F. Van Oystaeyen

The wedderburn-malcev theorem for comodule algebras

Let H be a Hopf algebra over a field k. Under some assumptions on H we state and prove a generalization of the Wedderburn-Malcev theorem for i7-comodule algebras. We show that our version of this theorem holds for a large enough class of Hopf algebras, such as coordinate rings of completely reducible affine algebraic groups, finite dimensional Hopf algebras over fields of characteristic 0 and group algebras. Some dual results are also included.

Graded almost noetherian rings and applications to coalgebras

It is well-known that the dual algebra of a coalgebra C is a topological algebra with the weak-∗ topology. In this paper we study some finiteness conditions relative to the topological structure of C∗ in terms of the category Rat(C∗M) of rational left C∗-modules. In particular, we focus on the problem whether Rat(C∗M) is closed under extensions. In torsion theoretic terms this raises the question of deciding when Rat(C∗M) is a torsion theory or a localizing subcategory in C∗M, the category of all left C∗-modules (the notion of localizing subcategory used here is as in [5], [19]). This problem has been previously treated in [9], [11], and [18], where a coalgebra satisfying this property is said to be a coalgebra having a torsion rat functor.

On Clifford systems and generalized crossed products

for additive subgroups R,, CJ E G. satisfying R,R, = R,, for all u, r E G. In [ 17 ] such a ring R is called an almost strongly graded ring of type G because it may be viewed as a direct generalization of a strongly graded ring or a generalized crossed product (in which case R = @,rc(; R, is assumed) or of the more common notion of a crossed product of R and G. Some properties of Clifford systems are strikingly similar to properties of normalizing extensions. Actually. using semigroups instead of groups. one can easily define a semigroup Clifford system in such a way that it becomes a common generalization of normalizing extensions. strongly graded rings. crossed products and (skew) groups rings. However this is not the aim of this note. Here we only presents some (new) properties of Clifford systems and strongly graded rings, e.g., Maschke’s theorem etc... . If e is the neutral element of G, then the Picard group of R,,. Pic(R,), plays a fundamental part in our results. Exactly the fact that we allow non-trivial elements of Pic(R,,) to enter the picture enables us to derive certain results without extra assumptions on R,. We have included some examples of possible applications, in particular to Azumaya algebras and the Brauer group of a ring. The methods used stem from the theory of normalizing extensions 14, 5 ] fixed rings for finite group actions, cf. S. Montgomery ] 141, and graded ring theory. cf. [ 17 ]. We also refer to E. Dade’s basic papers 16, 7 ].

Filtrations on simple Artinian rings

Abstract Filtrations on simple Artinian rings are related to pseudo-valuations by conditions on the associated graded rings.

Gradings of finite support. Application to injective objects

Let R be a group-graded ring. In this paper we study the relationship between injective objects in R-gr and in R-mod. It is well-known that gr-injectives, i.e. injective objects in R-gr, need not be injective in R-mod. However, in [lo], the second author showed that if R has finite support, then gr-injective modules with finite support are injective in R-mod. We generalize this result by showing that the restriction that R have finite support is unnecessary. Suppose M is a graded R-module with finite support, R also with finite support. In the first section we show that although sometimes one can grade M and R by a finite group, while preserving the homogeneous components of the grading, this is not always possible. Thus the theory of graded rings and modules with finite support does not coincide with the theory of finite group gradings. In Section 2, the full subcategory Cx of R-gr of graded R-modules with support in X s G is introduced. We show that the forgetful function Ux: Cx -+ R-mod, has a right adjoint, and if X is finite, also a left adjoint. We determine necessary and sufficient conditions for Cx to be equivalent to a module category. If X is finite, either Cx is zero or equivalent to be module category, S/I-mod, where S is Quinn’s smash product. Section 3 uses the category Cx to establish the main result, namely that gr-injectives with finite support are injective. As corollaries, we give necessary and sufficient conditions for an injective R-modules (injective indecomposable R-module) to be gradable if G is finite (supp R is finite). Finally, we show that if G is infinite, supp (R) finite and every gr-injective module is injective, then R is left noetherian, thus giving a converse to a result of the second author [lo].

