Quantum homogeneous spaces with faithfully flat module structures

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB. In the last few years many interesting examples of quantumG-spaces for concrete quantum groupsG have been constructured by Podleś, Noumi, Dijkhuizen and others (as analogs of classical compact symmetric spaces). In these examplesB consists of infinitesimal invariants of the function algebraA of the quantum group. As a consequence of a general theorem we show that in all these casesA as a left or rightB-module is faithfully flat. Moreover, the coalgebraA/AB ⁺ is cosemisimple.     

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σ _g∈G R _g be any associative algebra over a field such that the subspacesR _g satisfyR _g R _h ⊆R _gh . We prove that ifR ₁ satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withR ^H satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.     

P.I.G-rings and the contractability of primes

LetR=F{x ₁, …, x_k} be a prime affine p.i. ring andS a multiplicative closed set in the center ofR, Z(R). The structure ofG-rings of the formR _s is completely determined. In particular it is proved thatZ(R _s)—the normalization ofZ(R _s) —is a prüfer ring, 1≦k.d(R _s)≦p.i.d(R _s) and the inequalities can be strict. We also obtain a related result concerning the contractability ofq, a prime ideal ofZ(R) fromR. More precisely, letQ be a prime ideal ofR maximal to satisfyQϒZ(R)=q. Then k.dZ(R)/q=k.dR/Q, h(q)=h(Q) andh(q)+k.dZ(R)/q=k.dz(R). The last condition is a necessary butnot sufficient condition for contractability ofq fromR.     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

A reciprocity theorem for unitary representations of Lie groups

LetG be a Lie group,H a closed subgroup,L a unitary representation ofH andU ^L the corresponding induced representation onG. The main result of this paper, extending Ol’ŝanskii’s version of the Frobenius reciprocity theorem, expresses the intertwining number ofU ^L and an irreducible unitary representationV ofG in terms ofL and the restriction ofV _∞ toH.     

AnR danalogue of Valentine’s theorem on 3-convex sets

This paper deals with anR ^danalogue of a theorem of Valentine which states that a closed 3-convex setS in the plane is decomposable into 3 or fewer closed convex sets. In Valentine’s proof, the points of local nonconvexity ofS are treated as vertices of a polygonP contained in the kernel ofS, yielding a decomposition ofS into 2 or 3 convex sets, depending on whetherP has an even or odd number of edges. Thus the decomposition actually depends onc(P′), the chromatic number of the polytopeP′ dual toP. A natural analogue of this result is the following theorem: LetS be a closed subset ofR ^d, and letQ denote the set of points of local nonconvexity ofS. We require thatQ be contained in the kernel ofS and thatQ coincide with the set of points in the union of all the (d − 2)-dimensional faces of somed-dimensional polytopeP. ThenS is decomposable intoc(P′) closed convex sets.     

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG ⁿ →G ⁿ/Hom _R (M, G) is an injective precover if Ext _R ¹ (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depth_R R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depth_R R > 0.     

Subgroups of prescribed finite index in linear groups

LetR be a commutative ring,M a finitely generatedR-module andG a subgroup of Aut _R M. Under either of the following conditions, for every positive integerd there is a normal subgroupH ofG of finite index such thatG/H contains an element of orderd. (a)G is infinite and finitely generated. (b)R is finitely generated as a ring andG is not unipotent-by-finite. This extends recent work of A. Lubotzky.     

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerp ⁿ, theH _p ⁿ -subgroupH _pn (G) is defined byH _p ⁿ (G)=〈xεG|x ^pn≠1〉. A groupH≠1 is called aH _p ⁿ -group, if there is a finite groupG such thatH is isomorphic toH _p ⁿ (G) andH _p ⁿ (G)≠G. It is known that the Fitting length of a solvableH _p ⁿ -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH _p ⁿ -groups having Fitting lengthn, this result is “almost” best possible.     

Jacobson rings and rings strongly graded by an abelian group

LetR be ring strongly graded by an abelian groupG of finite torsion-free rank. Lete be the identity ofG, andR _e the component of degreee ofR. AssumeR _e is a Jacobson ring. We prove that graded subrings ofR are again Jacobson rings if eitherR _e is a left Noetherian ring orR is a group ring. In particular we generalise Goldie and Michlers’s result on Jacobson polycyclic group rings, and Gilmer’s result on Jacobson commutative semigroup rings of finite torsion-free rank.     

Unitary units in modular group algebras

Letp be a prime,K a field of characteristicp, G a locally finitep-group,KG the group algebra, andV the group of the units ofKG with augmentation 1. The anti-automorphismg→g ⁻¹ ofG extends linearly toKG; this extension leavesV setwise invariant, and its restriction toV followed byv→v ⁻¹ gives an automorphism ofV. The elements ofV fixed by this automorphism are calledunitary; they form a subgroup. Our first theorem describes theK andG for which this subgroup is normal inV. For each elementg inG, let denote the sum (inKG) of the distinct powers ofg. The elements 1+(g-1) withh,hεG are thebicyclic units ofKG. Our second theorem describes theK andG for which all bicyclic units are unitary. Research partly supported by the Hungarian National Foundation for Scientific Research grant no. T4265. The second author is indebted to the ‘Universitas’ Foundation and the Lajos Kossuth University of Debrecen, Hungary, for warm hospitality and generous support during the period when this work began. This article was processed by the authors using the Springer-Verlag T_EX mamath macro package 1990.     

