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1. CMB Online first

Medini, Andrea; van Mill, Jan; Zdomskyy, Lyubomyr S.
Infinite powers and Cohen reals
We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of $Z^\omega$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$.

Keywords:infinite power, zero-dimensional, first-countable, homogeneous, Cohen real, h-homogeneous, rigid
Categories:03E35, 54B10, 54G20

2. CMB Online first

Shaul, Liran
Homological dimensions of local (co)homology over commutative DG-rings
Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $I$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module $\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module. In this paper we generalize these facts to commutative noetherian DG-rings: let $A$ be a commutative non-positive DG-ring such that $\mathrm{H}^0(A)$ is a noetherian ring, and for each $i\lt 0$, the $\mathrm{H}^0(A)$-module $\mathrm{H}^i(A)$ is finitely generated. Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$, we show that the local cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ associated to $\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$ does not increase flat dimension.

Keywords:local cohomology, derived completion, homological dimension, commutative DG-ring
Categories:13B35, 13D05, 13D45, 16E45

3. CMB Online first

Bavula, V. V.; Lu, T.
Classification of simple weight modules over the Schrödinger algebra
A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple $C_{\mathcal{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $\mathcal{S}$ (over the centre). It is proved that some (prime) factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull minimal.

Keywords:weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra
Categories:17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50

4. CMB 2017 (vol 60 pp. 225)

Bahmanpour, Kamal; Naghipour, Reza
Faltings' Finiteness Dimension of Local Cohomology Modules Over Local Cohen-Macaulay Rings
Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$. As a consequence we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$ and if $\operatorname{mAss}_R(R/I)$ is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case $(R, \frak m)$ is a complete equidimensional local ring.

Keywords:Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomology
Categories:13D45, 14B15

5. CMB 2016 (vol 60 pp. 206)

Vetrik, Tomáš
The Metric Dimension of Circulant Graphs
A subset $W$ of the vertex set of a graph $G$ is called a resolving set of $G$ if for every pair of distinct vertices $u, v$ of $G$, there is $w \in W$ such that the~distance of $w$ and $u$ is different from the distance of $w$ and $v$. The~cardinality of a~smallest resolving set is called the metric dimension of $G$, denoted by $dim(G)$. The circulant graph $C_n (1, 2, \dots , t)$ consists of the vertices $v_0, v_1, \dots , v_{n-1}$ and the~edges $v_i v_{i+j}$, where $0 \le i \le n-1$, $1 \le j \le t$ $(2 \le t \le \lfloor \frac{n}{2} \rfloor)$, the indices are taken modulo $n$. Grigorious et al. [On the metric dimension of circulant and Harary graphs, Applied Mathematics and Computation 248 (2014), 47--54] proved that $dim(C_n (1,2, \dots , t)) \ge t+1$ for $t \lt \lfloor \frac{n}{2} \rfloor$, $n \ge 3$, and they presented a~conjecture saying that $dim(C_n (1,2, \dots , t)) = t+p-1$ for $n=2tk+t+p$, where $3 \le p \le t+1$. We disprove both statements. We show that if $t \ge 4$ is even, there exists an infinite set of values of $n$ such that $dim(C_n (1,2, \dots , t)) = t$. We also prove that $dim(C_n (1,2, \dots , t)) \le t + \frac{p}{2}$ for $n=2tk+t+p$, where $t$ and $p$ are even, $t \ge 4$, $2 \le p \le t$ and $k \ge 1$.

Keywords:metric dimension, resolving set, circulant graph, Cayley graph, distance
Categories:05C35, 05C12

6. CMB 2016 (vol 59 pp. 824)

Karpenko, Nikita A.
Incompressibility of Products of Pseudo-homogeneous Varieties
We show that the conjectural criterion of $p$-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties which includes the norm varieties associated to symbols in Galois cohomology of arbitrary degree.

Keywords:algebraic groups, projective homogeneous varieties, Chow groups and motives, canonical dimension and incompressibility
Categories:20G15, 14C25

7. CMB 2016 (vol 59 pp. 748)

Dolžan, David
The Metric Dimension of the Total Graph of a Finite Commutative Ring
We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields.

