Reduction of structure for torsors over semilocal rings

Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H ¹(R, S) → H ¹(R, G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H ¹(X, S) → H ¹(X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras and essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper. V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant. Z. Reichstein was partially supported by NSERC Discovery and Accelerator Supplement grants.     

On semilocal rings

We give several characterizations of semilocal rings and deduce that rationally closed subrings of semisimple artinian rings are semilocal, that artinian modules have semilocal endomorphism rings, and that artinian modules cancel from direct sums. Dedicated to the memory of Pere Menal     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

Group rings over Dedekind rings

Seghal posed the following question: IfA andB are rings, doesA[X,X ⁻¹] ℞B[X,X ⁻¹] implyA ℞B. In general the answer to this question is no. In this note we give an affirmative answer in the case thatA andB are Dedekind rings. The author is research assistant at the NFWO.     

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G (, R) of twisted type =A_l,l odd, D_l, E₆ over a commutative semilocal ring R with involution are in one-to-one correspondence with the -invariant parabolic nets of ideals of R of type , i.e., with the sets, of ideals of R such that: (l) whenever; (2) = for all ; (3) =R for > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 21–36, 1979.     

Lexicodes over rings

In this paper, we consider the construction of linear lexicodes over finite chain rings by using a \(B\) -ordering over these rings and a selection criterion. As examples we give lexicodes over \(\mathbb Z _4\) and \(\mathbb F _2+u\mathbb F _2\) . It is shown that this construction produces many optimal codes over rings and also good binary codes. Some of these codes meet the Gilbert bound. We also obtain optimal self-dual codes, in particular the octacode.     

Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

Counting codes over rings

We extend the definition of free codes to codes over local rings and arbitrary Frobenius rings. The number of free codes over finite Frobenius rings is determined by calculating the number for local rings and applying the Chinese Remainder Theorem. A formula for the number of codes of arbitrary type over a finite chain ring is given and this is applied to determine the number of linear codes over a finite principal ideal ring.     

FIm-modules over Noetherian rings

In this paper we study representation theory of the category ${\mathrm{FI}}^m$ introduced in [6], [7] which is a product of copies of the category FI, and show that quite a few interesting representation theoretic and homological properties of FI can be generalized to ${\mathrm{FI}}^m$ in a natural way. In particular, we prove the representation stability property of finitely generated ${\mathrm{FI}}^m$-modules over fields of characteristic 0.     

Modules over serial rings

This paper continues the study of Noetherian serial rings. General theorems describing the structure of such rings are proved. In particular, some results concerning π-projective and π-injective modules over serial rings are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 880–892 June, 1999.