Clifford theory for infinite dimensional modules

Clifford theory of possibly infinite dimensional modules is studied.     

Rings, modules, and algebras in infinite loop space theory

We give a new construction of the algebraic K-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory (elsewhere also called colored operad), a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in the Hovey-Shipley-Smith category of symmetric spectra, with an intermediate stop at a category of functors out of a particular wreath product.     

Relations between simplicial groups, 3-crossed modules and 2-quadratic modules

In this paper, we define two-quadratic module and explore the relations among two-quadratic modules, three-crossed modules and simplicial groups.     

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that , for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L^′. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Extn, for any integer n?0, using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L=L(λ), even allowing λ to vary, provided the p-adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.     

Local cohomology modules with infinite dimensional socles

In this paper we prove the following generalization of a result of Hartshorne: Let be a commutative Noetherian local ring of dimension at least two, , and . Let be a homogeneous element of such that the coefficients of form a system of parameters for . Then the socle of is infinite dimensional.

     

Orbit-equivalent infinite permutation groups

Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.     

Nil-algebras and infinite groups

We simplify our construction of nil-algebras by proving, for any integer d ≥ 2 and over any field , that there exists a residually nilpotent, nonnilpotent nil-algebra over generated by d elements. As a consequence, we obtain similar results for nonassociative algebras and groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 189–200, 2005.