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1.
Carlo Toffalori 《Mathematical Logic Quarterly》1996,42(1):369-378
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. 相似文献
2.
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. 相似文献
3.
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. 相似文献
4.
《Quaestiones Mathematicae》2013,36(4):271-275
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 mpn is finite for every prime p and natural number n. 相似文献
5.
《Quaestiones Mathematicae》2013,36(2):227-236
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. 相似文献
6.
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, Sp2m, Om 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. 相似文献
7.
Torbj?rn Tambour 《Arkiv f?r Matematik》1991,29(1):127-182
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. 相似文献
8.
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. 相似文献
9.
E. Ballico 《代数通讯》2013,41(7):2621-2637
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. 相似文献
10.
Sang Bum Lee 《代数通讯》2013,41(1):361-370
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. 相似文献
11.
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. 相似文献
12.
Sang Bum Lee 《代数通讯》2013,41(11):4420-4427
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. 相似文献
13.
David J. Hemmer 《代数通讯》2013,41(11):3292-3306
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. 相似文献
14.
Jeremy Rickard 《Publications Mathématiques de L'IHéS》1994,80(1):81-94
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 RΓc(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 Λc(X,Z/n) of direct summands of permutationZ/n[G]-modules, well-defined up to chain homotopy equivalence, which is isomorphic to RΓc(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. 相似文献
15.
We study modules over the group ring DG all proper submodules of which are finitely generated as D-modules. 相似文献
16.
《代数通讯》2013,41(4):1095-1102
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. 相似文献
17.
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. 相似文献
18.
Yoshinori Hamahata 《manuscripta mathematica》1993,79(1):307-327
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. 相似文献
19.
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). 相似文献
20.
A. A. Tuganbaev 《Journal of Mathematical Sciences》2008,152(2):298-303
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. 相似文献