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1.
Periodicity in Group Cohomology and Complete Resolutions   总被引:1,自引:0,他引:1  
A group G is said to have periodic cohomology with period qafter k steps, if the functors Hi(G, –) and Hi+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in Hq(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in Hq(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in Hq(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.  相似文献   

2.
Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.  相似文献   

3.
On Towers Approximating Homological Localizations   总被引:2,自引:0,他引:2  
Our object of study is the natural tower which, for any givenmap f:AB and each space X, starts with the localization of Xwith respect to f and converges to X itself. These towers canbe used to produce approximations to localization with respectto any generalized homology theory E*, yielding, for example,an analogue of Quillen's plus-construction for E*. We discussin detail the case of ordinary homology with coefficients inZ/p or Z[1/p]. Our main tool is a comparison theorem for nullificationfunctors (that is, localizations with respect to maps of theform f:Apt), which allows us, among other things, to generalizeNeisendorfer's observation that p-completion of simply-connectedspaces coincides with nullification with respect to a Moorespace M(Z[1/p], 1).  相似文献   

4.
5.
The author considers globally defined h Fourier integral operators(h FIO) with complex-valued phase functions. Symbolic calculusof h FIO is considered and, using a new complex Gauss transform,the composition of h pseudodifferential operators (h PDO) andh FIO is considered. For a self-adjoint h PDO A(h) and h PDOP(h) and Q(h) with compactly supported symbols, the resultsare applied to approximate the kernel of the operator by a single, globally defined h-oscillatoryintegral. 2000 Mathematics Subject Classification 81Q20, 35S30.  相似文献   

6.
A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, NG(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of NG(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.  相似文献   

7.
Let k be an algebraically closed field of characteristic 2,and let W be the ring of infinite Witt vectors over k. Supposethat D is a dihedral 2-group. We prove that the universal deformationring R(D, V) of an endo-trivial kD-module V is always isomorphicto W [/2x/2]. As a consequence, we obtain a similar result formodules V with stable endomorphism ring k belonging to an arbitrarynilpotent block with defect group D. This confirms, for suchV, conjectures on the ring structure of the universal deformationring of V that had previously been shown for V belonging tocyclic blocks or to blocks with Klein four defect groups.  相似文献   

8.
Let V be a commutative valuation domain of arbitrary Krull-dimension(rank), with quotient field F, and let K be a finite Galoisextension of F with group G, and S the integral closure of Vin K. If, in the crossed product algebra K * G, the 2-cocycletakes values in the group of units of S, then one can form,in a natural way, a ‘crossed product order’ S *G K * G. In the light of recent results by H. Marubayashi andZ. Yi on the homological dimension of crossed products, thispaper discusses necessary and/or sufficient valuation-theoreticconditions, on the extension K/F, for the V-order S * G to besemihereditary, maximal or Azumaya over V. 2000 MathematicsSubject Classification 16H05, 16S35.  相似文献   

9.
This paper is concerned with non-trivial solvability in p-adicintegers of systems of two and three additive forms. Assumingthat the congruence equation axk + byk + czk d (modp) has asolution with xyz 0(modp) we have proved that any system oftwo additive forms of odd degree k with at least 6k + 1 variables,and any system of three additive forms of odd degree k withat least 14k + 1 variables always has non-trivial p-adic solutions,provided p does not divide k. The assumption of the solubilityof the congruence equation above is guaranteed for example ifp > k4. In the particular case of degree k = 5 we have proved the followingresults. Any system of two additive forms with at least n variablesalways has non-trivial p-adic solutions provided n 31 and p> 101 or n 36 and p > 11. Furthermore any system of threeadditive forms with at least n variables always has non-trivialp-adic solutions provided n 61 and p > 101 or n 71 andp > 11. 2000 Mathematics Subject Classification 11D72, 11D79.  相似文献   

10.
If two operator algebras A and B are strongly Morita equivalent(in the sense of [5]), then their C*-envelopes C*(A) and C*(B)are strongly Morita equivalent (in the usual C*-algebraic sensedue to Rieffel). Moreover, if Y is an equivalence bimodule fora (strong) Morita equivalence of A and B, then the operation,YhA–, of tensoring with Y, gives a bijection between theboundary representations of C*(A) for A and the boundary representationsof C*(B) for B. Thus the ‘noncommutative Choquet boundaries’of Morita equivalent A and B are the same. Other important objectsassociated with an operator algebra are also shown to be preservedby Morita equivalence, such as boundary ideals, the Shilov boundaryideal, Arveson's property of admissability, and the latticeof C*-algebras generated by an operator algebra. 1991 MathematicsSubject Classification 47D25, 46L05, 46M99, 16D90.  相似文献   

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