scholarly journals Ring Constructions and Generation of the Unbounded Derived Module Category

2021 ◽  
Author(s):  
Charley Cummings
Keyword(s):  
Sufficient Conditions ◽  
Derived Category ◽  
Module Category ◽  
Injective Modules ◽  
Ring Extensions

AbstractWe consider the smallest triangulated subcategory of the unbounded derived module category of a ring that contains the injective modules and is closed under set indexed coproducts. If this subcategory is the entire derived category, then we say that injectives generate for the ring. In particular, we ask whether, if injectives generate for a collection of rings, do injectives generate for related ring constructions, and vice versa. We provide sufficient conditions for this statement to hold for various constructions including recollements, ring extensions and module category equivalences.

scholarly journals Injective modules under faithfully flat ring extensions

2015 ◽  
Vol 144 (3) ◽  
pp. 1015-1020 ◽  
Author(s):  
Lars Winther Christensen ◽  
Fatih Köksal
Keyword(s):  
Injective Modules ◽  
Flat Ring ◽  
Ring Extensions

Almost valuation property in bi-amalgamations and pairs of rings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950104 ◽  
Author(s):  
Najib Ouled Azaiez ◽  
Moutu Abdou Salam Moutui
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quotient Field ◽  
Integral Domains ◽  
Valuation Rings ◽  
Ring Extensions ◽  
Necessary And Sufficient

This paper examines the transfer of the almost valuation property to various constructions of ring extensions such as bi-amalgamations and pairs of rings. Namely, Sec. 2 studies the transfer of this property to bi-amalgamation rings. Our results cover previous known results on duplications and amalgamations, and provide the construction of various new and original examples satisfying this property. Section 3 investigates pairs of integral domains where all intermediate rings are almost valuation rings. As a consequence of our results, we provide necessary and sufficient conditions for a pair (R, T), where R arises from a (T, M, D) construction, to be an almost valuation pair. Furthermore, we study the class of maximal non-almost valuation subrings of their quotient field.

scholarly journals Finite homomorphic images of Bezout duo-domains

2014 ◽  
Vol 6 (2) ◽  
pp. 360-366 ◽  
Author(s):  
O.S. Sorokin
Keyword(s):  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Global Dimension ◽  
Necessary And Sufficient Conditions ◽  
Stable Range ◽  
Bezout Ring ◽  
Injective Modules ◽  
Elementary Divisor Ring ◽  
Necessary And Sufficient ◽  
Free Element

It is proved that for a quasi-duo Bezout ring of stable range 1 the duo-ring condition is equivalent to being an elementary divisor ring. As an application of this result a couple of useful properties are obtained for finite homomorphic images of Bezout duo-domains: they are coherent morphic rings, all injective modules over them are flat, their weak global dimension is either 0 or infinity. Moreover, we introduce the notion of square-free element in noncommutative case and it is shown that they are adequate elements of Bezout duo-domains. In addition, we are going to prove that these elements are elements of almost stable range 1, as well as necessary and sufficient conditions for being square-free element are found in terms of regularity, Jacobson semisimplicity, and boundness of weak global dimension of finite homomorphic images of Bezout duo-domains.

scholarly journals Split t-structures and torsion pairs in hereditary categories

2018 ◽  
Vol 17 (11) ◽  
pp. 1850218
Author(s):  
Ibrahim Assem ◽  
María José Souto-Salorio ◽  
Sonia Trepode
Keyword(s):  
Derived Category ◽  
Module Category ◽  
Hereditary Algebra ◽  
Tilted Algebra ◽  
Text Structures ◽  
Torsion Pairs

We construct a bijection between split torsion pairs in the module category of a tilted algebra having a complete slice in the preinjective component with corresponding [Formula: see text]-structures. We also classify split [Formula: see text]-structures in the derived category of a hereditary algebra.

