, we use dg algebra techniques and remodel results of André and Iyengar, proving them in setting of semi-free extensions, replacing the need for semi-free $\Gamma$-extensions. In this new setting, we show that free summands of the conormal module correspond to central elements of the relative homotopy Lie algebra ${\pi }^{\ast }(\phi )$ and, lastly, we provide an explicit splitting morphism in lower degrees.