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I will do my best to explain what goes in to proving the following theorem: if I is an ideal of finite projective dimension in a local ring R, and the conormal module I/I^2 has finite projective dimension over R/I, then I is generated by a regular sequence. This was conjectured by Vasconcelos, after he and (separately) Ferrand established the case that the conormal module is free.

The key tool is the homotopy Lie algebra. This is a graded Lie algebra naturally associated with any local homomorphism. It sits at the centre of a longstanding friendship between commutative algebra and rational homotopy theory, through which ideas and results have been passed back and forth for decades.

I'll go through the construction of the homotopy Lie algebra and how it's been used in commutative algebra in the past, before explaining how its structure detects when the conormal module has finite projective dimension. I'll also talk about ongoing work with Srikanth Iyengar comparing the cotangent complex with the homotopy Lie algebra.