Centralizers satisfying polynomial identities

The following results are proved: IfR is a simple ring with unit, and for someaεR witha ⁿ in the center ofR, anyn, such that the centralizer ofa inR satisfies a polynomial identity of degreem, thenR satisfies the standard identity of degreenm. WhenR is not simple,R will satisfy a power of the same standard identity, provided thata andn are invertible inR. These theorems are then applied to show that ifG is a finite solvable group of automorphisms of a ringR, and the fixed points ofG inR satisfy a polynomial identity, thenR satisfies a polynomial identity, providedR has characteristic 0 or characteristicp wherep✗|G|. This research was supported in part by NSF Grant No. GP 29119X.     

Extension of positive functionals on certain ordered spaces

An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL. A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism. These properties were introduced by Schmidt in [5]. In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM. An analogous theorem is derived for Riesz spaces with extension property (E2).     

Prime ideals in crossed products of finite groups

LetR * G be a crossed product of the finite groupG over the ringR. In this paper we discuss the relationship between the prime ideals ofR*G and theG-prime ideals ofR. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime idealsP ofR*G such thatP∩R is a fixedG-prime ideal ofR. As an application, we prove that ifG is a finite group of automorphisms ofR, then the prime (primitive) ranks ofR and of the fixed ringR ^G are equal provided •G•⁻∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group.     

On the rate of points in projective spaces

The rate of a standard gradedK-algebraR is a measure of the growth of the shifts in a minimal free resolution ofK as anR-module. It is known that rate(R)=1 if and only ifR is Koszul and that rate(R) ≥m(I)−1 wherem(I) denotes the highest degree of a generator of the defining idealI ofR. We show that the rate of the coordinate ring of certain sets of pointsX of the projective space P ⁿ is equal tom(I)−1. This extends a theorem of Kempf. We study also the rate of algebras defined by a space of forms of some fixed degreed and of small codimension.     

Universal finite group extensions and a non-splitting theorem

LetG andK be finite groups whose orders have a common prime divisor. Then there is a groupK ^* closely related toK for which there is a non-split extension ofK ^* byG. I wish to express thanks to the Mathematics Institute of the Hebrew University of Jerusalem for its hospitality from September to December 1972, and to Dr. Avinoam Mann for his helpful comments.     

On aschbacher’s localC(G; T) theorem

Aschbacher’s localC(G; T) theorem asserts that ifG is a finite group withF*(G)=O ₂(G), andTεSyl₂(G), thenG=C(G; T)K(G), whereC(G; T)=〈N _G (T ₀)|1≠T ₀ charT〉 andK(G) is the product of all near components ofG of typeL ₂(2 ⁿ ) orA ₂ ⁿ ₊₁. Near components are also known asχ-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher’s theorem in the case thatG is aK-group, i.e., in the case that every simple section ofG is isomorphic to one of the known simple groups. Our proof relies on a result of Meierfrankenfeld and Stroth [MS] on quadratic four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained. For John Thompson Supported in part by NSF grant #DMS 89-03124, by DIMACS, an NSF Science and Technology Center, funded under contract STC-88-09648, and by NSA grant #MDA-904-91-H-0043. Prof. Gorenstein died on August 26, 1992.     

Déformations formelles de revêtements: un principe local-global

LetC be a generically smooth, locally complete intersection curve defined over an algebraically closed fieldk of characteristicp≥0. LetG⊃ Aut _k C be a finite group of automorphisms ofC. We develop a theory ofG-equivariant deformations of the Galois coverC→C/G. We give a thorough study of the local obstructions, those localized at singular or widely ramified points, to deform equivariantly a cover. As an application, we discuss the case ofG-equivariant deformations of semistable curves.      

Algorithms for ham-sandwich cuts

Given disjoint setsP ₁,P ₂, ...,P _d inR ^d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP _i . We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets inR ^d−1 . With the current estimates, we get complexity close toO(n ^3/2 ) ford=3, roughlyO(n ^8/3 ) ford=4, andO(n ^d−1−a(d) ) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR ³ when the three sets are suitably separated. A preliminary version of the results of this paper appeared in [16] and [17]. Part of this research by J. Matoušek was done while he was visiting the School of Mathematics, Georgia Institute of Technology, Atlanta, and part of his work on this paper was supported by a Humboldt Research Fellowship. W. Steiger expresses gratitude to the NSF DIMACS Center at Rutgers, and his research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

Integral extensions of noncommutative rings

In this paper we study integral extensions of noncommutative rings. To begin, we prove that finite subnormalizing extensions are integral. This is done by proving a generalization of the Paré-Schelter result that a matrix ring is integral over the coefficient ring. Our methods are similar to those of Lorenz and Passman, who showed that finite normalizing extensions are integral. As corollaries we note that the (twisted) smash product over the restricted enveloping algebra of a finite dimensional restricted Lie algebra is integral over the coefficient ring and then prove a Going Up theorem for prime ideals in these ring extensions. Next we study automorphisms of rings. In particular, we prove an integrality theorem for algebraic automorphisms. Combining group gradings and actions, we show that if a ringR is graded by a finite groupG, andH is a finite group of automorphisms ofR that permute the homogeneous components, with the order ofH invertible inR, thenR is integral overR ₁ ^H , the fixed ring of the identity component. This, in turn, is used to prove our final result: Suppose that ifH is a finite dimensional semisimple cocommutative Hopf algebra over an algebraically closed field of positive characteristic. IfR is anH-module algebra, thenR is integral overR ^H , its subring of invariants.