Том 219, № 3 (2016)

In this paper, an explicit classification of formal A-modules over the ring of integers of a local field up to a strict isomorphism is obtained in the case of small ramification. Canonical representatives in each class of isomorphic formal A-modules are described. These results generalize the classification of ℤ_p-modules in small ramification.

The behavior of a product of commuting long and short root elements of the group of type B_r in p-restricted irreducible representations is investigated. For such representations with certain local properties of highest weights, it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. For a p-restricted representation with highest weight a₁ω₁ +· · ·+a_rω_r, this fact is proved when a_j ≠ p − 1 for some j < r − 1 and one of the following conditions holds: (1) \( {a}_r\ne p-1\kern0.75em and\kern0.5em {\displaystyle \sum_{i=1}^{r-2}{a}_i\ge p-1;} \)and (2) \( 2{a}_{r-1}+{a}_r or (r−3) (p-1) for a_r= p−1.

Let K be a multidimensional local field with characteristic different from the characteristic of its residue field, c be a unit of K, and F_c(X, Y) = X +Y +cXY be a polynomial formal group, which defines the formal module F_c(\( \mathfrak{M} \)) over the maximal ideal of the ring of integers in K. Assume that K contains the group of roots of the isogeny [p^m]_c(X), which we denote by μ_Fc,_m. Let be the multiplicative group of Cartier curves and _c be the formal analog of the module Fc(\( \mathfrak{M} \)). In the present paper, the formal symbol { ·, · }c : K_n()×_c → μ_Fc,_m is constructed and its basic properties are checked. This is the first step in the construction of an explicit formula for the Hilbert symbol.

For a localizing class \( \mathcal{S} \) of morphisms in a pretriangulated category \( \mathcal{D} \), a weak version of a sufficient condition that guarantees the transfer of the structure of a pretriangulated category onto the localization \( \mathcal{D}\left[{\mathcal{S}}^{-1}\right] \) is proposed. Moreover, the weakness of a similar sufficient condition is obtained in the context of triangulated categories.

Let Γ be a reductive algebraic group, and let P,Q ⊂ Γ be a pair of parabolic subgroups. Some properties of the intersection and incidence distances \( \begin{array}{c}\hfill {\mathrm{d}}_{\mathrm{in}}\left(P,Q\right)= \max \left\{ \dim P, \dim Q\right\}- \dim \left(P\cap Q\right),\hfill \\ {}\hfill {\mathrm{d}}_{\mathrm{in}\mathrm{c}}\left(P,Q\right)=\mathrm{mix}\left\{ \dim P, \dim Q\right\}- \dim \left(P\cap Q\right)\hfill \end{array} \) are considered (if P,Q are Borel subgroups, both numbers coincide with the Tits distance dist(P,Q) in the building Δ(Γ) of all parabolic subgroups of Γ). In particular, if Γ = GL(V ) and P = P_v,Q = P_u are stabilizers in GL(V ) of linear subspaces v, u ⊂ V , we obtain the formula d_in(P, Q) = − d² + a₁d + a₂, where d = din(v, u) = max{dim v, dim u} − dim(v ∩ u) is the intersection distance between the subspaces v and u and where a₁ and a₂ are integers expressed in terms of dim V , dim v, and dim u. Bibliography: 7 titles.

This paper is a sequel of the joint paper by the author with S. O. Ivanov, Yu. Volkov, and G. Zhou. In the present paper, the BV -structure, and therefore, the Gerstenhaber algebra structure on the Hochschild cohomology of local algebras of generalized quaternion type is completely described over a field of characteristic 2. The family of algebras under investigation contains group algebras of generalized quaternion groups for which the case of characteristic 2 is the only one where the calculation of Hochschild cohomology and structures on it is a highly nontrivial problem. Also the group algebras of generalized quaternion groups represent classes of Morita-equivalence of tame group blocks from K. Erdmann’s classification. In particular, the BV -structure on the Hochschild cohomology of group algebras of some noncommutative groups is described.

For p > 2, odd Jordan block sizes of the images of regular unipotent elements from subsystem subgroups of type A₂ in irreducible p-restricted representations for groups of type A_r over the field of characteristic p, the weights of which are locally small with respect to p, are found. The weight is called locally small if the double sum of its two neighboring coefficients is less than p. This result is part of a more general program investigating the behavior of unipotent elements in representations of the classical algebraic groups. It can be used to solve recognition problems for representations or linear groups by the presence of certain elements.