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We prove a duality theorem for certain graded algebras and show by various examples different kinds of failure of tameness of local cohomology.
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We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a logarithmic derivation module of an arrangement of hyperplanes with a multiplicity consisting of both positive and negative integers. We consider several properties of this module including Saitos criterion and reflexivity. As applications, we prove a shift isomorphism and duality of some Coxeter multiarrangements by using the primitive derivation.
We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local dualit
We prove that in a globally subanalytic family of convex bodies the set of zonoids is log-analytic, and in particular it is definable in the o-minimal structure generated by globally subanalytic sets and the graph of the exponential function.
Let $R$ be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair $(mathcal{F},mathcal{I})$ where $mathcal{F}$ is the class of flat $R$-modules and $mathcal{I}$ is the class of injective $R$-modules. For a general $R$, the AC-Gorenstein homological algebra of Bravo-Gillespie-Hovey is the one coming from the duality pair $(mathcal{L},mathcal{A})$ where $mathcal{L}$ is the class of level $R$-modules and $mathcal{A}$ is class of absolutely clean $R$-modules. Indeed we show here that the work of Bravo-Gillespie-Hovey can be extended to obtain similar abelian model structures on $R$-Mod from any a complete duality pair $(mathcal{L},mathcal{A})$. It applies in particular to the original duality pairs constructed by Holm-J{o} rgensen.
In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.
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