The monomial conjecture of Mel Hochster has been one of the most important open problems in commutative algebra for many years. The conjecture is as follows.rnConjecture. Let R be a Noetherian local ring of dimension d, and suppose that x_1,x_2,...,x_d is a system of parameters for R. Then for all integers t ≥ 0 we havernx_1~tx_2~t...x_d~t is not an element of (x_1~(t+1),...,x_d~(t+1), where (x_1~(t+1),..., x_d~(t+1)) denotes the ideal generated by x_1~(t+1),..., x_d~(t+1).
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