1. CMB Online first
 Shaul, Liran

Homological dimensions of local (co)homology over commutative DGrings
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
DGrings:
let $A$ be a commutative nonpositive DGring 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 DGring Categories:13B35, 13D05, 13D45, 16E45 

2. CMB 2017 (vol 60 pp. 705)
3. CMB 2017 (vol 60 pp. 478)
 Carrell, Jim; Kaveh, Kiumars

Springer's Weyl Group Representation via Localization
Let $G$ denote a reductive algebraic group over
$\mathbb{C}$
and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer
variety $\mathcal{B}_x$
is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing
the
Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable
property that
the Weyl group $W$ of $G$ admits a representation on the cohomology
of $\mathcal{B}_x$
even though $W$ rarely acts on $\mathcal{B}_x$ itself. Wellknown constructions
of this action
due to Springer et al use technical machinery from algebraic
geometry.
The purpose of this note is to describe an elementary approach
that gives this action
when $x$ is what we call parabolicsurjective. The idea is to
use localization to construct an action of $W$ on
the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic
subtorus of $G$.
This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and
gives the desired representation. The parabolicsurjective case
includes all nilpotents of type $A$ and,
more generally, all nilpotents for which it is known that $W$
acts on
$H_S^*(\mathcal{B}_x)$ for some torus $S$.
Our result is deduced from a general theorem describing
when a group action on the cohomology of the fixed point set of a
torus action
on a space lifts to the full cohomology algebra of the space.
Keywords:Springer variety, Weyl group action, equivariant cohomology Categories:14M15, 14F43, 55N91 

4. CMB 2017 (vol 60 pp. 225)
 Bahmanpour, Kamal; Naghipour, Reza

Faltings' Finiteness Dimension of Local Cohomology Modules Over Local CohenMacaulay Rings
Let $(R, \frak m)$ denote a local CohenMacaulay ring and $I$
a nonnilpotent 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 $\dim R1$.
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 59 pp. 682)
 Carlson, Jon F.; Chebolu, Sunil K.; Mináč, Ján

Ghosts and Strong Ghosts in the Stable Category
Suppose that $G$ is a finite group and $k$ is a field of characteristic
$p\gt 0$. A ghost map is a map in the stable category of
finitely generated $kG$modules which induces the zero map
in Tate cohomology in all degrees. In an earlier paper we showed
that the
thick subcategory generated by the trivial module
has no nonzero ghost maps if and only if
the Sylow $p$subgroup of $G$ is cyclic of order 2 or 3.
In this paper we introduce and study variations of ghost
maps.
In particular, we consider the behavior of ghost maps under
restriction
and induction functors. We find all groups satisfying a strong
form
of Freyd's generating hypothesis and show that ghosts can
be detected on a finite range of degrees of Tate cohomology.
We also
consider maps which mimic ghosts in high degrees.
Keywords:Tate cohomology, ghost maps, stable module category, almost split sequence, periodic cohomology Categories:20C20, 20J06, 55P42 

6. CMB 2016 (vol 59 pp. 483)
 Crooks, Peter; Holden, Tyler

Generalized Equivariant Cohomology and Stratifications
For $T$ a compact torus and $E_T^*$ a generalized $T$equivariant
cohomology theory, we provide a systematic framework for computing
$E_T^*$ in the context of equivariantly stratified smooth complex
projective varieties. This allows us to explicitly compute $E_T^*(X)$
as an $E_T^*(\text{pt})$module when $X$ is a direct limit of
smooth complex projective $T_{\mathbb{C}}$varieties with finitely
many $T$fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$,
$K_T^*$, and $MU_T^*$. We perform this computation on the affine
Grassmannian of a complex semisimple group.
Keywords:equivariant cohomology theory, stratification, affine Grassmannian Categories:55N91, 19L47 

7. CMB 2016 (vol 59 pp. 508)
 De Nicola, Antonio; Yudin, Ivan

Generalized Goldberg Formula
In this paper we prove a useful formula for the graded commutator
of the Hodge
codifferential with the left wedge multiplication by a fixed
$p$form acting on
the de Rham algebra of a Riemannian manifold. Our formula generalizes
a formula
stated by Samuel I. Goldberg for the case of 1forms. As first
examples of
application we obtain new identities on locally conformally KÃ¤hler
manifolds
and quasiSasakian manifolds. Moreover, we prove that under suitable
conditions
a certain subalgebra of differential forms in a compact manifold
is quasiisomorphic as a CDGA to the full de Rham algebra.
Keywords:graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasiSasakian manifold Categories:53C25, 53D35 

