Infinite sums of unstable Adams operations and cobordism

The elements of the ring of bidegree (0, 0) additive unstable operations in complex K-theory can be described explicitly as certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for complex cobordism MU, thus identifying the “Adams subring” of the corresponding ring of cobordism operations. We prove that the Adams subring is the centre of the ring of bidegree (0, 0) additive unstable cobordism operations.For an odd prime p, the analogous result in the p-local split setting is also proved.     

Invariants of Finite Group Schemes

Let G be a finite group scheme operating on an algebraic varietyX, both defined over an algebraically closed field k. The paperfirst investigates the properties of the quotient morphism X- X/G over the open subset of X consisting of points whose stabilizershave maximal index in G. Given a G-linearized coherent sheafon X, it describes similarly an open subset of X over whichthe invariants in the sheaf behave nicely in some way. The pointsin X with linearly reductive stabilizers are characterized inrepresentation theoretic terms. It is shown that the set ofsuch points is nonempty if and only if the field of rationalfunctions k(X) is an injective G-module. Applications of theseresults to the invariants of a restricted Lie algebra g operatingon the function ring k[X] by derivations are considered in thefinal section. Furthermore, conditions are found ensuring thatthe ring k[X]^g is generated over the subring of pth powers ink[X], where p=char,k>0, by a given system of invariant functionsand is a locally complete intersection.     

Schur multipliers and the Lazard correspondence

Let G be a finite p-group of nilpotency class less than p?1, and let L be the Lie ring corresponding to G via the Lazard correspondence. We show that the Schur multipliers of G and L are isomorphic as abelian groups and that every Schur cover of G is in Lazard correspondence with a Schur cover of L. Further, we show that the epicenters of G and L are isomorphic as abelian groups. Thus the group G is capable if and only if the Lie ring L is capable.     

Algebraic cobordism revisited

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson–Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson–Thomas theory: absolute, relative, and equivariant.     

For complex orientations preserving power operations, p-typicality is atypical

We show, for primes p?13, that a number of well-known MU(p)-rings do not admit the structure of commutative MU(p)-algebras. These spectra have complex orientations that factor through the Brown-Peterson spectrum and correspond to p-typical formal group laws. We provide computations showing that such a factorization is incompatible with the power operations on complex cobordism. This implies, for example, that if E is a Landweber exact MU(p)-ring whose associated formal group law is p-typical of positive height, then the canonical map MU(p)→E is not a map of H_∞ ring spectra. It immediately follows that the standard p-typical orientations on BP,E(n), and En do not rigidify to maps of E_∞ ring spectra. We conjecture that similar results hold for all primes.     

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory (basic examples are equivariant Chow groups of Edidin and Graham, Thomason??s equivariant K-theory and equivariant algebraic cobordism). Using the language of equivariant pretheories we generalize the theorem of Karpenko and Merkurjev on G-torsors and rational cycles. As an application, to every G-torsor E and a G-equivariant pretheory we associate a ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information about the motivic J-invariant of E, in the case of Grothendieck??s K ₀ indexes of the respective Tits algebras and in the case of algebraic cobordism ?? it gives a quotient of the cobordism ring of G.     

INVARIANT FIELDS AND LOCALIZED INVARIANT RINGS OF p-GROUPS

It is well known that for a p-group, the invariant field ispurely transcendental (T. Miyata, Invariants of certain groupsI, Nagoya Math. J. 41 (1971), 69–73). In this note, weshow that a minimal generating set of this field can be chosenas homogeneous invariants from the invariant ring. As a result,we show that the invariant ring localized at one suitable invariantis the localization of a polynomial subring at this same invariant.This second result is a generalization of a recent result ofthe first author for cyclic groups of order p (H. E. A. Campbell,Rings of invariants of representations of C_p in characteristicp, preprint, 2006). As well, we specialize these results tothis latter case.     

The permutation projective dimension of the quaternionic group

Abstract

Let R be a commutative Noetherian local Gorenstein ring with residue field k. We show that G(k), the Gorenstein injective envelope of k, is an artinian R-module, and we compute G(k) in the case where R = k[[S]] is a semigroup ring and S is symmetric. We also show that a certain subring of the endomorphism ring of G(k) is a complete local (but possibly non-commutative) ring.     

