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

$x_1^t \cdots x_d^t \not\in (x_1^{t+1},\dots,x_d^{t+1}). \,$

The statement can relatively easily be shown in characteristic zero.

References

1. http://www5a.biglobe.ne.jp/~tomari/hamana/roberts.pdf