An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander’s theorem on alg...

Based on an explicit representation of the Artin map for Kummer extensions, we present a method to compute arbitrary class fields. As in the proofs of the existence theorem, the problem is first reduced to the case where the field contains sufficiently many roots of unity. Using Kummer theory and an explicit version of the Artin reciprocity law we show how to compute class fields in this case. ...

We consider the kernel of the natural projection from the Artin group of dihedral type I2(k) to the associated Coxeter group, which we call a pure Artin group of dihedral type and write PI2(k). We show that the growth rates for both the spherical growth series and geodesic growth series of PI2(k) with respect to a natural generating set are Pisot numbers. 2010 Mathematics Subject Classification...

The goal of this paper is to define a new class of objects which we call triple affine Artin groups and to relate them with Cherednik’s double affine Hecke algebras. This has as immediate consequences new and simple descriptions of double affine Weyl and Artin groups, the double affine Hecke algebras as well as the corresponding elliptic objects. We also recover in an transparent and elementary...

Abstract. We construct several families of Artin-Schelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and CohenMacaulay domains. Many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an Ore extension of an Artin-Schelter regul...

Emil Artin was born on March 3, 1898 in Vienna, as the son of an art dealer and an opera singer, and he died on December 20, 1962 in Hamburg. He was one of the founding fathers of modern algebra. Van der Waerden acknowledged his debt to Artin and to Emmy Noether (1882–1935) on the title page of his Moderne Algebra (1930–31), which indeed was originally conceived to be jointly written with Artin...

The braid group of n-strings is the group of homotopy types of movements of n distinct points in the 2-plane R. It was introduced by E. Artin [1] in 1926 in order to study knots in R. He gave a presentation of the braid group by generators and relations, which are, nowadays, called the Artin braid relations. Since then, not only in the study of knots, the braid groups appear in several contexts...

In this paper, we study the bounded sum-of-digits discrete logarithm problem in finite fields. Our results concern primarily with fields Fqn where n|q− 1. The fields are called Kummer extensions of Fq. It is known that we can efficiently construct an element g with order greater than 2 in the fields. Let Sq(•) be the function from integers to the sum of digits in their q-ary expansions. We firs...

A simplicial complex L on n vertices determines a subcomplex TL of the ntorus, with fundamental group the right-angled Artin groupGL. Given an epimorphism χ : GL → Z, let T L be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of T L with coefficients in a field k, viewed as module...

In 1927, E. Artin proposed a conjectural density for the set of primes p for which a given integer g is a primitive root modulo p. After computer calculations in 1957 by D. H. and E. Lehmer showed unexpected deviations, Artin introduced a correction factor to explain these discrepancies. The modified conjecture was proved by Hooley in 1967 under assumption of the generalized Riemann hypothesis....