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1.
Chris Woodcock 《Journal of Pure and Applied Algebra》2010,214(8):1497-1500
In a recent paper entitled “A commutative analogue of the group ring” we introduced, for each finite group (G,⋅), a commutative graded Z-algebra R(G,⋅) which has a close connection with the cohomology of (G,⋅). The algebra R(G,⋅) is the quotient of a polynomial algebra by a certain ideal I(G,⋅) and it remains a fundamental open problem whether or not the group multiplication ⋅ on G can always be recovered uniquely from the ideal I(G,⋅).Suppose now that (G,×) is another group with the same underlying set G and identity element e∈G such that I(G,⋅)=I(G,×). Then we show here that the multiplications ⋅ and × are at least “almost equal” in a precise sense which renders them indistinguishable in terms of most of the standard group theory constructions. In particular in many cases (for example if (G,⋅) is Abelian or simple) this implies that the two multiplications are actually equal as was claimed in the previously cited paper. 相似文献
2.
Chris Woodcock 《Algebras and Representation Theory》2009,12(6):597-604
Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ?-algebra R G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring $H^\ast(G, \mathbb{F}_p)Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ℤ-algebra R
G
. This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity
element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring
H*(G, \mathbbFp)H^\ast(G, \mathbb{F}_p) of G has the same spectrum as the ring of invariants of R
G
mod p
(RG ?\mathbbZ \mathbbFp)G(R_G \otimes_{\mathbb{Z}} \mathbb{F}_p)^G where the action of G is induced by conjugation. 相似文献
3.
Let R be a commutative ring and G a free R-module with finite rank e>0. For any R-submodule E⊂G one may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components En, n≥0, of the image, that we shall call the n-th Rees powers of E (with respect to the embedding E⊂G). In this work we prove some asymptotic properties of the R-modules En, n≥0, which extend well known similar ones for the case of ideals, among them Burch’s inequality for the analytic spread. 相似文献
4.
Endre Makai Jr. 《Periodica Mathematica Hungarica》2012,65(1):107-123
The non-trivial hereditary monocoreflective subcategories of the Abelian groups are the following ones: {G ?? Ob Ab | G is a torsion group, and for all g ?? G the exponent of any prime p in the prime factorization of o(g) is at most E(p)}, where E(·) is an arbitrary function from the prime numbers to {0, 1, 2, ??,??}. (o(·) means the order of an element, and n ?? ?? means n < ??.) This result is dualized to the category of compact Hausdorff Abelian groups (the respective subcategories are {G ?? Ob CompAb | G has a neighbourhood subbase {G ?? } at 0, consisting of open subgroups, such that G/G ?? is cyclic, of order like o(g) above}), and is generalized to categories of unitary R-modules for R an integral domain that is a principal ideal domain. For general rings R with 1, an analogous theorem holds, where the hereditary monocoreflective subcategories of unitary left R-modules are described with the help of filters L in the lattice of the left ideals of the ring R. These subcategories consist of those left R-modules, for which the annihilators of all elements belong to L. If R is commutative, then this correspondence between these subcategories and these filters L is bijective. 相似文献
5.
Frédéric Latour 《Journal of Pure and Applied Algebra》2005,195(1):97-112
In this paper, we classify the irreducible representations of the rational Cherednik algebras of rank 1 in characteristic p>0. There are two cases. One is the “quantum” case, where “Planck's constant” is nonzero and generic irreducible representations have dimension pr, where r is the order of the cyclic group contained in the algebra. The other is the “classical” case, where “Planck's constant” is zero and generic irreducible representations have dimension r. 相似文献
6.
Jesse Elliott 《Advances in Mathematics》2006,203(1):319-363
The Witt-Burnside ring of a profinite group G over a commutative ring A generalizes both the Burnside ring of virtual G-sets and the rings of universal and p-typical Witt vectors over A. The Witt-Burnside ring of G over the monoid ring Z[M], where M is a commutative monoid, is proved isomorphic to the Grothendieck ring of a category whose objects are almost finite G-sets equipped with a map to M that is constant on G-orbits. In particular, if A is a commutative ring and A× denotes the set A as a monoid under multiplication, then the Witt-Burnside ring of G over Z[A×] is isomorphic to Graham's ring of “virtual G-strings with coefficients in A.” This result forms the basis for a new construction of Witt-Burnside rings and provides an important missing link between the constructions of Dress and Siebeneicher [Adv. in Math. 70 (1988) 87-132] and Graham [Adv. in Math. 99 (1993) 248-263]. With this approach the usual truncation, Frobenius, Verschiebung, and Teichmüller maps readily generalize to maps between Witt-Burnside rings. 相似文献
7.
