Formalization of Auslander--Buchsbaum--Serre criterion in Lean4

Abstract: We formalized a complete proof of the Auslander--Buchsbaum--Serre criterion in the Lean4 theorem prover. For a local ring, rather than following the well-known proof that considers the residue field as the quotient of the ring by a regular sequence to compute projective dimension and uses the Koszul complex to show the dimension of the cotangent space is at most the global dimension, we prove the criterion via maximal Cohen--Macaulay modules and a weakened version of the Ferrand--Vasconcelos theorem, which is more amenable to the formalization process and the current development of mathlib. Our formalization includes the construction of depth and of Cohen--Macaulay modules and rings, which are used frequently in the proof of the criterion. We also developed related results, including the unmixedness theorem for Cohen--Macaulay rings and Hilbert's Syzygy theorem.