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 be relatively easily shown in characteristic zero.

See also

• Homological conjectures in commutative algebra

References

1. "Local Cohomology and the Homological Conjectures in Commutative Algebra" (PDF). www5a.biglobe.ne.jp. Retrieved 2023-12-19.