DOIAbstract
AbstractLet R be a noetherian commutative local ring, and M,N be finitely generated R-modules. Then a generalized form of Serre's Vanishing conjecture can be stated as follows: if (1)length (M⊗RN)<∞,(2)pd(M),pd(N)<∞, and(3)dimM+dimN<dimR, thenχ(M,N)≔∑i=0∞(−1)ilength(ToriR(M,N))=0.It is known that Serre's Vanishing conjecture holds for a complete intersection ring R, but is not known for a Gorenstein ring R. We can make a similar conjecture replacing Tor by Ext, namely, if M and N satisfy the above three conditions, thenξ(M,N)≔∑i=0∞(−1)ilength(ExtRi(M,N))=0.In this paper, we will prove that, over a Gorenstein ring R, the Tor-version of the Vanishing conjecture, the Ext-version of the Vanishing conjecture, and the commutativity of the interse...
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