1. CMB Online first
 Lang, Honglei; Sheng, Yunhe; Wade, Aissa

$\mathsf{VB}$Courant algebroids, $\mathsf{E}$Courant algebroids and generalized geometry
In this paper, we first discuss the relation between $\mathsf{VB}$Courant
algebroids and $\mathsf{E}$Courant algebroids and construct some examples
of $\mathsf{E}$Courant algebroids. Then we introduce the notion of
a generalized complex
structure on an $\mathsf{E}$Courant algebroid, unifying the usual
generalized complex structures on evendimensional manifolds
and
generalized contact structures on odddimensional manifolds.
Moreover, we study generalized complex structures on an omniLie
algebroid in detail. In particular, we show that generalized
complex structures on an omniLie algebra $\operatorname{gl}(V)\oplus V$
correspond
to complex Lie algebra structures on $V$.
Keywords:$\mathsf{VB}$Courant algebroid, $\mathsf{E}$Courant algebroid, omniLie algebroid, generalized complex structure, algebroidNijenhuis structure Categories:53D17, 18B40, 58H05 

2. CMB 2017 (vol 60 pp. 879)
 Zheng, Yuefei; Huang, Zhaoyong

Triangulated Equivalences Involving Gorenstein Projective Modules
For any ring $R$, we show that, in the bounded derived category
$D^{b}(\operatorname{Mod} R)$ of left $R$modules,
the subcategory of complexes with finite Gorenstein projective
(resp. injective) dimension modulo the subcategory
of complexes with finite projective (resp. injective) dimension
is equivalent to
the stable category $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ (resp.
$\overline{\mathbf{GI}}(\operatorname{Mod} R)$)
of Gorenstein projective (resp. injective) modules. As a consequence,
we get that if $R$ is a left and right noetherian ring admitting
a dualizing complex,
then $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ and
$\overline{\mathbf{GI}}(\operatorname{Mod}
R)$ are equivalent.
Keywords:triangulated equivalence, Gorenstein projective module, stable category, derived category, homotopy category Categories:18G25, 16E35 

3. 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 

4. CMB 2015 (vol 58 pp. 824)
 Luo, XiuHua

Exact Morphism Category and Gorensteinprojective Representations
Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generated
by all arrows in $Q$, $A$ be a finitedimensional $k$algebra. The
category of all finitedimensional representations of $(Q, J^2)$ over
$A$ is denoted by $\operatorname{rep}(Q, J^2, A)$. In this paper, we
introduce the category $\operatorname{exa}(Q,J^2,A)$, which is a
subcategory of
$\operatorname{rep}{}(Q,J^2,A)$ of all exact representations.
The main result of this paper explicitly describes the Gorensteinprojective representations in $\operatorname{rep}{}(Q,J^2,A)$,
via the exact representations plus an extra condition.
As a corollary, $A$ is a selfinjective algebra, if
and only if the Gorensteinprojective representations are exactly the
exact representations of $(Q, J^2)$ over $A$.
Keywords:representations of a quiver over an algebra, exact representations, Gorensteinprojective modules Category:18G25 

5. CMB 2014 (vol 57 pp. 721)
 Bruillard, Paul; Galindo, César; Hong, SeungMoon; Kashina, Yevgenia; Naidu, Deepak; Natale, Sonia; Plavnik, Julia Yael; Rowell, Eric C.

Classification of Integral Modular Categories of FrobeniusPerron Dimension $pq^4$ and $p^2q^2$
We classify integral modular categories of dimension $pq^4$ and $p^2q^2$,
where
$p$ and $q$ are distinct primes. We show that such categories are always
grouptheoretical except for categories of dimension $4q^2$.
In these cases there are
wellknown examples of nongrouptheoretical categories, coming from
centers of
TambaraYamagami categories and quantum groups. We show that a
nongrouptheoretical integral modular category of dimension $4q^2$ is
equivalent to either one of these wellknown examples or is of dimension
$36$ and is twistequivalent to fusion categories arising from a
certain quantum group.
Keywords:modular categories, fusion categories Category:18D10 

6. CMB 2013 (vol 57 pp. 506)
 Galindo, César

On Braided and Ribbon Unitary Fusion Categories
We prove that every braiding over a unitary fusion category is
unitary and every unitary braided fusion category admits a unique
unitary ribbon structure.
Keywords:fusion categories, braided categories, modular categories Categories:20F36, 16W30, 18D10 

