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.

We prove the simple fact that the factor ring of a Koszul algebra by a regular, normal, quadratic element is a Koszul algebra. This fact leads to a new construction of quadratic Artin-Schelter regular algebras. This construction generalizes the construction of Artin-Schelter regular Clifford algebras. 1991 Mathematics Subject Classification. 16W50, 14A22.

We show that the Mathieu groups M22 and M11 can act on the supersingular K3 surface with Artin invariant 1 in characteristic 11 as symplectic automorphisms. More generally we show that all maximal subgroups of the Mathieu group M23 with three orbits on 24 letters act on a supersingular K3 surface with Artin invariant 1 in a suitable characteristic.

The Tits conjecture claims that the subgroup generated by the squares of the standard generators of an Artin group can be presented using only the obvious relations. We show that the Tits conjecture is true for the Artin groups of type Bn.

COMMENSURATORS OF RIGHT-ANGLED ARTIN GROUPS AND MAPPING CLASS GROUPS MATT CLAY, CHRISTOPHER J. LEININGER, AND DAN MARGALIT Abstract. We prove that, aside from the obvious exceptions, the mapping class We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to...

We describe the structure of semi-regular Auslander-Reiten components of artin algebras without external short paths in the module category. As an application we give a complete description of self-injective artin algebras whose Auslander-Reiten quiver admits a regular acyclic component without external short paths.

We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee [3]. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).

Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form small cancellation theory we show that the groups in this family are bi-automatic.

We define motivic Artin L-functions and show that they specialize to the usual Artin L-functions under the trace of Frobenius. In the last section we use our L-functions to prove a motivic analogue of the Chebotarev density theorem.

The invariance of the Artin root number under induction can be proved without special effort. In fact if one develops the properties of Artin Lfunctions in the usual way, then the inductivity of the root number is obvious. In barest outline the argument is as follows. First one proves the inductivity of Artin L-functions themselves, and then one combines Brauer’s induction theorem with the anal...