Decidability for ℤ[G]-Modules when G is Cyclic of Prime Order

We consider the decision problem for modules over a group ring ?[G], where G is a cyclic group of prime order. We show that it reduces to the same problem for a class of certain abelian structures, and we obtain some partial decidability results for this class. Mathematics Subject Classification: 03C60, 03B25.     

Abelian-by-G Groups,for G Finite,from the Model Theoretic Point of View

Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ?[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree. Mathematics Subject Classification: 03C60, 03B25.     

AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

We investigate the properties of categories of G _C -flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G _C -flat R-modules is part of a weak AB-context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander-Buchweitz approximations for R-modules of finite G _C -flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of G _C -flat R-modules yield only the modules in the original subcategories.     

A MAXIMAL CLASS OF ABELIAN GROUPS ARISING FROM THE ISOMORPHISM G = T[EXT[G.X)]

Abstract

We characterize the class of groups G having the property that G ? T[Ext(G,X)] for some group X. We show that for every prime p, the p-component of such a group G has the form T [π∞_n=1 Z(pn)^mpn] where m_pn is finite for every prime p and natural number n.     

A WEAKER FORM OF BAER'S SPLITTING PROBLEM OVER VALUATION DOMAINS

Abstract

Eklof-Fuchs [3] have shown that over an arbitrary valuation domain R, the modules B which satisfy Ext 1/R (B,T) = 0 for all torsion R-modules T are precisely the free R-modules. Here we modify the problem and describe all R-modules B for which Ext 1/R (B, T) vanishes for all bounded and for all divisible torsion R-modules T. It is well known that if R is a descrete rank one valuation domain then all torsion—free R-modules B have this property.     

Tilting modules for classical groups and howe duality in positive characteristics

We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to “Howe duality” in the exterior algebra. To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field k, we set in correspondence another series of classical groups (usually the same one). Denote byG ₁(m) the group of rankm from the first series and byG ₂(n) the group of rankn from the second series. For any pair (G ₁(m), G₂(n)) we construct theG ₁(m)×G₂(n)-module M(m, n). The construction of M(m, n) is independent of characteristic; for chark=0, the actions ofG ₁(m) andG ₂(n) on M(m, n) form a reductive dual pair in the sense of Howe. We prove that M(m, n) is a tiltingG ₁(m)-andG ₂(n)-module and that End _{G 1}(m) M(m, n) is generated byG ₂(n) and vice versa. The existence of such a module provides much information about the relations between the categoryK ₁(m, n) of rationalG ₁(m)-modules with highest weights bounded in a certain sense byn and the categoryK ₂(m, n) of rationalG ₂(n)-modules with highest weights bounded in the same sense bym. More specifically, we prove that there is a bijection of the set of dominant weights ofG ₁(m)-modules fromK ₁(m, n) to the set of dominant weights ofG ₂(m)-modules fromK ₂(m, n) such that Ext groups for inducedG ₁(m)-modules fromK ₁(m, n) are isomorphic to Ext groups for corresponding Weyl modules overG ₂(n). Moreover, the derived categoriesD ^bK₁(m, n) andD ^bK₂(m, n) appear to be equivalent. We also use our study of the modules M(m, n) to find generators and relations for the algebra of allG-invariants in , whereG=GL _m, Sp_2m, O_m and V is the naturalG-module. Research was supported in part by Grant M7N000/M7N300 from the International Science Foundation and Russian Government and by INTAS Grant 94-4720. Research was supported in part by Grant M8H000/M8H300 from the International Science Foundation and Russian Government and by INTAS Grant 94-4720.     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspaceĨ _d ^m of the noncommutative invariant algebraĨ _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition ofĨ _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Gorenstein Flat Covers and Gorenstein Cotorsion Modules Over Integral Group Rings

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module. Presented by A. Verschoren Mathematics Subject Classifications (2000) 20C05, 16E65.     

On projective embeddings of generic rational surfaces

Let R be a commutative ring and let ?(σ) be a Gabriel filter of R such that R is σ-noetherian. We discuss the decomposition of the σ ¹-torsion submodule of a σ-torsionfree R-module and characterize the σ ^l-injectivity of σ-closed R-modules through the σ _m-injectivity of modules over noetherian local rings (S, m). As an application, we obtain new criteria to determine injectivity of modules over noetherian rings, of finite Krull dimension, and Krull domains.     

