ON TWIN-TRIANGULAR ADDITIVE GROUP ACTIONS ON C4

This paper concerns some of the conditions satisfied by additive group actions on complex affine space which can be written locally as a translation of a variable. Assume X is the affine variety C ⁿ , G_a = (C, +), and σ : G_a × X → X is the action defined by a group monomorphism G _a → Aut _C X. If σ is locally trivial, then the action satisfies what is termed a “GICO” condition.

It will be shown that a large class of G_a -actions on C ⁴, that is, fixed-point free, “twin-triangular” actions with finitely-generated rings of invariants, are at least GICO. Remaining questions are discussed.     

A note on hopfian modules *

A proper regular action of the additive group of complex numbers on complex affine space is known to be locally trivial if the coordinate ring of affine space is a flat extension of the ring of G _a invariants. Conditions under which a separated action is locally trivial are examined (Section 2) and an algorithm to determine the local triviality of any given action is presented (Section 3).     

G a Invariants and Slices

Criteria for local triviality of algebraic actions of the additive group of complex numbers on complex affine space are extended to more general varieties. Finite generation of rings of invariants of locally trivial actions on factorial affine varieties is dicussed, giving some sufficient conditions for finite generation and examples where finite generation fails. A missing hypothesis in a theorem of Miyanishi is identified, and an example is given to demonstrate the necessity of the hypothesis. The corrected theorem is shown to hold for a class of triangular G _a actions on C⁴, with the consequence that all these actions are conjugate to translations. A new criterion for an action to be conjugate to a global translation is given.     

Triangular actions onC 3

Free algebraic actions of a connected algebraic groupG onC ³ which can be triangularized are shown to be trivial, that isC ³ is equivariantly isomorphic toGxC ^3–dimG . This result follows directly from the case of the additive groupG=G _a and is shown to hold for quasi-algebraic actions as well. Connections with the classification of homogeneous affine varieties are discussed.Partially supported by NSF grant DMS 8420315     

The biweight enumerator of self-orthogonal binary codes

Let C be a binary code of length n and let J_C (a, b, c, d) be its biweight enumerator. If n is even and C is self-dual, then J_C is an element of the ring R^$\text{G}$ of absolute invariants of a certain group $\text{G}$. Under the additional assumption that all codewords of C have weight divisible by 4, a similar result holds with a different group. If n is odd and C is maximal self-orthogonal, then J_C is an element of a certain R^$\text{G}$-module. Again a similar result holds if the codewords of C have weights divisible by 4. The groups involved are related to finite groups generated by reflections. In this paper the structure of these groups is described, and polynomial bases for the rings and modules in question are obtained. This answers a question posed in The Theory of Error- correcting Codes by F.J. MacWilliams and N.J.A. Sloane.     

The automorphism group of certain factorial threefolds and a cancellation problem

A new class of counterexamples to a generalized cancellation problem for affine varieties is presented. Each member of the class is an affine factorial complex threefold admitting a locally trivial action of the additive group, hence the total space for a principal G _a bundle over a quasiaffine base. The automorphism groups for these varieties are also determined.     

Conjugacy Classes and Invariant Subrings of R-Automorphisms of R[x]

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ?₂ and m ² = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.     

Affine normal surfaces with simply-connected smooth locus

Given a normal affine surface V defined over \mathbbC{\mathbb{C}}, we look for algebraic and topological conditions on V which imply that V is smooth or has at most rational singularities. The surfaces under consideration are algebraic quotients \mathbbCⁿ/G{\mathbb{C}^n/G} with an algebraic group action of G and topologically contractible surfaces. Theorem 3.6 can be considered as a global version of the well-known result of Mumford giving a smoothness criterion for a germ of a normal surface in terms of the local fundamental group.     

Affine \mathbb{T} -varieties of complexity one and locally nilpotent derivations

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus \mathbbT \mathbb{T} of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine \mathbbZⁿ {\mathbb{Z}^n} -graded domain A, so that ∂ generates a k ₊-action on X that is normalized by the \mathbbT \mathbb{T} -action.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

Specialization of a generic hyperplane section through a rational point of an algebraic variety

Summary Let V/k be an irreducible algebraic variety of dimension ≥3, defined over a field k of characteristic O, passing through the origin (O) of the ambient affine n-space. Let p be the prime ideal of V in the polynomial ring k[X₁, ..., X_n]. If V/k is k-normal at (O), then for almost all hyperplanes H_a: a₁X₁+...+a_nX_n=0 with a₁, ..., a_n ε k the ideal (p, a₁X₁+...+a_nX_n) is a prime ideal. If the generic hyperplane section of V through (O) is normal over the ground field of the generic section of V, then the section V ∩ H_a is k-normal at (O) for almost all H_a. Entrata in Redazione il 1 ottobre 1971.     

ON A CONNECTION BETWEEN THE SCHUR MULTIPLIER AND THE BRAUER GROUP OF A FIELD

ABSTRACT

Let k be a field and suppose ζ_n is a primitive n-th root of unity contained in k. We denote by Sr(k) = H² (G_k,C^x) the Schur multiplier of k. We observe: If the n-part of Sr(k) is trivial, then every element of order n in the Brauer group Br(k) can be represented as a cyclic algebra of the form

(E,σζ_n)

with the same ζn throughout. We apply this in particular to the case of a global field k and conclude by discussing the Schur multipliers of certain other types of fields.     

ON CERTAIN NEW RADICALS ARISING FROM A GIVEN RADICAL

Abstract

If R is a ring and n is an integer weMaydefine a ring T_n (R) on the same underlying additive abelian group by using the formula a * b = nab to define a new multiplication. T_n , is a functor on the category of associative rings. If C is a class of rings then, for each n, the class C_n , is defined to consist of all rings R such that T_n (R) is in C. If C is a radical class then each class C_n , is also a radical class. We consider the properties of the radical class C which are inherited by C_n , and relationships between these classes C _n as n varies.     

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999     

Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Let A(C) be the coordinate ring of a monomial curve C⊆Aⁿ corresponding to the numerical semigroup S minimally generated by a sequence a₀,…,a_n. In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr_m(A) with respect to the maximal ideal m of A=A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr_m(A) in the case a₀,…,a_n is a generalized arithmetic sequence.     

On the groups of unitriangular automorphisms of relatively free groups

We describe the structure of the group U _n of unitriangular automorphisms of the relatively free group G _n of finite rank n in an arbitrary variety C of groups. This enables us to introduce an effective concept of normal form for the elements and present U _n by using generators and defining relations. The cases n = 1, 2 are obvious: U ₁ is trivial, and U ₂ is cyclic. For n ?? 3 we prove the following: If G _n?1 is a nilpotent group then so is U _n . If G _n?1 is a nilpotent-by-finite group then U _n admits a faithful matrix representation. But if the variety C is different from the variety of all groups and G _n?1 is not nilpotent-by-finite then U _n admits no faithful matrix representation over any field. Thus, we exhaustively classify linearity for the groups of unitriangular automorphisms of finite rank relatively free groups in proper varieties of groups, which complements the results of Olshanskii on the linearity of the full automorphism groups AutG _n . Moreover, we introduce the concept of length of an automorphism of an arbitrary relatively free group G _n and estimate the length of the inverse automorphism in the case that it is unitriangular.     

Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=G _R with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {u _t} ofG and anyg∈G, the time spent inC by the {u _t}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g ⁻¹u_tg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].