7. CMB 2013 (vol 57 pp. 318)
 Huang, Zhaoyong

Duality of Preenvelopes and Pure Injective Modules
Let $R$ be an arbitrary ring and $()^+=\operatorname{Hom}_{\mathbb{Z}}(,
\mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers
and $\mathbb{Q}$ is the ring of rational numbers, and let
$\mathcal{C}$ be a subcategory of left $R$modules and $\mathcal{D}$
a subcategory of right $R$modules such that $X^+\in \mathcal{D}$
for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure
injective. Then a homomorphism $f: A\to C$ of left $R$modules with
$C\in \mathcal{C}$ is a $\mathcal{C}$(pre)envelope of $A$ provided
$f^+: C^+\to A^+$ is a $\mathcal{D}$(pre)cover of $A^+$. Some
applications of this result are given.
Keywords:(pre)envelopes, (pre)covers, duality, pure injective modules, character modules Categories:18G25, 16E30 

8. CMB 2011 (vol 55 pp. 319)
 Jardine, J. F.

The Verdier Hypercovering Theorem
This note gives a simple cocycletheoretic proof of the Verdier
hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the
homotopy category of simplicial sheaves or presheaves by simplicial
homotopy classes of maps, in the case where $Y$ is locally fibrant. The
statement proved in this paper is a generalization of the standard
Verdier hypercovering result in that it is pointed (in a very broad
sense) and there is no requirement for the source object $X$ to be
locally fibrant.
Keywords:simplicial presheaf, hypercover, cocycle Categories:14F35, 18G30, 55U35 

9. CMB 2010 (vol 53 pp. 425)
 Chapoton, Frédéric

Free PreLie Algebras are Free as Lie Algebras
We prove that the $\mathfrak{S}$module $\operatorname{PreLie}$ is a free Lie algebra in
the category of $\mathfrak{S}$modules and can therefore be written as the
composition of the $\mathfrak{S}$module $\operatorname{Lie}$ with a new $\mathfrak{S}$module
$X$. This implies that free preLie algebras in the category of
vector spaces, when considered as Lie algebras, are free on
generators that can be described using $X$. Furthermore, we define a
natural filtration on the $\mathfrak{S}$module $X$. We also obtain a
relationship between $X$ and the $\mathfrak{S}$module coming from the
anticyclic structure of the $\operatorname{PreLie}$ operad.
Categories:18D50, 17B01, 18G40, 05C05 

10. CMB 2009 (vol 52 pp. 273)
 MacDonald, John; Scull, Laura

Amalgamations of Categories
We consider the pushout of embedding functors in $\Cat$, the
category of small categories.
We show that if the embedding functors satisfy a 3for2
property, then the induced functors to the pushout category are
also embeddings. The result follows from the connectedness of
certain associated slice categories. The condition is motivated
by a similar result for maps of semigroups. We show that our
theorem can be applied to groupoids and to inclusions of full
subcategories. We also give an example to show that the theorem
does not hold when the
property only holds for one of the inclusion functors, or when it
is weakened to a onesided condition.
Keywords:category, pushout, amalgamation Categories:18A30, 18B40, 20L17 

11. CMB 2008 (vol 51 pp. 310)
 Witbooi, P. J.

Relative Homotopy in Relational Structures
The homotopy groups of a finite partially ordered set (poset) can be
described entirely in the context of posets, as shown in a paper by
B. Larose and C. Tardif.
In this paper we describe the relative version of such a
homotopy theory, for pairs $(X,A)$ where $X$ is a poset and $A$ is a
subposet of $X$. We also prove some theorems on the relevant version
of the notion of weak homotopy equivalences for maps of pairs of such
objects. We work in the category of reflexive binary relational
structures which contains the posets as in the work of Larose and
Tardif.
Keywords:binary reflexive relational structure, relative homotopy group, exact sequence, locally finite space, weak homotopy equivalence Categories:55Q05, 54A05;, 18B30 

12. 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 

13. CMB 2007 (vol 50 pp. 440)
 Raghuram, A.

A KÃ¼nneth Theorem for $p$Adic Groups
Let $G_1$ and $G_2$ be $p$adic groups. We describe a decomposition of
${\rm Ext}$groups in the category of smooth representations of
$G_1 \times G_2$ in terms of ${\rm Ext}$groups for $G_1$ and $G_2$.
We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal
representation
$\pi$ of a $p$adic group $G$. We also consider an example of
identifying
the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain
representations of $p$adic ${\rm GL}_{2n}$.
Categories:22E50, 18G15, 55U25 

14. CMB 2007 (vol 50 pp. 182)
15. CMB 2006 (vol 49 pp. 407)
 Jardine, J. F.

Intermediate Model Structures for Simplicial Presheaves
This note shows that any set of cofibrations containing the standard
set of generating projective cofibrations determines a cofibrantly
generated proper closed model structure on the category of simplicial
presheaves on a small Grothendieck site, for which the weak
equivalences are the local weak equivalences in the usual sense.
Categories:18G30, 18F20, 55U35 

