In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s.
They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth.
The following list given by Melvin Hochster is considered definitive for this area.
refer to Noetherian commutative rings;
will be a local ring with maximal ideal