Homological conjectures in commutative algebra

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. In the sequel, formula_1, and formula_2 refer to Noetherian commutative rings; formula_3 will be a local ring with maximal ideal formula_4, and formula_5 and formula_6 are finitely generated formula_3-modules.