16. CMB 2004 (vol 47 pp. 321)
 Bullejos, M.; Cegarra, A. M.

Classifying Spaces for Monoidal Categories Through Geometric Nerves
The usual constructions of classifying spaces for monoidal categories
produce CWcomplexes with
many cells that, moreover, do not have any proper geometric meaning.
However, geometric nerves of
monoidal categories are very handy simplicial sets whose simplices
have
a pleasing geometric
description: they are diagrams with the shape of the 2skeleton of
oriented standard simplices. The
purpose of this paper is to prove that geometric realizations of
geometric nerves are classifying
spaces for monoidal categories.
Keywords:monoidal category, pseudosimplicial category,, simplicial set, classifying space, homotopy type Categories:18D10, 18G30, 55P15, 55P35, 55U40 

17. CMB 2003 (vol 46 pp. 429)
 Sastry, Pramathanath; Tong, Yue Lin L.

The Grothendieck Trace and the de Rham Integral
On a smooth $n$dimensional complete variety $X$ over ${\mathbb C}$ we
show that the trace map ${\tilde\theta}_X \colon\break
H^n (X,\Omega_X^n)
\to {\mathbb C}$ arising from Lipman's version of Grothendieck duality
in \cite{ast117} agrees with
$$
(2\pi i)^{n} (1)^{n(n1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb
C}) \to {\mathbb C}
$$
under the Dolbeault isomorphism.
Categories:14F10, 32A25, 14A15, 14F05, 18E30 

18. CMB 2002 (vol 45 pp. 180)
 Connolly, Francis X.; Prassidis, Stratos

On the Exponent of the ${\nk}_0$Groups of Virtually Infinite Cyclic Groups
It is known that the $K$theory of a large class of groups can be
computed from the $K$theory of their virtually infinite cyclic
subgroups. On the other hand, Nilgroups appear to be the obstacle in
calculations involving the $K$theory of the latter. The main
difficulty in the calculation of Nilgroups is that they are
infinitely generated when they do not vanish. We develop methods for
computing the exponent of ${\nk}_0$groups that appear in the
calculation of the $K_0$groups of virtually infinite cyclic groups.
Categories:18F25, 19A31 

19. CMB 2001 (vol 44 pp. 459)
 Kahl, Thomas

LScatÃ©gorie algÃ©brique et attachement de cellules
Nous montrons que la Acat\'egorie d'un espace simplement connexe de
type fini est inf\'erieure ou \'egale \`a $n$ si et seulement si son
mod\`ele d'AdamsHilton est un r\'etracte homotopique d'une alg\`ebre
diff\'erentielle \`a $n$ \'etages. Nous en d\'eduisons que
l'invariant $\Acat$ augmente au plus de 1 lors de l'attachement
d'une cellule \`a un espace.
We show that the Acategory of a simply connected space of finite type
is less than or equal to $n$ if and only if its AdamsHilton model is
a homotopy retract of an $n$stage differential algebra. We deduce
from this that the invariant $\Acat$ increases by at most 1 when a
cell is attached to a space.
Keywords:LScategory, strong category, AdamsHilton models, cell attachments Categories:55M30, 18G55 

20. CMB 2000 (vol 43 pp. 162)
 Foth, Philip

Moduli Spaces of Polygons and Punctured Riemann Spheres
The purpose of this note is to give a simple combinatorial
construction of the map from the canonically compactified moduli
spaces of punctured complex projective lines to the moduli spaces
$\P_r$ of polygons with fixed side lengths in the Euclidean space
$\E^3$. The advantage of this construction is that one can obtain a
complete set of linear relations among the cycles that generate
homology of $\P_r$. We also classify moduli spaces of pentagons.
Categories:14D20, 18G55, 14H10 

21. CMB 2000 (vol 43 pp. 138)
 Boyd, C.

Exponential Laws for the Nachbin Ported Topology
We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet
spaces $E$ and $F$ that we have the topological identity
$$
\bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H}
\Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega
\biggr).
$$
Analogous results for the compact open topology have long been
established. We also give an example to show that the (Qno)
hypothesis on both $E$ and $F$ is necessary.
Categories:46G20, 18D15, 46M05 

22. CMB 2000 (vol 43 pp. 3)
 Adin, Ron; Blanc, David

Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and
free Lie algebras in the category of nonassociative algebras. These
resolutions derive in both cases from geometric objects, which in turn
reflect the combinatorics of suitable collections of leaflabeled
trees.
Keywords:resolutions, homology, Lie algebras, associative algebras, nonassociative algebras, Jacobi identity, leaflabeled trees, associahedron Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50 

23. CMB 1997 (vol 40 pp. 39)
 Zhao, Dongsheng

On projective $Z$frames
This paper deals with the projective objects in the category of all
$Z$frames, where the latter is a common generalization of
different types of frames. The main result obtained here is that a
$Z$frame is ${\bf E}$projective if and only if it is stably
$Z$continuous, for a naturally arising collection ${\bf E}$ of morphisms.
Categories:06D05, 54D10, 18D15 
