Milnor K-theory of Smooth Varieties

Let k be a field and X a smooth projective variety of dimension d over k. Generalizing a construction of Kato and Somekawa, we define a Milnor-type group which is isomorphic to the ordinary Milnor We prove that is isomorphic to both the higher Chow group CH^d+s (X,s) and the Zariski cohomology group      

On universal nilpotent groups

AbstractWe show that some classes of nilpotent groups have countable universal members and others do not.Presented by Bjarni Jónsson.     

On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

On stable torsion-free nilpotent groups

Summary We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z _k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.Supported by the Deutsche Forschungsgemeinschaft     

On multipliers of torsion-free nilpotent groups

The notion of the Schur multiplier is carried over to torsion-free nilpotent groups of finite rank, and the relation between the rank of a torsion-free nilpotent group and the rank of its multiplier is determined, [3].Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 541–544, May, 1969.     

On group rings of nilpotent groups

It is shown that ifG is a non-abelian torsion free nilpotent group andF is a field, then the classical skew field of fractionsF(G) of the group ring,F[G] contains a noncommutative free subalgebra. The author is supported by NSF Grant No. MCS-8201115.     

On some near to nilpotent groups

Locally (soluble-by-finite) groups each of whose subgroups is either pronormal or subnormal are studied. It is proven that such torsion-free groups are nilpotent, and a sketch of the structure of locally finite groups of this type is obtained.     

On the existence of stably nilpotent subgroups in locally nilpotent groups

A condition for the existence of stably nilpotent subgroups of a given nilpotency class in a locally nilpotent group is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 361–368, September, 1968.     

On subgroups of residually nilpotent metabelian groups

ABSTRACT

We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006 Faith , C. ( 2006 ). Factor rings of pseudo-Frobenius rings . J. Algebra and Its Applications 6 :(to appear). [CSA] [Web of Science ®] , [Google Scholar]). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5).

These results also generalize results of the author in Faith (1985 Faith , C. ( 1985 ). The maximal regular ideal of self-injective and continuous rings splits off . Arch. Math. 44 : 511 – 521 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring.

The results are applied in Section 6 to extend theorems of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) characterizing VNR rings, and, as the title of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) suggests, extend the conjecture of Shamsuddin.     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].     

On Coleman automorphisms of finite nilpotent groups by cyclic groups

Let G be a finite group and let N be a nilpotent normal subgroup of G such that G/N is cyclic. It is shown that under some conditions all Coleman automorphisms of G are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.