The Auslander-Buchsbaum formula is a powerful tool in commutative algebra connecting the concepts of depth and projective dimension. One of its most meaningful applications is the famous result which states that a local commutative ring is regular if and only if it has finite global dimension.
For regular rings, the Auslander-Buchsbaum formula can be obtained by regarding regular rings as Calabi-Yau algebras. Regular rings admit many non-commutative generalisations, one of them being the class of higher Auslander algebras.

In this talk, we present a non-commutative generalisation of the Auslander-Buchsbaum formula for higher Auslander algebras. We then use this formula to characterise ring theoretical properties of higher Auslander algebras and to provide criteria to check representation-finiteness of higher Auslander algebras.
This is joint work with R. Marczinzik.