MAXIMAL IDEALS OF THE ENDOMORPHISM RING OF AN INJECTIVE MODULE

2014 ◽  
Vol 13 (04) ◽  
pp. 1350131 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
MAI HOANG BIEN
Keyword(s):  
Endomorphism Ring ◽  
Injective Module ◽  
Maximal Ideals

In this paper, we describe the maximal ideals of the endomorphism ring of an injective module.

scholarly journals Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology

2009 ◽  
Vol 145 (4) ◽  
pp. 954-992 ◽  
Author(s):  
Catharina Stroppel
Keyword(s):  
Endomorphism Ring ◽  
Cohomology Ring ◽  
Injective Module ◽  
Duke Math ◽  
Khovanov Homology ◽  
Perverse Sheaves ◽  
Parabolic Subalgebra ◽  
Principal Block ◽  
Category O

AbstractFor a fixed parabolic subalgebra 𝔭 of $\mathfrak {gl}(n,\mathbb {C})$ we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H*(ℬ𝔭) of the corresponding Springer fibre. We give a diagrammatic description of 𝒪0𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category PervB(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra ℋn is realised as the endomorphism ring of some object from PervB(G(n,n)) which corresponds under localisation and the Riemann–Hilbert correspondence to a full projective–injective module in the corresponding category 𝒪0𝔭. From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547–596] by restriction.

Abelian Groups Quasi-Injective Over their Endomorphism Rings

1972 ◽  
Vol 24 (4) ◽  
pp. 617-621 ◽  
Author(s):  
George D. Poole ◽  
James D. Reid
Keyword(s):  
Endomorphism Ring ◽  
Abelian Groups ◽  
Invariant Subgroup ◽  
Injective Module ◽  
Endomorphism Rings ◽  
Additive Structure ◽  
Direct Sums ◽  
Divisible Groups

L. Fuchs has posed the problem of identifying those abelian groups that can serve as the additive structure of an injective module over some ring [1, p. 179], and in particular of identifying those abelian groups which are injective as modules over their endomorphism rings [1, p. 112]. Richman and Walker have recently answered the latter question, generalized in a non-trivial way [7], and have shown that the groups in question are of a rather restricted structure.In this paper we consider abelian groups which are quasi-injective over their endomorphism rings. We show that divisible groups are quasi-injective as are direct sums of cyclic p-groups. Quasi-injectivity of certain direct sums (products) is characterized in terms of the summands (factors). In general it seems that the answer to the question of whether or not a group G is quasinjective over its endomorphism ring E depends on how big HomE(H, G) is, with H a fully invariant subgroup of G.

ON M-C-INJECTIVE AND SELF-C-INJECTIVE MODULES

2010 ◽  
Vol 03 (03) ◽  
pp. 387-393 ◽  
Author(s):  
A. K. Chaturvedi ◽  
B. M. Pandeya ◽  
A. M. Tripathi ◽  
O. P. Mishra
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Injective Module ◽  
Injective Modules ◽  
Factor Module ◽  
Closed Submodule

Let M1 and M2 be two R-modules. Then M2 is called M1-c-injective if every homomorphism α from K to M2, where K is a closed submodule of M1, can be extended to a homomorphism β from M1 to M2. An R-module M is called self-c-injective if M is M-c-injective. For a projective module M, it has been proved that the factor module of an M -c-injective module is M -c-injective if and only if every closed submodule of M is projective. A characterization of self-c-injective modules in terms endomorphism ring of an R-module satisfying the CM-property is given.

scholarly journals KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

2010 ◽  
Vol 52 (A) ◽  
pp. 69-82 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
ŞULE ECEVIT ◽  
M. TAMER KOŞAN
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Weak Form ◽  
Indecomposable Module ◽  
Endomorphism Rings ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Injective Modules ◽  
Maximal Ideals

AbstractWe show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

MAXIMAL IDEALS IN PREADDITIVE CATEGORIES AND SEMILOCAL CATEGORIES

2011 ◽  
Vol 10 (01) ◽  
pp. 1-27 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
MARCO PERONE
Keyword(s):  
Endomorphism Ring ◽  
Maximal Ideal ◽  
Maximal Ideals ◽  
Natural Divisor ◽  
Krull Monoid

The first aim of this article is to study maximal ideals of a preadditive category [Formula: see text]. Maximal ideals, which do not exist in general for arbitrary preadditive categories, are associated to a maximal ideal of the endomorphism ring of an object and always exist when the category is semilocal. If [Formula: see text] is additive and semilocal, any skeleton [Formula: see text] of [Formula: see text] is a Krull monoid and we are able to characterize the essential valuations of [Formula: see text] and provide some natural divisor homomorphisms and divisor theories of [Formula: see text].

M-SP-INJECTIVE MODULES

2012 ◽  
Vol 05 (01) ◽  
pp. 1250005
Author(s):  
V. Kumar ◽  
A. J. Gupta ◽  
B. M. Pandeya ◽  
M. K. Patel
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Injective Module ◽  
Injective Modules ◽  
Finite Dimensional

In this paper we study M-small principally injective (in short, M-sp-injective) module which is the generalization of M-principally injective module. We prove that if M is finite dimensional and quasi-sp-injective then its endomorphism ring S is semi-local ring. We characterize the M-sp-injective module with the help of epi-retractable modules.

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

Sign in / Sign up

Export Citation Format

Share Document