Monomial conjecture and complete intersections
Let \((A,{\mathfrak m})\) be a \(d\)-dimensional local ring. Let \(\{x_ 1, \dots, x_ d\}\) denote a system of parameters of the ring \(A\). The monomial conjecture (MC) [see \textit{M. Hochster}, Nagoya Math. J. 51, 25- 43 (1973; Zbl 0268.13004)], states that \(x_ 1^ t \dots x_ d^ t \notin (x_ 1^{t+1}, \dots, x_ d^{t+1})A\) for all \(t \geq 1\) for all systems of parameters in any local ring \(A\). It is proved by M. Hochster [see the cited paper], that (MC) is true, e.g., whenever \(A\) is an equicharacteristic local ring. Moreover he showed also that (MC) is equivalent to the direct summand conjecture. In the present paper the authors prove an interesting equivalent condition for (MC) which says: For a local complete intersection ring \(R\) and an ideal \({\mathfrak a}\) consisting of zero divisors the annihilator \(0:_ R {\mathfrak a}\) of \({\mathfrak a}\) is not contained in any parameter ideal of \(R\). This follows by elaborating on local cohomology modules.
scientific article; zbMATH DE number 492813
scientific article