Coefficient rings of numerical semigroup algebras

Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show that arithmetic and set-theoretic properties of a numerical semigroup ring are properties of an equi-gcd numerical semigroup algebra.

     

Cohomology of projective space seen by residual complex

A subcomplex of a residual complex on projective space is constructed for computing the cohomology modules of locally free sheaves. A constructive new proof of the Bott formula is given by explicitly exhibiting bases for the cohomology modules.

     

Applications of residues to combinatorial identities

A concrete aspect of Grothendieck Duality is used to give local cohomology proofs of combinatorial identities including MacMahon's master theorem, Grosswald identity, identity of Shoo, Tepper identity, and others.

     

Finitely continuous differentials on generalized power series

Let κ[[e ^G ]] be the field of generalized power series with exponents in a totally ordered Abelian group G and coefficients in a field κ. Given a subgroup H of G such that G/H is finitely generated, we construct a vector space Ω _G/H of differentials as a universal object in certain category of κ[[e ^H ]]-derivations on κ[[e ^G ]]. The vector space Ω _G/H together with logarithmic residues gives rise to a framework for certain combinatorial phenomena, including the inversion formula for diagonal delta sets.     

Theory of residues on the projective plane

An elementary theory of residues on the projective plane is given, which includes a canonical residual complex, a trace map, the residue theorem, and intersection numbers. Some classical theorems about plane projective geometry, such as the Reiss relation, the Cayley-Bacharach theorem, and the Newton theorem, are reproved via this approach of residues.     

Factorizations in numerical semigroup algebras

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to study whether a flat rectangular algebra is a complete intersection. Along this direction, special types of algebras generated by few monomials are worked out in detail.     

Residues for Akizuki's one-dimensional local domain

For a one-dimensional local domain constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of is described using these residue maps.

     

Changes of Parameters for Generalized Power Series

A new notion of differentials is introduced to the method of generating functions for generalized power series with exponents in a totally ordered Abelian group. A logarithmic analogue of cohomology residues is defined to serve as a tool in the technique of equating coefficients.