scholarly journals Relative global Gorenstein dimensions

2021 ◽  
pp. 2250208
Author(s):  
Víctor Becerril
Keyword(s):  
Admissible Pair ◽  
Abelian Category ◽  
Projective Dimension ◽  
Dimension Formula

Let [Formula: see text] be an abelian category. In this paper, we investigate the global [Formula: see text]-Gorenstein projective dimension [Formula: see text], associated to a GP-admissible pair [Formula: see text]. We give homological conditions over [Formula: see text] that characterize it. Moreover, given a GI-admissible pair [Formula: see text], we study conditions under which [Formula: see text] and [Formula: see text] are the same.

Homological Dimensions Relative to Special Subcategories

2021 ◽  
Vol 28 (01) ◽  
pp. 131-142
Author(s):  
Weiling Song ◽  
Tiwei Zhao ◽  
Zhaoyong Huang
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Homological Dimensions ◽  
Category Of Modules ◽  
The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

scholarly journals FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

2019 ◽  
Vol 62 (2) ◽  
pp. 383-439 ◽  
Author(s):  
LEONID POSITSELSKI
Keyword(s):  
Associative Ring ◽  
Abelian Category ◽  
Projective Dimension ◽  
Countable Base ◽  
Linear Topology ◽  
Countable Type ◽  
Flat Ring ◽  
Abelian Categories ◽  
Induced Topology ◽  
Strongly Flat

AbstractLet R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

n-Matlis cotorsion modules and n-matlis domains

2019 ◽  
Vol 19 (07) ◽  
pp. 2050139
Author(s):  
Yongyan Pu ◽  
Gaohua Tang ◽  
Fanggui Wang
Keyword(s):  
Projective Dimension ◽  
Negative Integer ◽  
Dimension Formula

Let [Formula: see text] be a domain with its field [Formula: see text] of quotients, [Formula: see text] an [Formula: see text]-module and [Formula: see text] a fixed non-negative integer. Then [Formula: see text] is called [Formula: see text]-Matlis cotorsion if [Formula: see text] for any integer [Formula: see text]. Also [Formula: see text] is said to be [Formula: see text]-Matlis flat if [Formula: see text] for any [Formula: see text]-Matlis cotorsion [Formula: see text]-module [Formula: see text]. We proved that [Formula: see text] is a complete hereditary cotorsion theory, where [Formula: see text] (respectively, [Formula: see text]) denotes the class of all [Formula: see text]-Matlis flat (respectively, [Formula: see text]-Matlis cotorsion) [Formula: see text]-modules. In this paper, it is proved that [Formula: see text] is an [Formula: see text]-Matlis domain if and only if epic images of [Formula: see text]-Matlis cotorsion [Formula: see text]-modules are again [Formula: see text]-Matlis cotorsion if and only if [Formula: see text]-Matlis flat [Formula: see text]-modules are of projective dimension [Formula: see text] if and only if [Formula: see text] if and only if [Formula: see text].

scholarly journals Torsion pairs and filtrations in abelian categories with tilting objects

2015 ◽  
Vol 14 (08) ◽  
pp. 1550121
Author(s):  
Jason Lo
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Explicit Description ◽  
Homological Dimension ◽  
Abelian Categories ◽  
Dimension 2 ◽  
Tilting Object ◽  
Category Of Modules ◽  
Torsion Pairs ◽  
Tilting Objects

Given a noetherian abelian k-category [Formula: see text] of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category [Formula: see text] and the abelian category of modules over End (T) op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that [Formula: see text] has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

scholarly journals Triangulated Matlis equivalence

2018 ◽  
Vol 17 (04) ◽  
pp. 1850067 ◽  
Author(s):  
Leonid Positselski
Keyword(s):  
Commutative Ring ◽  
Projective Dimension ◽  
Derived Categories ◽  
Multiplicative System ◽  
Module Category ◽  
Dimension Formula ◽  
Abelian Categories ◽  
Torsion Modules ◽  
Cohomology Modules

This paper is a sequel to [L. Positselski, Dedualizing complexes and MGM duality, J. Pure Appl. Algebra 220(12) (2016) 3866–3909, arXiv:1503.05523 [math.CT]; Contraadjusted modules, contramodules, and reduced cotorsion modules, preprint (2016), arXiv:1605.03934 [math.CT]]. We extend the classical Harrison–Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules and contramodules over a Matlis domain. This generalizes to the case of any commutative ring [Formula: see text] with a fixed multiplicative system [Formula: see text] such that the [Formula: see text]-module [Formula: see text] has projective dimension [Formula: see text]. The latter equivalence connects complexes of [Formula: see text]-modules with [Formula: see text]-torsion and [Formula: see text]-contramodule cohomology modules. It takes a nicer form of an equivalence between the derived categories of abelian categories when [Formula: see text] consists of nonzero-divisors or the [Formula: see text]-torsion in [Formula: see text] is bounded.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

Modules of finite projective dimension and cocovers

1996 ◽  
Vol 306 (1) ◽  
pp. 445-457 ◽  
Author(s):  
Dieter Happel ◽  
Luise Unger
Keyword(s):  
Projective Dimension ◽  
Finite Projective Dimension
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