Nilpotent Elements and Skew Polynomial Rings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 821-840 ◽  
Author(s):  
A. Alhevaz ◽  
A. Moussavi ◽  
E. Hashemi
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Special Cases ◽  
Ring Extensions ◽  
Skew Polynomial

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring.

scholarly journals The homological projective dual of

2020 ◽  
Vol 156 (3) ◽  
pp. 476-525
Author(s):  
Jørgen Vold Rennemo
Keyword(s):  
Complete Intersection ◽  
Invariant Theory ◽  
Clifford Algebras ◽  
Derived Categories ◽  
Derived Category ◽  
Module Category ◽  
Geometric Invariant ◽  
Projective Duality ◽  
Clifford Module ◽  
Category Of Modules

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

An alternative perspective on flatness of modules

2016 ◽  
Vol 15 (08) ◽  
pp. 1650145 ◽  
Author(s):  
Yılmaz Durğun
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Injective Modules ◽  
Alternative Perspective ◽  
Necessary And Sufficient

Given modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be absolutely [Formula: see text]-pure if [Formula: see text] is a monomorphism for every extension [Formula: see text] of [Formula: see text]. For a module [Formula: see text], the absolutely pure domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is absolutely [Formula: see text]-pure. As an opposite to flatness, a module [Formula: see text] is said to be f-indigent if its absolutely pure domain is smallest possible, namely, consisting of exactly the fp-injective modules. Properties of absolutely pure domains and off-indigent modules are studied. In particular, the existence of f-indigent modules is determined for an arbitrary rings. For various classes of modules (such as finitely generated, simple, singular), necessary and sufficient conditions for the existence of f-indigent modules of those types are studied. Furthermore, f-indigent modules on commutative Noetherian hereditary rings are characterized.

Triangulated Equivalences Involving Gorenstein Projective Modules

2017 ◽  
Vol 60 (4) ◽  
pp. 879-890 ◽  
Author(s):  
Yuefei Zheng ◽  
Zhaoyong Huang
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Projective Modules ◽  
Injective Dimension ◽  
Stable Category ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Left And Right

AbstractFor any ring R, we show that, in the bounded derived category Db(Mod R) of left R-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category (resp. ) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if R is a left and right noetherian ring admitting a dualizing complex, then and are equivalent.

On almost valuation ring pairs

2020 ◽  
pp. 2150182
Author(s):  
Noômen Jarboui ◽  
David E. Dobbs
Keyword(s):  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Text Type ◽  
Intermediate Ring ◽  
Valuation Domains ◽  
Divided Domain ◽  
Integrally Closed ◽  
Ring Extensions ◽  
Ring Pair

If [Formula: see text] are (commutative) rings, [Formula: see text] denotes the set of intermediate rings and [Formula: see text] is called an almost valuation (AV)-ring pair if each element of [Formula: see text] is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let [Formula: see text] be rings, with [Formula: see text] denoting the integral closure of [Formula: see text] in [Formula: see text]. Then [Formula: see text] is an AV-ring pair if and only if both [Formula: see text] and [Formula: see text] are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions [Formula: see text]. If [Formula: see text] is an AV-ring pair, then [Formula: see text] is a P-extension. The AV-ring pairs [Formula: see text] arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of [Formula: see text] type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for [Formula: see text] being an AV-ring pair to entail that [Formula: see text] is an overring of [Formula: see text], but there exist domain-theoretic counter-examples to such a conclusion in general. If [Formula: see text] is an AV-ring pair and [Formula: see text] satisfies FCP, then each intermediate ring either contains or is contained in [Formula: see text]. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

scholarly journals Module categories for group algebras over commutative rings

2013 ◽  
Vol 11 (2) ◽  
pp. 297-329 ◽  
Author(s):  
Dave Benson ◽  
Srikanth B. Iyengar ◽  
Henning Krause
Keyword(s):  
Main Idea ◽  
Commutative Rings ◽  
Compact Objects ◽  
Homotopy Category ◽  
Module Category ◽  
Stable Module Category ◽  
Injective Modules ◽  
Stable Module ◽  
A Finite Group ◽  
Finitely Presented

AbstractWe develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.

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