The Brauer splitting theorem and projective representations of finite groups over rings

In the theory of representations of finite groups the projective representations arise naturally when one studies the relations between the representations of the group and representations of certain subgroups. The role played by the group rings in representation theory is taken by the twisted groups rings when one considers projective representations. For representations over algebraically closed fields the theory of Schur multipliers provides a very satisfactory tool that may be used to lift projective representations of a finite group to usual representations of a finite central extension of that group. Furthermore there exists a satisfactory theory of projective characters, block-theory for projective representations, etc. . . . . for which we may refer to Karpilovsky’s recent book, [3]. Since for finite groups, the twisted group rings defined over commutative rings are Azumaya algebras (when a mild condition holds), it is natural to ask for an equivalent of the Brauer splitting theorem for group rings. In this paper we derive this splitting theorem for twisted group rings over commutative rings without nontrivial idempotents. This is obtained by introducing a suitable extension of the Schur multiplier theorem in such a way that this result may be applied to a single representation and over arbitrary fields or rings of the forementioned type. Along the way we get involved in some cocycle manipulations that will lead to the determination of the center of a twisted group ring in terms of ray-classes. The results derived in this paper should be of some use in the study of properties and

Global dimension and regularity of rees rings for non-zariskian filtrations

If x is a central regular element of the Noetherian ring A such that A/xA has finite global dimension then gldimA=max{1+gldimA/xA,gldim Ax} and we apply this to filtered rings. We study the regularity of the Rees ring when both the filtered and the associated graded ring are regular rings.

Quantum sections and Gauge algebras

Using quantum sections of filtered rings and the associated Rees rings one can lift the scheme structure on Proj of the associated graded ring to the Proj of the Rees ring. The algebras of interest here are positively filtered rings having a non-commutative regular quadratic algebra for the associated graded ring; these are the so-called gauge algebras obtaining their name from special examples appearing in E. Wittens gauge theories. The paper surveys basic definitions and properties but concentrates on the development of several concrete examples.

Generalized Rees Rings and Arithmetical Graded Rings

0. INTRODUCTION In this paper we study graded rings with an arithmetical ideal theory for the graded ideals, e.g., Gr-Dedekind and Gr-principal ideal rings. If these rings are positively graded rings, then the structure of Gr-Dedekind and of Gr-principal ideal rings is easily investigated and it is much like the structure of the ungraded equivalents. For arbitrary Z-gradations, however, the new classes of rings introduced here have an interesting structure relating to the class group of the part of degree zero. The main results in Section 2 determine the structure of Gr-Dedekind rings. First, if R is a Gr-Dedekind ring such that RR, = R, then R is a generalised Rees ring (and vice versa). These are obtained as follows: let I be a fractional ideal of a Dedekind domain R,, consider the graded ring l?,(I) = CnCZ I”X” which is a graded subring of K,[X, X-I], K, being the field of fractions of R,. Nate that classically, the Rees ring of an ideal I of a domain R was defined to be the graded ring R(~)=ROIO...OI”O...rR+I;Y+I’X*+...+ p/y* + . . . . Now, for an arbitrary Gr-Dedekind ring R, the part of degree 0, R, say, is a Dedekind ring and in Theorem 2.10 we establish that there is an e E n\l such that R(‘) is a generalized Rees ring, where R(‘) is the graded ring defined by (R @J)k = R,,. The closing paragraphs of Section 2 deal with the study of the class groups of Gr-Dedekind rings R and the reiations between these and the class groups of R, the part of degree 0. In particular, we pay attention to some connections between the structure of class groups and the ramification in R of prime ideals of R,, containing a certain ideal 6(R), called the discriminator of R, i.e., 6(R) = R _, R i. In [5 1, we give some applications of these ideas, in particular the constructions of a “graded” Zeta-function etc... _ Another application is to the theory of the Brauer group of a commutative ring, where arithmetical graded rings play a very peculiar role. We hope ‘ti present this material in a forthcoming paper. ! 8.5

The simple modules of the Lie superalgebra osp(1,2)

Abstract A classification (up to irreducible elements of a certain Euclidean ring) of the simple modules of the Lie superalgebra osp (1,2) (over an uncountable algebraically closed field of characteristic zero) is presented.