Abelian class groups of reductive group schemes

We introduce the abelian class group C _ab (G) of a reductive group scheme G over a ring A of arithmetical interest and study some of its basic properties. For example, we show that if the fraction field of A is a global field without real primes, then there exists a surjection C(G) ? C _ab (G), where C(G) is the class set of G.     

On the Grothendieck group of simply connected semisimple algebraic groups

The K₀ groups of simply connected semisimple algebraic groups are calculated. The triviality of the Chow groups CH¹ and CH² of such groups is obtained as a consequence. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 223–235.     

Infinitesimal group schemes as iterative differential Galois groups

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes. Furthermore, this gives a tool to tell whether all purely inseparable ID-extensions are in fact Picard-Vessiot extensions.     

Minimal subschemes of the group association schemes of Mathieu groups

Minimal subschemes of the group association schemes of Mathieu groupsM _n (n = 11, 12, 22, 23, 24) are determined. It is proved that for eachM _n (n = 11, 12, 22, 23, 24), there is a unique minimal subscheme of. The class numbers of these minimal subschemes are 7, 11, 9, 11 and 20 respectively. A general computer program to determine subschemes of group association schemes of relatively small class numbers is discussd.     

Structure of the Adele group of a product of algebraic groups

Translated from Matematicheskie Zametki, Vol. 50, No. 4, pp. 33–37, October, 1991.     

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.     

Homological algebra for the representation Green functor for abelian groups

In this paper we compute some derived functors of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.

When the group is a cyclic -group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor .

When the group is , we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired functors.

     

On the existence and exactness of the associated sheaf functor

Let $\text{C}$ be a category with inverse limits. A category $\text{x}$is called an $\text{A}$-topos if there is a site (?, τ), i.e. a small category ? together with a Grothendieck topology τ such that $\text{x}$is equivalent to the category Sh_τ[$\text{C}$⁰. $\text{A}$] of τ-sheaves on $\text{C}$ with values in $\text{C}$. If $\text{x}$is an $\text{C}$-topos, then so is Sh_τ'[?⁰, $\text{x}$] for any site (?', τ'). It is shown that if for every site (?,τ) the associated sheaf functor from presheaves to τ-sheaves with values in $\text{A}$ exists (and preserves finite inverse limits), then the same holds if $\text{A}$is replaced by any $\text{A}$-topos $\text{x}$. Roughly speaking, the main result is that for a site (?,τ) the associated sheaf functor [?⁰, $\text{A}$] → Sh_τ [?⁰, $\text{A}$] exists and preserves finite inverse limits, provided $\text{A}$ has filtered direct limits which commute with finite inverse limits, e.g. if $\text{A}$ is a Grothendieck category or a category of sheaves with values in a locally finitely presentable category [8. 7.1]. Analogous results hold in the additive case.     

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group is largely determined by a linearly reductive subgroup scheme of SL(2), with the McKay quiver of relative to its standard module being the Gabriel quiver of the principal block . The graphs underlying these quivers are extended Dynkin diagrams of type or , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.     

Commuting algebraic correspondences and groups

In this paper, the naturalness of the appearance of the group approach in the study of problems of algebraic commutation is considered. New, previously unknown solutions are obtained. Special attention is paid to an example of commutation obtained from the icosahedral equation. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 662–670, May, 1997. Translated by A. I. Shtern