In dimension d, Q-factorial Gorenstein toric Fano varieties with Picard number ρX correspond to simplicial reflexive polytopes with ρX+d vertices. Casagrande showed that any d-dimensional simplicial reflexive polytope has at most 3d vertices, if d is even, respectively, 3d − 1, if d is odd. Moreover, for d even there is up to unimodular equivalence only one such polytope with 3d vertices, corre...

Let Λ and Γ be left and right noetherian rings and ΛU a Wakamatsu tilting module with Γ = End(ΛT ). We introduce a new definition of U -dominant dimensions and show that the U -dominant dimensions of ΛU and UΓ are identical. We characterize k-Gorenstein modules in terms of homological dimensions and the property of double homological functors preserving monomorphisms. We also study a generaliza...

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We prove that if M , N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext R (M,N) for n ≫ 0 is equivalent to the vanishing of Ext R (N,M) for n ≫ 0. Furthermore, if b R has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.