5,542,343 research outputs found
Model structures on modules over Ding-Chen rings
An -FC ring is a left and right coherent ring whose left and right self
FP-injective dimension is . The work of Ding and Chen in \cite{ding and chen
93} and \cite{ding and chen 96} shows that these rings possess properties which
generalize those of -Gorenstein rings. In this paper we call a (left and
right) coherent ring with finite (left and right) self FP-injective dimension a
Ding-Chen ring. In case the ring is Noetherian these are exactly the Gorenstein
rings. We look at classes of modules we call Ding projective, Ding injective
and Ding flat which are meant as analogs to Enochs' Gorenstein projective,
Gorenstein injective and Gorenstein flat modules. We develop basic properties
of these modules. We then show that each of the standard model structures on
Mod-, when is a Gorenstein ring, generalizes to the Ding-Chen case. We
show that when is a commutative Ding-Chen ring and is a finite group,
the group ring is a Ding-Chen ring.Comment: 12 page
Improved Chen-Ricci inequality for curvature-like tensors and its applications
We present Chen-Ricci inequality and improved Chen-Ricci inequality for
curvature like tensors. Applying our improved Chen-Ricci inequality we study
Lagrangian and Kaehlerian slant submanifolds of complex space forms and
C-totally real submanifolds of Sasakian space forms
Chen Lie algebras
The Chen groups of a finitely-presented group G are the lower central series
quotients of its maximal metabelian quotient, G/G''. The direct sum of the Chen
groups is a graded Lie algebra, with bracket induced by the group commutator.
If G is the fundamental group of a formal space, we give an analog of a basic
result of D. Sullivan, by showing that the rational Chen Lie algebra of G is
isomorphic to the rational holonomy Lie algebra of G modulo the second derived
subalgebra. Following an idea of W.S. Massey, we point out a connection between
the Alexander invariant of a group G defined by commutator-relators, and its
integral holonomy Lie algebra.
As an application, we determine the Chen Lie algebras of several classes of
geometrically defined groups, including surface-like groups, fundamental groups
of certain classical link complements, and fundamental groups of complements of
complex hyperplane arrangements. For link groups, we sharpen Massey and
Traldi's solution of the Murasugi conjecture. For arrangement groups, we prove
that the rational Chen Lie algebra is combinatorially determined.Comment: 23 page
Chen ranks and resonance
The Chen groups of a group are the lower central series quotients of the
maximal metabelian quotient of . Under certain conditions, we relate the
ranks of the Chen groups to the first resonance variety of , a jump locus
for the cohomology of . In the case where is the fundamental group of
the complement of a complex hyperplane arrangement, our results positively
resolve Suciu's Chen ranks conjecture. We obtain explicit formulas for the Chen
ranks of a number of groups of broad interest, including pure Artin groups
associated to Coxeter groups, and the group of basis-conjugating automorphisms
of a finitely generated free group.Comment: final version, to appear in Advances in Mathematic
- …