Monomial conjecture

In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following:^[1]

Let A be a Noetherian local ring of Krull dimension d and let x₁, ..., x_d be a system of parameters for A (so that A/(x₁, ..., x_d) is an Artinian ring). Then for all positive integers t, we have

${\displaystyle x_{1}^{t}\cdots x_{d}^{t}\not \in (x_{1}^{t+1},\dots ,x_{d}^{t+1}).\,}$

The statement can relatively easily be shown in characteristic zero.

• Homological conjectures in commutative algebra