George Szeto 《Journal of Pure and Applied Algebra》1976,9(1):65-71
Let R be a commutative ring with identity 1, and A a finitely generated R-algebra. It is shown that A is an Azumaya R-algebra if and only if every stalk of the Pierce sheaf induced by A is an Azumaya algebra. 相似文献
8.
The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p−3 conjecture.” 相似文献
9.
Let R be a commutative ring with identity, and let | | be a non-trivial absolute value on R. We study the R-algebra norms on the polynomial algebra R[X
1, ... , X
r
] which extend | |.
Received: March 22, 2007. Revised: October 2, 2007. 相似文献
10.
Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ?(R) and ?(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that:
-
If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ?(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/?(R t G) ? R t 1 (G/G p ) as R-algebras;
-
If R is a ring of prime characterisitic p and R* is p-divisible, then ?(R t G) = 0 if and only if ?(R) = 0 and G p = 1; and
-
If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ? R t 1 H 0 as R-algebras.
11.
Mikhail Tkachenko 《Journal of Pure and Applied Algebra》2010,214(7):1103-1109
We show that every Abelian group G with r0(G)=|G|=|G|ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm-Kaplansky invariant fp,k of G satisfies (fp,k)ω=fp,k provided that fp,k is infinite and fp,k>fp,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups. 相似文献
12.
M. A. Turmanov 《Journal of Mathematical Sciences》2006,137(6):5336-5345
Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented
as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that
the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently
used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its
endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness
in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules
over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module
E
(G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary
ring R with the identity element. Indeed, if
R
M is an arbitrary unitary left module and M
+ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M
+) under the canonical ring homomorphism R → E(M
+). Then it holds that if
E(M+)
N is a pure submodule in
E(M+)
M
+, then
R
N is a pure submodule in
R
M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free
groups of finite rank will be investigated.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238,
2004. 相似文献
13.
Hamid Reza Maimani 《Journal of Pure and Applied Algebra》2008,212(1):168-174
Let R be a commutative ring with identity and let I be an ideal of R. Let R?I be the subring of R×R consisting of the elements (r,r+i) for r∈R and i∈I. We study the diameter and girth of the zero-divisor graph of the ring R?I. 相似文献
14.
We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra. 相似文献
15.
The Brauer-Clifford group BrClif(Z,G) corresponding to a finite group G and a finite-dimensional semisimple G-algebra Z was recently introduced by Alexandre Turull in the course of his work on character correspondence conjectures in group representation theory. This Brauer-Clifford group is a group of equivalence classes of Azumaya algebras over Z whose G-algebra structure agrees on restriction to the fixed (and usually nontrivial) G-algebra structure of Z. In this paper we extend the notion of the Brauer-Clifford group to the case of (S,H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is a commutative H-module algebra. These Brauer-Clifford groups turn out to be an example of the Brauer group of the symmetric monoidal category of S # H-modules, a perspective which allows one to construct a dual Brauer-Clifford group for the category of S-modules with compatible right H-comodule structure. 相似文献
16.
Ian Craw 《Journal of Functional Analysis》1978,27(2):170-178
Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X. 相似文献
17.
Timur I. Nezhmetdinov 《代数通讯》2013,41(12):4669-4681
Let R be a finite commutative ring with identity and ? p d be the cyclic group of prime power order. Define R? p d to mean the group ring of ? p d over R. We determine the structure of the group of units of R? p d in the case when R is generated by an element whose order is not divisible by p. 相似文献
18.
Éric Gaudron 《Journal of Number Theory》2007,127(2):220-261
We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called rational case. More precisely, let k be a number field and v0 be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u∈Lie(G(Cv0)) a logarithm of a point p∈G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)k⊗Cv0. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height loga of p, removing a polynomial term in logloga. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy. 相似文献
19.
N. A. Koreshkov 《Mathematical Notes》2010,88(1-2):39-47
In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ αεG A α graded by some group G, over a field of characteristic zero, has a nonzero component A 1 (where 1 stands for the identity element of G), and A 1 is a semisimple associative algebra. Let B = ⊕ αεG B α be a finite-dimensional semisimple Lie algebra over a prime field F p , and let B be graded by a commutative group G. If B = F p ? ? A L , where A L is the commutator algebra of a ?-algebra A = ⊕ αεG A α ; if ? ? ? A is an algebra of associative type, then the 1-component of the algebra K ? ? B, where K stands for the algebraic closure of the field F p , is the sum of some algebras of the form gl(n i ,K). 相似文献
20.
Erling Størmer 《Journal of Functional Analysis》1974,15(2):202-215
We study the group properties of the spectrum of a strongly continuous unitary representation of a locally compact Abelian group G implementing an ergodic group of 1-automorphisms of a von Neumann algebra . It is shown that in many cases the spectrum equals the dual group of G; e.g. if G is the integers and not finite dimensional and Abelian, then the spectrum is the circle group. 相似文献