A Note on the Monomial Conjecture

Abstract

Several cases of the monomial conjecture are proved. An equivalent form of the direct summand conjecture is discussed. Let (A, m, k) be a noetherian local ring of dimension n, m its maximal ideal and k = A/rn. The Monomial Conjecture (henceforth MC) of Hochster asserts that, given any system of parameters (henceforth s.o.p.) xI, .. ., xn of A, (XIX2 ... Xn) ? Xv***)Xn)bt>O Hochster proved the conjecture in the equicharacteristic case [HI], [H2]. He also established the fact that in the mixed and the positive characteristic cases the Direct Summand Conjecture (henceforth DSC) and hence MC is equivalent to the Canonical Element Conjecture [H12] (henceforth CEC). Thus MC occupies a central position in the study of several homological conjectures. In [I1l] Hochster pointed out that, given any s.o.p. xl,...,xn of A, xt, ... , xt satisfies MC for t sufficiently large. Next Goto [G] proved MC for Buchsbaum rings and Koh proved DSC for degree p extensions [K]. Several special cases of CEC were proved in [DI] when depthA = dimA 1. In [D2], the following result was established: If Ji AnnH;-j(A) and J JiJ2 ... Jr where r dimA depthA > 0, then Xl,... ,xn satisfies MC if J ? (xl,... , xn). This in turn implies that, given any s.o.p. xl, . . , xn in a complete local normal domain A, XI, X2, X3) ... , xt satisfies MC for t > 0. We will have several more applications of this result in Section 2. We recall that there is no loss of generality in assuming A to be a complete local normal domain. In our most recent work we established the validity of CEC over i) A/xA when A is a complete local normal domain and x E mJ1 [D3], ii) any almost complete intersection domain A or any almost complete intersection ring A over which p (= the mixed characteristic) is a non-zero-divisor [D4], iii) rings of the form R/Q, where R is a complete Gorenstein ring such that the complete local normal domain A is a homomorphic image of R, dimR = dimA, and Q is the canonical module of A [D4], and iv) complete local normal domains A for which Q is S3. Thus MC holds in all the above cases. The main results of this paper are arranged in the following way. In Section 1, first we reduce the study of MC to almost complete intersection rings. Recall that, due to the results mentioned earlier, the problem boils down to almost complete intersection rings of depth 0. Next we prove the following: Received by the editors July 20, 1996. 1991 Mathematics Subject Classification. Primary 13D02; Secondary 13H1.0. This work was partially supported by an NSA grant and an NSF grant. (?)1998 American Mathematical Society