ترغب بنشر مسار تعليمي؟ اضغط هنا

Modules over posets: commutative and homological algebra

126   0   0.0 ( 0 )
 نشر من قبل Ezra Miller
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English
 تأليف Ezra Miller




اسأل ChatGPT حول البحث

The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multiple real parameters. The algebraic theory yields computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology indexed by arbitrary posets. The exposition focuses on the nature and ramifications of a suitable finiteness condition to replace the noetherian hypothesis. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both the theoretical and computational purposes: it guarantees finite primary decompositions, as well as various finite presentations and resolutions all related by a syzygy theorem, and the data structures thus produced are computable in addition to being interpretable. The tameness condition and its resulting theory are new even in the finitely generated discrete setting, where being tame is materially weaker than being noetherian.



قيم البحث

اقرأ أيضاً

Let fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd_{fa}(M,N) denote the supremum of the is such that H^i_{fa}(M,N) eq 0. First, by using the theory of Gorenstein homological dimensions, we obtai n several upper bounds for cd_{fa}(M,N). Next, over a Cohen-Macaulay local ring (R,fm), we show that cd_{fm}(M,N)=dim R-grade(Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.
Let $R$ be a commutative ring. We investigate $R$-modules which can be written as emph{finite} sums of {it {second}} $R$-submodules (we call them emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be have a (minima l) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (emph{main}) emph{second attached prime ideals} related to a module with such a presentation.
Let $R$ be a commutative ring and $M$ a non-zero $R$-module. We introduce the class of emph{pseudo strongly hollow submodules} (emph{PS-hollow submodules}, for short) of $M$. Inspired by the theory of modules with emph{secondary representations}, we investigate modules which can be written as emph{finite} sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of emph{minimal} PS-hollow strongly representations of modules over Artinian rings.
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.
168 - Francois Couchot 2011
It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $widehat{R}$ of $R$ is of finite rank over the completion $widetilde{R}$ of $R$ in the $R$-topology. In t his case, for each indecomposable injective module $E$, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over one Krull-dimensional local Noetherian rings has finite Malcev rank. The preservation of Goldie dimension finiteness by localization is investigated too.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا