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Register a new userDuality, a-invariants and canonical modules of rings arising from linear optimization problems
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The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and Gorenstein property--as well as the canonical module and the a-invariant--of Rees algebras and subrings arising from systems with the integer rounding property. We relate the algebraic properties of Rees algebras and monomial subrings with integer rounding properties and present a duality theorem.
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In the present paper, we consider the problem when the toric ring arising from an integral cyclic polytope is Cohen-Macaulay by discussing Serres condition and we give a complete characterization when that is Gorenstein. Moreover, we study the normality of the other semigroup ring arising from an integral cyclic polytope but generated only with its vertices.
In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.
Let $Delta$ be a triangulated homology ball whose boundary complex is $partialDelta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $Delta$, $mathbb F[Delta]$, is isomorphic to the Stanley--Reisner module of the pair $(Delta, partialDelta)$, $mathbb F[Delta,partial Delta]$. This result implies that an Artinian reduction of $mathbb F[Delta,partial Delta]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $mathbb F[Delta]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.
We study properties of the Stanley-Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley-Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of non-homologically isolated singularities, providing many examples in which the Stanley-Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.
The behaviour under coarsening functors of simple, entire, or reduced graded rings, of free graded modules over principal graded rings, of superfluous monomorphisms and of homological dimensions of graded modules, as well as adjoints of degree restriction functors, are investigated.
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