Koszul property was generalized to homogeneous algebras of degree N > 2 in [5], and related to N -complexes in [7]. We show that if the N -homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincaré duality between Hochschild homology and cohomology of A, as for N = 2. Mathem...

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.

Abstract. Given arbitrary integers k and d with 0 ≤ 2k ≤ d, we construct a Gorenstein Fano polytope P ⊂ R of dimension d such that (i) its Ehrhart polynomial i(P , n) possesses d distinct roots; (ii) i(P , n) possesses exactly 2k imaginary roots; (iii) i(P , n) possesses exactly d − 2k real roots; (iv) the real part of each of the imaginary roots is equal to −1/2; (v) all of the real roots belo...

We present our recent understanding on resolutions of Gorenstein orbifolds, which involves the finite group representation theory. We concern only the quotient singularity of hypersurface type. The abelian group Ar (n) for A-type hypersurface quotient singularity of dimension n is introduced. For n = 4, the structure of Hilbert scheme of group orbits and crepant resolutions of Ar (4)-singularit...

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the onl...

Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open pr...

Among all projective algebraic varieties, Fano varieties (those with ample anticanonical bundle) can be considered the simplest. Birkar, Cascini, Hacon and McKernan showed that the Cox ring of a Fano variety, the ring of all sections of all line bundles, is finitely generated [4]. This implies a fundamental fact about the birational geometry of a Fano variety: there are only finitely many small...