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1. 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

2. CMB 2017 (vol 60 pp. 705)

Benelkourchi, Slimane
Envelope Approach to Degenerate Complex Monge-Ampère Equations on Compact Kähler Manifolds
We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

Keywords:degenerate complex Monge-Ampère equation, compact Kähler manifold, big cohomology, plurisubharmonic function
Categories:32W20, 32Q25, 32U05

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. Well-known 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 parabolic-surjective. 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 parabolic-surjective 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 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 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 1-forms. As first examples of application we obtain new identities on locally conformally Kähler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a CDGA to the full de Rham algebra.

Keywords:graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasi-Sasakian manifold
Categories:53C25, 53D35

8. CMB 2016 (vol 59 pp. 271)

Dehghani-Zadeh, 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 $\operatorname{\mathsf{G_C-fd}}_RX$ 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

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 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

11. 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, Mináċ, and Tân, 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

12. 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 non-trivial 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 $s=1-n$.

Keywords:motivic cohomology, regulator, Artin L-functions
Categories:11R42, 11R70, 14F42, 19F27

13. 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 Fukukawa-Harada-Masuda, which was only for Lie type $A$.

Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula
Categories:55N91, 14N15

14. 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

15. 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

16. 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

17. 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 finite-dimensional $kG$-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

18. CMB 2011 (vol 55 pp. 81)

Divaani-Aazar, Kamran; Hajikarimi, Alireza
Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals
Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $\dim R/\mathfrak a\leq 1$, then all generalized local cohomology modules $H^i_{\mathfrak a}(M,N)$ are $\mathfrak a$-cofinite.

Keywords:cofinite modules, generalized local cohomology modules, local cohomology modules
Categories:13D45, 13E05, 13E10

19. 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

20. CMB 2011 (vol 54 pp. 619)

Dibaei, Mohammad T.; Vahidi, Alireza
Artinian and Non-Artinian 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 Cohen-Macaulay 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 non-empty 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

21. 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

22. CMB 2010 (vol 53 pp. 577)

Asgharzadeh, Mohsen; Tousi, Massoud
A Unified Approach to Local Cohomology Modules using Serre Classes
This paper discusses the connection between the local cohomology modules and the Serre classes of $R$-modules. This connection has provided a common language for expressing some results regarding the local cohomology $R$-modules that have appeared in different papers.

Keywords:associated prime ideals, local cohomology modules, Serre class

23. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
Homotopy Formulas for Cyclic Groups Acting on Rings
The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

Keywords:group cohomology, norm map, cyclic group, homotopy
Categories:20J06, 20K01, 16W22, 18G35

24. CMB 2007 (vol 50 pp. 598)

Lorestani, Keivan Borna; Sahandi, Parviz; Yassemi, Siamak
Artinian Local Cohomology Modules
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\H^i_\fa(M)$ is finitely generated for all $i
Keywords:local cohomology module, Artinian module, reflexive module
Categories:13D45, 13E10, 13C05

25. CMB 2007 (vol 50 pp. 567)

Joshi, Kirti
Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence
In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

Keywords:threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsion
Categories:14F30, 14J30
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