Monomial conjecture

{{one source |date=May 2024}}

In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following:{{Cite web |title=Local Cohomology and the Homological Conjectures in Commutative Algebra |url=https://www5a.biglobe.ne.jp/~tomari/hamana/roberts.pdf |access-date=2023-12-19 |website=www5a.biglobe.ne.jp}}

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

: $x\_1^t\; \backslash cdots\; x\_d^t\; \backslash not\backslash in\; (x\_1^\{t+1\},\backslash dots,x\_d^\{t+1\}).\; \backslash ,$

The statement can relatively easily be shown in characteristic zero.