Low dimensional orders of finite representation type

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.

Extended plus closure in complete local rings

Skew power series rings with general commutation formula

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation

On a q-Analogue of the McKay Correspondence and the ADE Classification of [formula omitted]2 Conformal Field Theories

This paper is a natural continuation of the study of skew power series rings $A=R[[t;\sigma,\delta]]$ initiated in an earlier work. We construct skew Laurent series rings $B$ and show the existence of some canonical Ore sets $S$ for the skew power series rings $A$ such that a certain completion of the localization $A_S$ is isomorphic to $B.$ This is applied to certain Iwasawa algebras. Finally we introduce subrings of overconvergent skew Laurent series rings.

Given a complete, positively filtered ring (R, f) and a compatible skew derivation (σ, δ), we may construct its skew power series ring R[[x; σ, δ]]. Due to topological obstructions, even if δ is an inner σ-derivation, in general we cannot “untwist” it, i.e., reparametrise to find a filtered isomorphism R[[x; σ, δ]] ≌ R[[x′; σ]], as might be expected from the theory of skew polynomial rings; similarly when σ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of R[[x; σ, δ]] in the simplest case when R is a matrix ring over a (noncommutative) noetherian discrete valuation ring.

Many well-known local rings, including soluble Iwasawa algebras and certain completed quantum algebras, arise naturally as iterated skew power series rings. We calculate their Krull and global dimensions, obtaining lower bounds to complement the upper bounds obtained by Wang. In fact, we show that many common such rings obey a stronger property, which we call triangularity, and which allows us also to calculate their classical Krull dimension (prime length). Finally, we correct an error in the literature regarding the associated graded rings of general iterated skew power series rings, but show that triangularity is enough to recover this result.

On the codimension of modules over skew power series rings with applications to Iwasawa algebras

In this note we study the structures of power-serieswise Ar- mendariz rings and IFP rings when they are skewed by ring endomor- phisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the pro- cess. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings. Throughout this paper every ring is an associative ring with identity. Let σ be an endomorphism of a ring R. We use R((x;σ)) (resp. R(x;σ)) to denote the skew power series ring (resp. skew polynomial ring) with an indeterminate x over a ring R, subject to the relation xr = σ(r)x for r ∈ R. Note that σ(1) = 1 for any skew power series ring (skew polynomial ring) R((x;σ)) (R(x;σ)), since 1x n = x n = x1x n−1 = σ(1)x n for any n ≥ 1 where 1 is the identity of R. Armendariz (2, Lemma 1) proved that whenever polynomials f(x) = m X i=0 aix i , g(x) = n X j=0 bjx j

We classify Borel–Schur algebras having finite representation type. We also determine Auslander–Reiten sequences for a large class of simple modules over Borel–Schur algebras. A partial information on the structure of the socles of Borel-Schur algebras is given.

In this note we give a description of those infinitesimal group schemes of characteristic 2 whose Hopf algebras have finite representation type.

We prove that an artin algebra A is of finite representation type if and only if almost all finitely generated indecomposable A-modules are cycle-finite. Moreover, we prove that every cycle-finite module category of an artin algebra A of infinite representation type contains infinitely many left stable and infinitely many right stable directing modules. In particular, the main results of the paper provide solution of the open problem concerning infinity of directing modules over cycle-finite artin algebras of infinite representation type.

Integral representations of finite groups with finite representation type

Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation.Let A be a complete noetherian local ring, u, its maximal ideal, and k = A/u.the residue field.(In most applications A is k itself, or a ring of Witt vectors.)Let C be the category of Artin local A-algebras with residue field k.A covariant functor F from C to Sets is called pro-representable if it has the formwhere R is a complete local A-algebra such that R/mn is in C, all n.(m is the maximal ideal in R.)In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, )-> F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism.Theorem 2.11, 2, gives a criterion for F to have a hull, and also a simple criterion for pro-representability which avoids the use of Grothendieck's techniques of nonflat descent [3], in some cases.Grothendieck's program is carried out by Levelt in [4].3 contains a few geometric applications of these results.To avoid awkward terminology, I have used the word " pro-representable " in a more restrictive sense than Grothendieck [3] has.He considers the category of A-algebras of finite length and allows R to be a projective limit of such rings.The methods of this paper are a simple extension of those used by David Mumford in a proof (unpublished) of the existence of formal moduli for polarized Abelian varieties.I am indebted to Mumford and to John Tte for many valuable suggestions.1.The category CA.Let A be a complete noetherian local ring, with maximal ideal u. and residue field k = A/u.. We define C= CA to be the category of Artinian local A-algebras having residue field k. (That is, the "structure morphism" A -> ,4 of such a ring A induces a trivial extension of residue fields.)Morphisms in C are local homomorphisms of A-algebras.