R. Hartshorne and A. Hirschowitz proved that a generic collection of lines on ℙn, n≥3, has bipolynomial Hilbert function. We extend this result to a specialization of the collection of generic lines, by considering a union of lines and 3-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).
We show how to use the extension and torsion functors in order to compute the torsion submodule of a differential module associated with a multidimensional control system. In particular, we show that the concept of the weak primeness of matrices corresponds to the torsion-freeness of a certain module.
There is a classical result known as Baer’s Lemma that states that an -module is injective if it is injective for . This means that if a map from a submodule of , that is, from a left ideal of to can always be extended to , then a map to from a submodule of any -module can be extended to ; in other words, is injective. In this paper, we generalize this result to the category consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma...
Let be a commutative Noetherian ring and an ideal of . We introduce the concept of -weakly Laskerian -modules, and we show that if is an -weakly Laskerian -module and is a non-negative integer such that is -weakly Laskerian for all and all , then for any -weakly Laskerian submodule of , the -module is -weakly Laskerian. In particular, the set of associated primes of is finite. As a consequence, it follows that if is a finitely generated -module and is an -weakly...
Let R be a commutative Noetherian ring. Let and be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the -finiteness dimension of relative to in the context of generalized local cohomology modules as , where M is an R-module. We also show that for any R-module M. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.