An Upper Bound for the w-Weak Global Dimension of Pullbacks

2021 ◽  
Vol 28 (04) ◽  
pp. 689-700
Author(s):  
Jin Xie ◽  
Gaohua Tang
Keyword(s):  
Commutative Ring ◽  
Upper Bound ◽  
Global Dimension ◽  
Factor Ring ◽  
Dimension Formula ◽  
Milnor Square

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] an ideal of [Formula: see text]. We introduce and study the [Formula: see text]-weak global dimension [Formula: see text] of the factor ring [Formula: see text]. Let [Formula: see text] be a [Formula: see text]-linked extension of [Formula: see text], and we also introduce the [Formula: see text]-weak global dimension [Formula: see text] of [Formula: see text]. We show that the ring [Formula: see text] with [Formula: see text] is exactly a field and the ring [Formula: see text] with [Formula: see text] is exactly a [Formula: see text]. As an application, we give an upper bound for the [Formula: see text]-weak global dimension of a Cartesian square [Formula: see text]. More precisely, if [Formula: see text] is [Formula: see text]-linked over [Formula: see text], then [Formula: see text]. Furthermore, for a Milnor square [Formula: see text], we obtain [Formula: see text].

THE NONEQUIVALENCE AND DIMENSION FORMULA FOR ATTRACTORS OF LORENZ-TYPE SYSTEMS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350200 ◽  
Author(s):  
YUMING CHEN ◽  
QIGUI YANG
Keyword(s):  
Upper Bound ◽  
Analytical Formula ◽  
Type Systems ◽  
Lyapunov Dimension ◽  
Dimension Formula ◽  
Lü System ◽  
Suitable Parameter ◽  
Lu System

In this paper, Lü system with a set of chaotic parameters is proved to be smoothly nonequivalent to Chen and Lorenz systems with any parameter. The analytical formula for the upper bound of Lyapunov dimension of attractors in Lorenz-type systems are presented under some suitable parameter conditions. These properties studied in this paper may contribute to a better understanding of the Lorenz-type systems.

scholarly journals Noncommutative motives of Azumaya algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 379-403 ◽  
Author(s):  
Gonçalo Tabuada ◽  
Michel Van den Bergh
Keyword(s):  
Commutative Ring ◽  
Global Dimension ◽  
Azumaya Algebras ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Noncommutative Motives ◽  
Finite Dimensional Algebras ◽  
The Way

AbstractLet $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$-scheme, and $A$ a sheaf of Azumaya algebras over $X$ of rank $r$. Under the assumption that $1/r\in R$, we prove that the noncommutative motives with $R$-coefficients of $X$ and $A$ are isomorphic. As an application, we conclude that a similar isomorphism holds for every $R$-linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.

The Krull and global dimension of the tensor product of quantum tori

2016 ◽  
Vol 15 (09) ◽  
pp. 1650174
Author(s):  
Ashish Gupta
Keyword(s):  
Tensor Product ◽  
Upper Bound ◽  
Tensor Products ◽  
Global Dimension ◽  
Base Field ◽  
Quantum Torus ◽  
Common Value ◽  
Quantum Tori ◽  
Special Cases ◽  
The Common

An [Formula: see text]-dimensional quantum torus is defined as the [Formula: see text]-algebra generated by variables [Formula: see text] together with their inverses satisfying the relations [Formula: see text], where [Formula: see text]. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of [Formula: see text] that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

scholarly journals Classification of pointed fusion categories of dimension p3 up to weak Morita Equivalence

2020 ◽  
pp. 2140008
Author(s):  
Kevin Maya ◽  
Adriana Mejía Castaño ◽  
Bernardo Uribe
Keyword(s):  
Complete Classification ◽  
Global Dimension ◽  
Morita Equivalence ◽  
Equivalence Classes ◽  
Fusion Categories ◽  
Dimension Formula ◽  
Equivalent Categories

We give a complete classification of pointed fusion categories over [Formula: see text] of global dimension [Formula: see text] for [Formula: see text] any odd prime. We proceed to classify the equivalence classes of pointed fusion categories of dimension [Formula: see text] and we determine which of these equivalence classes have equivalent categories of modules.

Sheaves over infinite posets

2017 ◽  
Vol 16 (02) ◽  
pp. 1750026 ◽  
Author(s):  
J. Asadollahi ◽  
P. Bahiraei ◽  
R. Hafezi ◽  
R. Vahed
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Topological Structure ◽  
Ordered Set ◽  
Global Dimension ◽  
Partially Ordered ◽  
Acyclic Complexes ◽  
Gorenstein Injective

In this paper, we study the category of sheaves over an infinite partially ordered set with its natural topological structure. Totally acyclic complexes in this category will be characterized in terms of their stalks. This leads us to describe Gorenstein projective, injective and flat sheaves. As an application, we get an analogue of a formula due to Mitchell, giving an upper bound on the Gorenstein global dimension of such categories. Based on these results, we present situations in which the class of Gorenstein projective sheaves is precovering as well as situations in which the class of Gorenstein injective sheaves is preenveloping.