8. CMB 2016 (vol 59 pp. 271)
 DehghaniZadeh, Fatemeh

Artinianness of Composed Graded Local Cohomology Modules
Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with
local base ring $(R_{0}, \mathfrak{m}_{0})$ and let
$R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated
graded $R$modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We
show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M,
N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and
$j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal
of
$R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$primary ideal.
Keywords:generalized local cohomology, Artinian, graded module Categories:13D45, 13E10, 16W50 

9. 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
$\operatorname{\mathsf{G_Cfd}}_RX$ 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 CohenMacaulay modules, dualizing
modules and Gorenstein rings.
Keywords:flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative CohenMacaulay module, semidualizing module Categories:13D05, 13D45, 18G20 

10. 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 nonzero
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 

11. CMB 2015 (vol 59 pp. 159)
 MacColl, Joseph

Rotors in Khovanov Homology
Anstee, Przytycki, and Rolfsen introduced the idea of rotants,
pairs of links related by a generalised form of link mutation.
We exhibit infinitely many pairs of rotants which can be distinguished
by Khovanov homology, but not by the Jones polynomial.
Keywords:geometric topology, knot theory, rotants, khovanov homology, jones polynomial Categories:57M27, 57M25 

12. CMB 2015 (vol 58 pp. 730)
 Efrat, Ido; Matzri, Eliyahu

Vanishing of Massey Products and Brauer Groups
Let $p$ be a prime number and $F$ a field containing a root of
unity of order $p$.
We relate recent results on vanishing of triple Massey products
in the mod$p$ Galois cohomology of $F$,
due to Hopkins, Wickelgren, MinÃ¡Ä, and TÃ¢n, to classical
results in the theory of central simple algebras.
For global fields, we prove a stronger form of the vanishing
property.
Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields Categories:16K50, 11R34, 12G05, 12E30 

13. CMB 2015 (vol 58 pp. 620)
 Sands, Jonathan W.

$L$functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups
Let $n$ be a positive even integer, and let $F$ be a totally real
number field and $L$ be an abelian Galois extension which is totally
real or CM.
Fix a finite set $S$ of primes of $F$ containing the infinite primes
and all those which ramify in
$L$, and let $S_L$ denote the primes of $L$ lying above those in
$S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$integers of $L$.
Suppose that $\psi$ is a quadratic character of the Galois group of
$L$ over $F$. Under the assumption of the motivic Lichtenbaum
conjecture, we obtain a nontrivial annihilator of the motivic
cohomology group
$H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the
$S$modified Artin $L$function $L_{L/F}^S(s,\psi)$ at $s=1n$.
Keywords:motivic cohomology, regulator, Artin Lfunctions Categories:11R42, 11R70, 14F42, 19F27 

14. CMB 2014 (vol 58 pp. 80)
 Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya

The Equivariant Cohomology Rings of Peterson Varieties in All Lie
Types
Let $G$ be a complex semisimple linear algebraic group and let
$Pet$ be the associated Peterson variety in the flag
variety $G/B$.
The main theorem of this note gives an efficient presentation
of the equivariant cohomology ring $H^*_S(Pet)$ of the
Peterson variety as a quotient of a polynomial ring by an ideal
$J$ generated by quadratic polynomials, in the spirit of the
Borel presentation of the cohomology of the flag variety. Here
the group $S \cong \mathbb{C}^*$ is a certain subgroup of a maximal
torus $T$ of $G$.
Our description of the ideal $J$ uses the Cartan matrix and is
uniform across Lie types. In our arguments we use the Monk formula
and Giambelli formula for the equivariant cohomology rings of
Peterson varieties for all Lie types, as obtained in the work
of Drellich. Our result generalizes a previous theorem of FukukawaHaradaMasuda,
which was only for Lie type $A$.
Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula Categories:55N91, 14N15 

15. CMB 2013 (vol 57 pp. 225)
 Adamaszek, Michał

Small Flag Complexes with Torsion
We classify flag complexes on at most $12$ vertices with torsion in
the first homology group. The result is moderately computeraided.
As a consequence we confirm a folklore conjecture that the smallest
poset whose order complex is homotopy equivalent to the real
projective plane (and also the smallest poset with torsion in the
first homology group) has exactly $13$ elements.
Keywords:clique complex, order complex, homology, torsion, minimal model Categories:55U10, 06A11, 55P40, 5504, 0504 