Weak-Injective Modules

Pure-injective and RD-injective R-modules over domains R have been investigated by many authors. We introduce another class of R-modules, called weak-injective modules, which turn out to be useful in addressing several unanswered questions between the two classes of modules. We also find that this class is an envelope class over any domain, giving a partial answer to the existence of envelope classes in the hierarchy of injective and divisible modules.

Communicated by I. Swanson.     

Injective and Projective Properties of <Emphasis Type="Italic">R</Emphasis>[<Emphasis Type="Italic">x</Emphasis>]-Modules

We study whether the projective and injective properties of left R-modules can be implied to the special kind of left R[x]-modules, especially to the case of inverse polynomial modules and Laurent polynomial modules.     

Weak-Injective Modules of Weak Dimension at Most One

We consider modules over integral domains R. A main purpose is to show that certain module properties assumed on R-modules of weak dimension ≤1 imply that these properties are shared by all modules in the category of R-modules.

Also we prove several results involving modules of weak dimension ≤1.     

Symmetric Group Modules with Specht and Dual Specht Filtrations

The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for GL _n (k). This article is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology.

Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for GL _n (k), these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of GL _n (k)-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be indecomposable self dual trivial source modules.     

Finite group actions and étale cohomology

If a finite group G acts on a quasi-projective variety X, then H*_c(X,Z/n), the étale cohomology with compact support of X with coefficients inZ/n, has aZ/n[G]-module structure. It is well known that there is a finer invariant, an object RΓ_c(X,Z/n) of the derived category ofZ/n[G]-modules, whose cohomology is H*_c(X,Z/n). We show that there is a finer invariant still, a bounded complex Λ_c(X,Z/n) of direct summands of permutationZ/n[G]-modules, well-defined up to chain homotopy equivalence, which is isomorphic to RΓ_c(X,Z/n) in the derived category. This complex has many properties analogous to those of the simplicial chain complex of a simplicial complex with a group action. There are similar results forl-adic cohomology.     

Modules over Group Rings with Certain Finiteness Conditions

We study modules over the group ring DG all proper submodules of which are finitely generated as D-modules.     

Tilting modules and ∗-modules

The relation between ?-modules-studied in [MO], [D], [C], [DH], [CM], [Z] and [T]-and Tiltng modules over an arbitrary ring is analyzed. In particular we prove that Tilting modules are exactly the faithful and finendo ?-modules. This answers a question of Trlifaj [T, Problem 1.5], showing that for any ring R the class of ?-modules generating the injectives and that one of Tiltings coincides. As a first application, we give an easy proof of the fact that every faithful ?-module over a finite-dimensional K-algebra is a classical Tilting module (see [DH, Theorem 1]). As a second application, we characterise the Tiltings as those modules which induce an equivalence between two categories with suitable dual properties.     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

Tensor products of Drinfeld modules andv-adic representations

We define the tensor product ϕ ⊗ ψ and relatedt-modules Sym²(ϕ), and ∧²(ϕ) for Drinfeld modules ϕ, ψ defined over the rational function fieldK=F _q (T), and describe thev-adic Tate modules of theset-modules by using those of ϕ, ψ.     

Auslander–Bridger Modules

Classically, the Auslander–Bridger transpose finds its best applications in the well-known setting of finitely presented modules over a semiperfect ring. We introduce a class of modules over an arbitrary ring R, which we call Auslander–Bridger modules, with the property that the Auslander–Bridger transpose induces a well-behaved bijection between isomorphism classes of Auslander–Bridger right R-modules and isomorphism classes of Auslander–Bridger left R-modules. Thus we generalize what happens for finitely presented modules over a semiperfect ring. Auslander–Bridger modules are characterized by two invariants (epi-isomorphism class and lower-isomorphism class), which are interchanged by the transpose. Via a suitable duality, we find that kernels of morphisms between injective modules of finite Goldie dimension are also characterized by two invariants (mono-isomorphism class and upper-isomorphism class).     

Modules with many direct summands

We study rings over which all right modules are I ₀-modules. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 233–241, 2006.