Monomial conjecture

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

$x_1^t\cdots x_d^t\in ̸(x_1^{t+1},\dots ,x_d^{t+1}).$

The statement can relatively easily be shown in characteristic zero.

See also

• Homological conjectures in commutative algebra