On Quasi-Regular Torsion-Free Modules

A torsion-free module is called quasi-regular if each cyclic submodule is a quasi-summand. This article characterizes torsion-free Abelian groups that are quasi-regular as modules over a subring of their endomorphism ring. In particular, if G is a torsion-free Abelian group such that its ring Q E of quasi-endomorphisms is Artinian, then the left E-module G is quasi-regular if and only if the left C-module G is quasi-regular, where C is the center of its endomorphism ring E.     

Skew Group Rings which are Semi-Hereditary Orders and Prüfer Orders in Simple Artinian Rings

Let R be a commutative ring, let G be a finite group acting on R as automorphisms of R and let R * G be the skew group ring. By using the decomposition subgroups of G, the inertial subgroups of G, the properties of the coefficient ring R and the properties of the fixed subring R ^G , some necessary and sufficient conditions for R * G to be a prime Goldie ring, a semi-hereditary order in a simple Artinian ring, or a Prüfer order in a simple Artinian ring are given.     

Embedding Properties of Metabelian Lie Algebras and Metabelian Discrete Groups

We show that for every natural number m a finitely generatedmetabelian group G embeds in a quotient of a metabelian groupof type FP_m. Furthermore, if m 4, the group G can be embeddedin a metabelian group of type FP_m. For L a finitely generatedmetabelian Lie algebra over a field K and a natural number mwe show that, provided the characteristic p of K is 0 or p >m, then L can be embedded in a metabelian Lie algebra of typeFP_m. This result is the best possible as for 0 < p m everymetabelian Lie algebra over K of type FP_m is finite dimensionalas a vector space.     

Shephard–Todd–Chevalley Theorem for Skew Polynomial Rings

We prove the following generalization of the classical Shephard–Todd–Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring \(A:=k_{p_{ij}}[x_1,\cdots,x_n]\). Then the fixed subring A ^G has finite global dimension if and only if G is generated by quasi-reflections. In this case the fixed subring A ^G is isomorphic to a skew polynomial ring with possibly different p _ij ’s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.     

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.     

Locally Finite Finitary Skew Linear Groups

Let V be a vector space over the division ring D of infinitedimension. We study locally finite, primitive groups G of finitarylinear automorphisms of V. We show that the derived group G'of G is infinite, simple, and lies in every non-trivial normalsubgroup of G, and that G' G Aut G'. Moreover if char D =0, then G is either the finitary symmetric group or the alternatinggroup on some infinite set. If D is commutative, that is, ifD is a field, then all these results are known and are the consequenceof the collective work of a number of people: in particular(in alphabetical order) V. V. Belyaev, J. I. Hall, F. Leinen,U. Meierfrankenfeld, R. E. Phillips, O. Puglisi, A. Radfordand quite probably others. 2000 Mathematics Subject Classification:20H25, 20H20, 20F50.     

Constructing Witt-Burnside rings

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.     

Symmetric Elements Under Oriented Involutions in Group Rings

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ? of the group G, an oriented invoultion ^σ? of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.     

Towers of MU-Algebras and the Generalized Hopkins-Miller theorem

Our results are of three types. First, we describe a generalprocedure of adjoining polynomial variables to A-ring spectrawhose coefficient rings satisfy certain restrictions. A hostof examples of such spectra is provided by killing a regularideal in the coefficient ring of MU, the complex cobordism spectrum.Second, we show that the algebraic procedure of adjoining rootsof unity carries over in the topological context for such spectra.Third, we use the developed technology to compute the homotopytypes of spaces of strictly multiplicative maps between suitableK(n)-localizations of such spectra. This generalizes the famousHopkins–Miller theorem and gives strengthened versionsof various splitting theorems. 2000 Mathematics Subject Classification55N20, 55S35, 55T25 (primary), 16E40, 13D10 (secondary).     

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_q[G] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_q[G] is Noetherian recovers aresult due to Joseph [10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.     

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P(G) over some topological group G if and only if S embeds into the semigroup exp(G)\exp(G) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F(G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F(G) contains a G-invariant element (which is an analogue of the Haar measure on G).