On the Girth of Graphs Critical with Respect to Edge-Colourings

A graph G with maximum valency r is called critical if r + 1colours are needed for an edgecolouring, but every proper subgraphrequires at most r. In this note we consider the minimum orderf(r, g) of a critical graph of maximum valency r and girth g.We show that f(r, 3) = r+1 or r+2 according as r is even orodd, f(r, 4) = 2r+1,f(3, 5) = 9 and f(3, 6) = 15.     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

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.     

Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+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 H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(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 H^q(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.     

Deformation of Complete Intersections in the Plane

We study deformations of zero-dimensional complete intersectionsin the plane, and prove the following results. (1) Two complexnon-singular curves intersecting at r points with multiplicitiesd₁,...,d_r can be deformed into curves intersecting (at somepoints) with multiplicities d'₁,...,d'_s which are arbitraryprescribed partitions of the numbers d₁,...,d_r. (2) Two realcurves intersecting with multiplicity at most 2 at each of theirreal common points can be deformed so that all real multipleintersection points split into real simple intersection points.1991 Mathematics Subject Classification 14M10, 14P05.     

On Towers Approximating Homological Localizations

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).     

Exact Sequences of Semistable Vector Bundles on Algebraic Curves

Let X be a smooth complex projective curve of genus g 1. Ifg 2, then assume further that X is either bielliptic or withgeneral moduli. Fix integers r, s, a, b with r > 1, s >1 and as br. Here we prove the existence of an exact sequence [formula] of semistable vector bundles on X with rk(H) = r, rk(Q) = s,deg(H) = a and deg(Q) = b. 1991 Mathematics Subject Classification14H60.     

Universal deformation rings and dihedral 2-groups

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.     

Elliptic and Hyperelliptic Curves Over Supersimple Fields

It is proved that if F is an infinite field with characteristicdifferent from 2, whose theory is supersimple, and C is an ellipticor hyperelliptic curve over F with generic ‘modulus’,then C has a generic F-rational point. The notion of generityhere is in the sense of the supersimple field F.     

Morita Equivalence of Operator Algebras and Their C*-Envelopes

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,Y_hA–, 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.     

Conditions for the Solvability of Systems of Two and Three Additive Forms Over p-Adic Fields

This paper is concerned with non-trivial solvability in p-adicintegers of systems of two and three additive forms. Assumingthat the congruence equation ax^k + by^k + cz^k 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.     

Hyperkahler Structures and Group Actions

A 4n-dimensional Riemannian manifold (M, g) is hyperkählerif it possesses three anti-commuting complex structures I, J,K such that the metric g is Kähler with respect to eachof them. The reduced holonomy group of such a manifold is necessarilya subgroup of Sp(n) so the Ricci tensor of g vanishes and (M,g) can be regarded as a positive definite solution to Einstein'sequations in vacuum.     

Crossed Product Orders over Valuation Rings

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.     

Splendid Derived Equivalences for Blocks of Finite Groups

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, N_G(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 N_G(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.     

Actions of Commutative Hopf Algebras

We show that actions of finite-dimensional semisimple commutativeHopf algebras H on H-module algebras A are essentially group-gradings.Moreover we show that the centralizer of H in the smash productA # H equals A^H H. Using these we invoke results about groupgraded algebras and results about centralizers of separablesubalgebras to give connections between the ideal structureof A, A^H and A # H. Examples of the above occur naturally when one considers: (1) finite abelian groups G of automorphisms of an algebra Awith | G |^–1 A; (2) G-graded algebras, for finite groups G; (3) finite-dimensional restricted Lie algebras L, with semisimplerestricted enveloping algebra u(L), acting as derivations onan algebra A.     

Dualizing Complex and the Canonical Element Conjecture II

In this paper we continue our study of the Canonical ElementConjecture (henceforth C.E.C.) via the dualizing complex. Throughoutthe work (A, m, k) will denote a noetherian complete local ringA of dimension n, m its maximal ideal and k=A/m. Since A iscomplete, we can find a complete local Gorenstein ring (R, m_R,k) (complete intersection) such that dim R=dim A and A=R/I.Let denote the canonical module of A, that is, =Hom_R (A, R),which may be identified with the annihilator of I in R, an idealof R. When A is a domain, we change notation and denote I byP; in this case P is a height 0 prime ideal of R.     

The Hausdorff Dimension of Julia Sets of Entire Functions IV

It is known that, for a transcendental entire function f, theHausdorff dimension of J(f) satisfies 1 dimJ(f) 2. For eachd (1, 2), an example of a transcendental entire function fwith dimJ(f) = d is given. It is then indicated how this functioncan be modified to produce a transcendental meromorphic functionF with one pole with dimJ(F) = d. These appear to be the firstexamples of Julia sets with non-integer dimensions whose dimensionshave been calculated exactly.     

Global h Fourier Integral Operators with Complex-Valued Phase Functions

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.     

Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line

We study the operator Lu(t):= u'(t) – A(t) u(t) on L^p(R; X) for sectorial operators A(t), with t R, on a Banachspace X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreniconditions, and have the property of maximal L^p-regularity.We establish optimal regularity on the time interval R showingthat L is closed on its minimal domain. We further give conditionsfor ensuring that L is a semi-Fredholm operator. The Fredholmproperty is shown to persist under A(t)-bounded perturbations,provided they are compact or have small A(t)-bounds. We applyour results to parabolic systems and to generalized Ornstein–Uhlenbeckoperators. 2000 Mathematics Subject Classification 35K20, 35K90,47A53.     

The Steinberg Lattice of a Finite Chevalley Group and its Modular Reduction

Let p be a prime and let q = p^a, where a is a positive integer.Let G 7equals; G(F_q) be a Chevalley group over F_q, with associatedsystem of roots and Weyl group W. Steinberg showed in 1957that G has an irreducible complex representation whose degreeequals the p-part of |G| [11]. This representation, now knownas the Steinberg representation, has remarkable properties,which reflect the structure of G, and there have been many researchpapers devoted to its study. The module constructed in [11]is in fact a right ideal in the integral group ring ZG of G,and is thus a ZG-lattice, which we propose to call the Steinberglattice of G. It should be noted that lattices not integrallyisomorphic to the Steinberg lattice may also afford the Steinbergrepresentation, and such lattices may differ considerably intheir properties compared with the Steinberg lattice.