The phantom cover of a module

A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor₁(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor₁(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.     

The homological degree of a module

A homological degree of a graded module is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when is not Cohen-Macaulay. We construct a degree, , that behaves well under hyperplane sections and the modding out of elements of finite support. When carried out in a local algebra this degree gives a simulacrum of complexity à la Castelnuovo-Mumford's regularity. Several applications for estimating reduction numbers of ideals and predictions on the outcome of Noether normalizations are given.

     

The essential ideal is a Cohen-Macaulay module

Let be a finite -group which does not contain a rank two elementary abelian -group as a direct factor. Then the ideal of essential classes in the mod- cohomology ring of is a Cohen-Macaulay module whose Krull dimension is the -rank of the centre of . This basically answers in the affirmative a question posed by J. F. Carlson (Question 5.4 in Problems in the calculation of group cohomology, 1999).

     

The dual of a finitely generated multiplication module

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in 1986.     

The homotopy categorical crossed module of a CW-complex

Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.     

The f-depth of an ideal on a module

Let be an ideal of a Noetherian local ring and a finitely generated -module. The f-depth of on is the least integer such that the local cohomology module is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules.

     

The Koszul dual of a weakly Koszul module

We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give an example of a quasi-Koszul module which is not weakly Koszul.     

The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.     

The classifying space of a categorical crossed module

Any pointed CW‐complex X has associated a categorical crossed module WX whose homotopy groups coincide with those of the space up to dimension 3. Here we associate WX more closely with the homotopy 3‐type of X. We introduce the nerve of a categorical crossed module L and define its classifying space BL as the geometrical realization of the nerve. Then we prove that there is a map X → BWX inducing isomorphism of the homotopy groups π_i for i ≤ 3. Finally, comparison with other algebraic models of 3‐types is achieved (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The structure of bialgebra with a Hopf module

Let H be a Hopf algebra, B a bialgebra, and (B, ?, ρ) a right H-Hopf module. Assume that (B, ρ) is a right H-comodule algebra, (B, ?) is a right H-module coalgebra, and let A = B ^{co H} = {a ∈ B | ρ(a) = a ? 1}. Then we prove that B has a factorization of A□ _ρ ^? (the underlying space is A ? H) as a bialgebra, which generalizes Radford’s factorization of bialgebras with projection [12].     

The genus of a type of graph

Based on the joint tree model introduced by Liu, the genera of further types of graphs not necessary to have certain symmetry can be obtained. In this paper, we obtain the genus of a new type of graph with weak symmetry. As a corollary, the genus of complete tripartite graph K n,n,l (l≥n≥2) is also derived. The method used here is more direct than those methods, such as current graph, used to calculate the genus of a graph and can be realized in polynomial time.