Keywords:total graph, finite ring, metric dimension
Categories:13M99, 05E40

8. CMB 2016 (vol 59 pp. 403)

Zargar, Majid Rahro; Zakeri, Hossein
On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules
Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\mathfrak{d}_R X$, $\operatorname{\mathsf{Gfd}}_R X$ and $\operatorname{\mathsf{G_C-fd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$-module such that $\operatorname{H}_{\mathfrak{a}}^i(M)=0$ for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then $\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings.

Keywords:flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative Cohen-Macaulay module, semidualizing module
Categories:13D05, 13D45, 18G20

9. CMB 2015 (vol 58 pp. 741)

Gao, Zenghui
Homological Properties Relative to Injectively Resolving Subcategories
Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$-modules. A left $R$-module $M$ (resp. right $R$-module $N$) is called $\mathcal{E}$-injective (resp. $\mathcal{E}$-flat) if $\operatorname{Ext}_R^1(G,M)=0$ (resp. $\operatorname{Tor}_1^R(N,G)=0$) for any $G\in\mathcal{E}$. Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$-module $M$ is $\mathcal{E}$-injective if and only if $M$ is a direct sum of an injective left $R$-module and a reduced $\mathcal{E}$-injective left $R$-module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$-modules such that $\mathcal{E}^+\subseteq\mathcal{F}$ and $\mathcal{F}^+\subseteq\mathcal{E}$. It is shown that a finitely presented right $R$-module $M$ is $\mathcal{E}$-flat if and only if $M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right $R$-module. In addition, we introduce and investigate the $\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary rings and $\mathcal{E}$-von Neumann regular rings and characterize them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat modules.

Keywords:injectively resolving subcategory, \mathcal{E}-injective module (dimension), \mathcal{E}-flat module (dimension), cover, preenvelope, \mathcal{E}-(semi)hereditary ring
Categories:16E30, 16E10, 16E60

10. CMB 2015 (vol 59 pp. 144)

Laterveer, Robert
A Brief Note Concerning Hard Lefschetz for Chow Groups
We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.

Keywords:algebraic cycles, Chow groups, finite-dimensional motives
Categories:14C15, 14C25, 14C30

11. CMB 2015 (vol 58 pp. 787)

Kitabeppu, Yu; Lakzian, Sajjad
Non-branching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups
In this paper, we generalize the finite generation result of Sormani to non-branching $RCD(0,N)$ geodesic spaces (and in particular, Alexandrov spaces) with full support measures. This is a special case of the Milnor's Conjecture for complete non-compact $RCD(0,N)$ spaces. One of the key tools we use is the Abresch-Gromoll type excess estimates for non-smooth spaces obtained by Gigli-Mosconi.

Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally Hilbertian
Categories:53C23, 30L99

12. CMB 2015 (vol 58 pp. 664)

Vahidi, Alireza
Betti Numbers and Flat Dimensions of Local Cohomology Modules
Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$--module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some inequalities between the Betti numbers of $X$ and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of $X$ in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.

Keywords:Betti numbers, flat dimensions, local cohomology modules
Categories:13D45, 13D05

13. CMB 2015 (vol 58 pp. 519)

Kang, Su-Jeong
Refined Motivic Dimension
We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.

Keywords:motivic dimension, generalized Hodge conjecture
Categories:14C30, 14C25

14. CMB 2015 (vol 58 pp. 402)

Tikuisis, Aaron Peter; Toms, Andrew
On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras
We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple $\mathrm{C}^*$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first-named author, this shows that slow dimension growth coincides with $\mathcal Z$-stability, for approximately subhomogeneous algebras.

Keywords:nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growth
Categories:46L35, 46L05, 46L80, 47L40, 46L85

15. CMB 2014 (vol 57 pp. 814)

Hou, Ruchen
On Global Dimensions of Tree Type Finite Dimensional Algebras
A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k-$algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum of possible global dimensions is $n-1$.

Keywords:global dimension, tree type finite dimensional $k-$algebra, quiver
Categories:16D40, 16E10, , 16G20

16. CMB 2014 (vol 57 pp. 673)

Ahmadi, S. Ruhallah; Gilligan, Bruce
Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two
If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$ acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification $\widehat{G}$ of $G$ on $X$ except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of $X$.