scholarly journals Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 251
Author(s):  
Anastasis Kratsios
Keyword(s):  
Vector Bundle ◽  
Upper Bound ◽  
Cohomological Dimension ◽  
Global Dimension ◽  
Homological Dimension ◽  
Free Algebras ◽  
Commutative Algebras ◽  
Flat Dimension ◽  
Smooth Affine ◽  
Affine Scheme

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.

scholarly journals The EA-Dimension of a Commutative Ring

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Mosbah Eljeri
Keyword(s):  
Commutative Ring ◽  
Upper Bound ◽  
Zero Divisors ◽  
A Chain

An elementary annihilator of a ring A is an annihilator that has the form (0:a)A; a∈R∖(0). We define the elementary annihilator dimension of the ring A, denoted by EAdim(A), to be the upper bound of the set of all integers n such that there is a chain (0:a0)⊂⋯⊂(0:an) of annihilators of A. We use this dimension to characterize some zero-divisors graphs.

scholarly journals A dimension formula relating to algebraic groups

1971 ◽  
Vol 4 (2) ◽  
pp. 241-245 ◽  
Author(s):  
Su-shing Chen
Keyword(s):  
Upper Bound ◽  
Algebraic Groups ◽  
Rational Points ◽  
Unitary Representations ◽  
Cusp Forms ◽  
Harmonic Forms ◽  
Dimension Formula ◽  
Irreducible Unitary Representations ◽  
Semisimple Algebraic Groups

An upper bound is given of the dimension of certain spaces of cusp harmonic forms of arithmetic subgroups Γ of semisimple algebraic groups G in terms of the multiplicities of corresponding irreducible unitary representations of the group GR of real rational points of G in the space 0L2(GR/Γ) of cusp forms.

Pro- Iwahori–Hecke algebras are Gorenstein

2013 ◽  
Vol 13 (4) ◽  
pp. 753-809 ◽  
Author(s):  
Rachel Ollivier ◽  
Peter Schneider
Keyword(s):  
Upper Bound ◽  
Reductive Group ◽  
Global Dimension ◽  
Arbitrary Field ◽  
Graded Rings ◽  
Injective Dimension ◽  
Locally Compact ◽  
Invariant Vectors ◽  
Residue Characteristic ◽  
Iwahori Subgroup

AbstractLet$\mathfrak{F}$be a locally compact nonarchimedean field with residue characteristic$p$, and let$\mathrm{G} $be the group of$\mathfrak{F}$-rational points of a connected split reductive group over$\mathfrak{F}$. For$k$an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke$k$-algebra${\mathrm{H} }^{\prime } $and of the pro-$p$Iwahori–Hecke$k$-algebra$\mathrm{H} $of$\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of$\mathrm{G} $. If$\mathrm{G} $is semisimple, we also show that this upper bound is sharp, that both$\mathrm{H} $and${\mathrm{H} }^{\prime } $are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of$\mathrm{H} $(respectively${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to$\mathrm{H} $and${\mathrm{H} }^{\prime } $.When$k$has characteristic$p$, we prove that in ‘most’ cases$\mathrm{H} $and${\mathrm{H} }^{\prime } $have infinite global dimension. In particular, we deduce that the category of smooth$k$-representations of$\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$generated by their invariant vectors under the pro-$p$Iwahori subgroup has infinite global dimension (at least if$k$is algebraically closed).

scholarly journals LOCC protocols with bounded width per round optimize convex functions

2021 ◽  
pp. 2150013
Author(s):  
Debbie Leung ◽  
Andreas Winter ◽  
Nengkun Yu
Keyword(s):  
Convex Functions ◽  
Convex Combination ◽  
Local System ◽  
Global Dimension ◽  
Coarse Grained ◽  
State Transformation ◽  
Convex Decomposition ◽  
Dimension Formula ◽  
State Discrimination ◽  
Output Dimension

We start with the task of discriminating finitely many multipartite quantum states using LOCC protocols, with the goal to optimize the probability of correctly identifying the state. We provide two different methods to show that finitely many measurement outcomes in every step are sufficient for approaching the optimal probability of discrimination. In the first method, each measurement of an optimal LOCC protocol, applied to a [Formula: see text]-dimensional local system, is replaced by one with at most [Formula: see text] outcomes, without changing the probability of success. In the second method, we decompose any LOCC protocol into a convex combination of a number of “slim protocols” in which each measurement applied to a [Formula: see text]-dimensional local system has at most [Formula: see text] outcomes. To maximize any convex functions in LOCC (including the probability of state discrimination or fidelity of state transformation), an optimal protocol can be replaced by the best slim protocol in the convex decomposition without using shared randomness. For either method, the bound on the number of outcomes per measurement is independent of the global dimension, the number of parties, the depth of the protocol, how deep the measurement is located, and applies to LOCC protocols with infinite rounds, and the “measurement compression” can be done “top-down” — independent of later operations in the LOCC protocol. The second method can be generalized to implement LOCC instruments with finitely many classical outcomes: if the instrument has [Formula: see text] coarse-grained final measurement outcomes, global input dimension [Formula: see text] and global output dimension [Formula: see text] for [Formula: see text] conditioned on the [Formula: see text]th outcome, then one can obtain the instrument as a convex combination of no more than [Formula: see text] slim protocols; that is, [Formula: see text] bits of shared randomness suffice.

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