Let 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 then [Formula: see text] 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, then [Formula: see text] 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 intersection multiplicity defined in Mori and Smith (J. Pure Appl. Algebra 157 (2001) 279) are all equivalent. Further, we will prove that a certain Ext-version of the Vanishing conjecture holds for a large class of noncommutative projective schemes, typically including all commutative projective schemes over a field, by extending Bézout's Theorem.