Keywords:homogeneous complex manifold, non-compact dimension two, complexification

17. CMB 2013 (vol 57 pp. 245)

Brodskiy, N.; Dydak, J.; Lang, U.
Assouad-Nagata Dimension of Wreath Products of Groups
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: \par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.

Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groups
Categories:54F45, 55M10, 54C65

18. CMB 2013 (vol 57 pp. 335)

Karassev, A.; Todorov, V.; Valov, V.
Alexandroff Manifolds and Homogeneous Continua
ny homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a $V^n$-continuum in the sense of Alexandroff. We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.

Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuum
Categories:54F45, 54F15

19. CMB 2013 (vol 56 pp. 795)

MacDonald, Mark L.
Upper Bounds for the Essential Dimension of $E_7$
This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type $E_7$ over a field of characteristic not 2 or 3. In particular, $\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$.

Keywords:$E_7$, essential dimension, stabilizer in general position
Categories:20G15, 20G41

20. CMB 2013 (vol 57 pp. 159)

Oral, Kürşat Hakan; Özkirişci, Neslihan Ayşen; Tekir, Ünsal
Strongly $0$-dimensional Modules
In a multiplication module, prime submodules have the property, if a prime submodule contains a finite intersection of submodules then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodule. A multiplication module in which every prime submodule is strongly prime will be called strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Q-modules and strongly 0-dimensional modules.

Keywords:strongly 0-dimensional rings, Q-module, Von Neumann regular module
Categories:13C99, 16D10

21. CMB 2013 (vol 56 pp. 745)

Fu, Xiaoye; Gabardo, Jean-Pierre
Dimension Functions of Self-Affine Scaling Sets
In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$.

Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension function

22. CMB 2012 (vol 56 pp. 737)

Elliott, George A.; Niu, Zhuang
On the Radius of Comparison of a Commutative C*-algebra
Let $X$ be a compact metric space. A lower bound for the radius of comparison of the C*-algebra $\operatorname{C}(X)$ is given in terms of $\operatorname{dim}_{\mathbb{Q}} X$, where $\operatorname{dim}_{\mathbb{Q}} X $ is the cohomological dimension with rational coefficients. If $\operatorname{dim}_{\mathbb{Q}} X =\operatorname{dim} X=d$, then the radius of comparison of the C*-algebra $\operatorname{C}(X)$ is $\max\{0, (d-1)/2-1\}$ if $d$ is odd, and must be either $d/2-1$ or $d/2-2$ if $d$ is even (the possibility of $d/2-1$ does occur, but we do not know if the possibility of $d/2-2$ also can occur).

Keywords:Cuntz semigroup, comparison radius, cohomology dimension, covering dimension

23. CMB 2012 (vol 56 pp. 683)

Nikseresht, A.; Azizi, A.
Envelope Dimension of Modules and the Simplified Radical Formula
We introduce and investigate the notion of envelope dimension of commutative rings and modules over them. In particular, we show that the envelope dimension of a ring, $R$, is equal to that of the $R$-module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a ring is no more than its envelope dimension and characterize Noetherian rings for which these two dimensions are equal. Moreover we generalize and study the concept of simplified radical formula for modules, which we defined in an earlier paper.

Keywords:envelope dimension, simplified radical formula, prime submodule
Categories:13A99, 13C99, 13C13, 13E05

24. CMB 2012 (vol 56 pp. 551)

Handelman, David
Real Dimension Groups
Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like, i.e., it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over the a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In the Appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolation
Categories:46A40, 06F20, 13J25, 19K14

25. CMB 2012 (vol 56 pp. 491)

Bahmanpour, Kamal
A Note on Homological Dimensions of Artinian Local Cohomology Modules
Let $(R,{\frak m})$ be a non-zero commutative Noetherian local ring (with identity), $M$ be a non-zero finitely generated $R$-module. In this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that $ \operatorname{{\rm injdim_{_{R_{\frak p}}}}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and ${\rm fd}_{R_{\p}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ are bounded from above by $ \operatorname{{\rm injdim_{_{R}}}} H^i_{\frak m}(M)$ and $ {\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$.

Keywords:cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology
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