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This conjecture has assumed a central role since it has a very simple statement and it implies several other important conjectures, notably the Canonical Element Conjecture, for rings of positive or mixed characteristic. In fact, when this conjecture was first announced, it had numerous further consequences, some of which, such as the New Intersection Conjecture, were proved later by different means. We refer to Hochster [6] and [10] for descriptions of these conjectures and their status at various times. The Monomial Conjecture is almost trivial for rings that contain the rational numbers and is not difficult for rings of positive charateristic, but it is still an open problem for rings of mixed characteristic. The most recent advance was made by Ray Heitmann [5], who proved it in mixed characteristic in dimension 3.

Almost regular sequences and the monomial conjecture