Factorization of transformations over a local ring

Let V be a finitely generated free module over a local ring R, and π an invertible linear transformation for V. Then π is a product of simple mappings. In addition, the determinant of each simple factor but one can be chosen to be any given unit in R. The smallest number of factors needed is not greater than r+1, where r is the codimension of the vector space that is associated with the module of vectors fixed under π.     

Shapes of free resolutions over a local ring

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal ${\mathbb {Q}}$ -Betti sequences.     

Automorphisms of the symplectic group over a local ring

Let R denote a commutative local ring with maximal ideal m and residue field K = R/m. Let V be a symplectic space over R. In this paper we determine the group automorphisms of the symplectic group Sp_n(V) when n 6, the characteristic of k is not 2, and k is not the finite field of three elements.     

Normal structure of the unitary group over a local ring

A new approach is used to describe the normal subgroups of unitary groups over the ring of integers of an unramified quadratic extension of a local field of characteristic 2. This method is based on the author's results concerning subgroups of unitary groups normalized by diagonal unitary matrices. By means of this method it has been possible to remove restrictions imposed by other authors in a number of papers pertaining to the same question.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 150–163, 1982.The results in this paper were obtained by the author while he was a post-graduate student under the direction of Z. I. Borevich. The author would like to express his sincere gratitude to Professor Borevich.     

The orthogonal group over a local ring is 4-reflectional

LetV be a free finite-dimensional module over a commutative local ringR where 2 is a unit. Letf:V × V R be a regular symmetric bilinear form. We prove that every element of the orthogonal group O(V) is a product of four involutions in O(V).     

Strong cleanness of generalized matrix rings over a local ring

Let K_s(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K_s(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K_s(R) to be strongly clean and, respectively, for the ring K_s(R) to be strongly clean. Specializing to s = 1 yields some known results. New families of strongly clean rings are presented.     

The rees valuations of a module over a two-dimensional regular local ring

In this paper we show that the Rees valuation rings, of a finitely generated, torsion-free module M over a two-dimensional regular local ring are precisely the Rees valuation rings of the rank(M)-th Fitting invariant of M. The technical tools used are quadratic transforms and Buchsbaum-Rim multiplicity.     

Symmetric powers of complete modules over a two-dimensional regular local ring

Let be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free -module , write for the th symmetric power of , mod torsion. We study the modules , , when is complete (i.e., integrally closed). In particular, we show that , for any minimal reduction and that the ring is Cohen-Macaulay.

     

Every local ring is dominated by a one-dimensional local ring

Let be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring of such that (i) dominates , (ii) the residue field of is a finite purely transcendental extension of , (iii) every associated prime of (0) in contracts in to an associated prime of (0), and (iv) . In addition, it is shown that can be obtained so that either is the maximal ideal of or is a localization of a finitely generated -algebra.

     

The Brauer group of a curve over a strictly local discrete valuation ring

LetK be the field of fractions of a curve overR whereR is the henselization of a regular local ring on an algebraic curve over a field which is algebraically closed and has characteristic 0. ThenK has the exponent=degree property for division algebras. In fact every central finite dimensionalK-division algebra with exponentn is a cyclic algebra of degreen. In memory of Professor S. A. Amitsur     

Note on Diophantine inequality and Linear Artin Approximation over a local ring

G. Rond [G. Rond, Approximation diophantienne dans les corps de séries en plusieurs variables, Ann. Institut Fourier 56 (2) (2006) 299–308. [10]] has proved Linear version of Artin Approximation theorem (LAA) and Diophantine inequality for a single homogeneous polynomial equation in two unknowns with coefficients in a formal (or convergent) power series ring over a field. M. Hickel and H. Ito, S. Izumi have generalized Rond's result to certain good local domains, independently, in 2008. This is a complementary Note to theirs. The most important point is that we can delete the equicharacteristic assumption in both papers. To cite this article: M. Hickel et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On Hardy-Bessel potential spaces over the ring of integers in a local field

Штрихарц [3] дал характе ристику пространствL ^p (R ⁿ ) бесселевых потенциа лов порядка α функций из п ространстваL ^p (R ⁿ ) с пом ощьюL ^p -норм функционалов $$D_\alpha f(x) = \left( {\smallint _0^\infty \left( {\smallint _{\rm B} |f(x + rt) - f(x)|dt} \right)^2 r^{ - 2\alpha - 1} dr} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ для 0<α<1, гдеB обозначает ед иничный шар. Целью нас тоящей статьи является изучение пр остранств потенциал а Харди-БесселяF _α ^p (P ⁰) (0<p<1, α>1/р?1) в терминах функц ионаловS _r ^α f(x) (1τ≤2), которые в случаеR ⁿ } соответствуютD _α f (x), гдеР ⁰ - кольцо целых в локальном поле. Получ ено приложение, относяще еся к регулярности бе сселевых потенциалов.     

Faltings' theorem for the annihilation of local cohomology modules over a Gorenstein ring

In this paper we study the Annihilator Theorem and the Local-global Principle for the annihilation of local cohomology modules over a (not necessarily finite-dimensional) Noetherian Gorenstein ring.

     

Indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring

In this paper, we construct indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring. The modules are quite explicitly constructed from a given complete monomial ideal. We also give structural and numerical results on integrally closed modules. These are used in the proof of indecomposability of the modules. As a consequence, we have a large class of indecomposable integrally closed modules of arbitrary rank whose ideal is not necessarily simple. This extends the original result on the existence of indecomposable integrally closed modules and strengthens the non-triviality of the theory developed by Kodiyalam.