16. 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 finitedimensional 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 $\check{H}^{n1}(K;G)=0$ and $\dim_G K\leq
n1$. This provides a partial answer to a question of
KallipolitiPapasoglu
whether any twodimensional homogeneous Peano continuum cannot be separated by arcs.
Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$continuum Categories:54F45, 54F15 

17. CMB 2013 (vol 57 pp. 210)
18. 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 $\max\{0, (d1)/21\}$ if $d$ is odd, and must be either $d/21$ or $d/22$ if $d$ is even (the possibility of $d/21$ does occur, but we do not know if the possibility of $d/22$ also can occur).
Keywords:Cuntz semigroup, comparison radius, cohomology dimension, covering dimension 

19. CMB 2012 (vol 56 pp. 491)
 Bahmanpour, Kamal

A Note on Homological Dimensions of Artinian Local Cohomology Modules
Let $(R,{\frak m})$ be a nonzero commutative Noetherian local ring
(with identity), $M$ be a nonzero 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 Category:13D45 

20. CMB 2011 (vol 55 pp. 48)
 Chebolu, Sunil K.; Christensen, J. Daniel; Mináč, Ján

Freyd's Generating Hypothesis for Groups with Periodic Cohomology
Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$
divides
the order of $G$.
Freyd's generating hypothesis for the stable module category of
$G$ is the statement that a map between finitedimensional
$kG$modules in the thick subcategory generated by $k$ factors through a
projective if the induced map on Tate cohomology is trivial. We show that if
$G$
has periodic cohomology, then the generating hypothesis holds if and only if
the Sylow
$p$subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions
that are equivalent to the GH
for groups with periodic cohomology.
Keywords:Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology Categories:20C20, 20J06, 55P42 

21. CMB 2011 (vol 55 pp. 81)
22. CMB 2011 (vol 55 pp. 153)
 Mafi, Amir; Saremi, Hero

Artinianness of Certain Graded Local Cohomology Modules
We show that if
$R=\bigoplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring
with local base ring $(R_0,\mathfrak{m}_0)$, irrelevant ideal $R_+$, and
$M$ a finitely generated graded $R$module, then
$H_{\mathfrak{m}_0R}^j(H_{R_+}^t(M))$ is Artinian for $j=0,1$ where
$t=\inf\{i\in{\mathbb{N}_0}: H_{R_+}^i(M)$ is not finitely
generated $\}$. Also, we prove that if $\operatorname{cd}(R_+,M)=2$, then for
each $i\in\mathbb{N}_0$, $H_{\mathfrak{m}_0R}^i(H_{R_+}^2(M))$ is
Artinian if and only if $H_{\mathfrak{m}_0R}^{i+2}(H_{R_+}^1(M))$ is
Artinian, where $\operatorname{cd}(R_+,M)$ is the cohomological dimension of $M$
with respect to $R_+$. This improves some results of R. Sazeedeh.
Keywords:graded local cohomology, Artinian modules Categories:13D45, 13E10 

23. CMB 2011 (vol 54 pp. 619)
 Dibaei, Mohammad T.; Vahidi, Alireza

Artinian and NonArtinian Local Cohomology Modules
Let $M$ be a finite module over a commutative noetherian ring $R$.
For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between
cohomological dimensions of $M$ with respect to $\mathfrak{a},
\mathfrak{b}$,
$\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is
shown that $M$ is generalized CohenMacaulay if there exists an
ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with
respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer
such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the
nonempty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M)
$ is not artinian for
some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers
of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.
Keywords:local cohomology modules, cohomological dimensions, Bass numbers Categories:13D45, 13E10 

24. CMB 2010 (vol 53 pp. 667)
 Khashyarmanesh, Kazem

On the Endomorphism Rings of Local Cohomology Modules
Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal
of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also
prove that, for a nonnegative integer $n$ such that
$H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for
all $i >0$ and $z \in \mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic
image of $R$, where $R_z$ is the ring of fractions of $R$ with
respect to a multiplicatively closed subset $\{z^j \mid j \geqslant
0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z
\in \mathfrak{a}$,
then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$
is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$.
Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence Categories:13D45, 13D07, 13D25 

25. CMB 2010 (vol 53 pp. 577)