We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study projective degrees rational maps . use a general framework determine map via computation multiplicity saturated special fiber ring. As specific applications, we provide explicit formulas all determined perfect ideals height two or Gorenstein three.

Tate local cohomology and Gorenstein theory, which are important generalizations of the classical cohomology, has been investigated. It found that they have such vanishing properties long exact sequences. However, for homology, what about duality? In this paper we concerned with homology homology. first part generalize as study properties, artinianness some sequence modules. We find an artiania...

Let X be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by F a smooth model of a generic irreducible element in fibers of φ1 and so F is a curve or a smooth surface. The main result is that there is a computable constant K independent of X such that g(F ) ≤ 647 or pg(F ) ≤ 38 whenever pg(X) ≥ K.

Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.

Let X be an n-dimensional normal projective variety with terminal, Gorenstein, Qfactorial singularities. Let L be an ample line bundle on X . Let t be the nefvalue of ðX ;LÞ. Then we classify ðX ;LÞ, describing the structure of the nefvalue morphism of ðX ;LÞ, when t satisfies n k < t < n k þ 1 and nd 2k 3, kd 4. In the smooth case, we discuss the case n 1⁄4 2k 4, kd 5, as well.

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected t...

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.

Ladders of recollements abelian categories are introduced, and used to address three general problems. a certain height allow construct triangulated categories, involving derived singularity from ones. also tilt recollements, ladders guarantee that properties like Gorenstein projective or injective preserved by some functors in recollements. Breaking symmetry is crucial developing this theory.