Self-duality of rank 2 reflexive modules

Every finitely generated rank 2 third syzygy module over a regular local ring is known to be self-dual. We show more generally that any finitely generated rank 2 reflexive module is self-dual, and that the isomorphism is skew symmetric. We use this ismorphism to estimate how large k may be if the module is a kth syzygy, and how closely allied rank 3 modules are related to their duals.     

On modules of finite upper rank

For a group and a prime , the upper -rank of is the supremum of the sectional -ranks of all finite quotients of . It is unknown whether, for a finitely generated group , these numbers can be finite but unbounded as ranges over all primes. The conjecture that this cannot happen if is soluble is reduced to an analogous `relative' conjecture about the upper -ranks of a `quasi-finitely-generated' module for a soluble minimax group . The main result establishes a special case of this relative conjecture, namely when the module is finitely generated and the minimax group is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)     

Verma modules for rank two Heisenberg-Virasoro algebra

Let■ be a compatible total order on the additive group Z~2,and L be the rank two HeisenbergVirasoro algebra.For any c=(c_1,c_2,c_3,c_4) ∈ C~4,we define a Z~2-graded Verma module M(c,■) for L.A necessary and sufficient condition for M(c,■) to be irreducible is provided.Moreover,the maximal Z~2-graded submodules of M(c,■) are characterized when M(c,■) is reducible.     

Quasi-finite modules for Lie superalgebras of infinite rank

We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras , and , and determine the necessary and sufficient conditions for such modules to be unitarizable. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.

     

Koszul differential graded modules

The concept of Koszulity for differential graded (DG, for short) modules is introduced. It is shown that any bounded below DG module with bounded Ext-group to the trivial module over a Koszul DG algebra has a Koszul DG submodule (up to a shift and truncation), moreover such a DG module can be approximated by Koszul DG modules (Theorem 3.6). Let A be a Koszul DG algebra, and Dc(A) be the full triangulated subcategory of the derived category of DG A-modules generated by the object AA. If the trivial DG module...     

On primitive roots for rank one Drinfeld modules

Let k be a global function field over a finite field and let A be the ring of the elements in k regular outside a fixed place ∞. Let K be a global A-field of finite A-characteristic and let ? be a rank one Drinfeld A-module over K. Given any α∈K, we show that the set of places P of K for which α is a primitive root modulo P under the action of ? possesses a Dirichlet density. We also give conditions for this density to be positive.     

Discriminant algebras of finite rank algebras and quadratic trace modules

Based on the construction of the discriminant algebra of an even-ranked quadratic form and Rost’s method of shifting quadratic algebras, we give an explicit rational construction of the discriminant algebra of finite-rank algebras and, more generally, of quadratic trace modules, over arbitrary commutative rings. The discriminant algebra is a tensor functor with values in quadratic algebras, and a symmetric tensor functor with values in quadratic algebras with parity. The automorphism group of a separable quadratic trace module is a smooth, but in general not reductive, group scheme admitting a Dickson type homomorphism into the constant group scheme Z ₂.     

Indecomposable modules of large rank over Cohen-Macaulay local rings

A commutative Noetherian local ring is called Dedekind-like provided is one-dimensional and reduced, the integral closure is generated by at most 2 elements as an -module, and is the Jacobson radical of . If is an indecomposable finitely generated module over a Dedekind-like ring , and if is a minimal prime ideal of , it follows from a classification theorem due to L. Klingler and L. Levy that must be free of rank 0, 1 or 2.

Now suppose is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let be the minimal prime ideals of . The main theorem in the paper asserts that, for each non-zero -tuple of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated -modules satisfying for each .

     

Computations in differential and difference modules

Constructive methods based on the Gröbner bases theory have been used many times in commutative algebra over the past 20 years, in particular, they allow the computation of such important invariants of manifolds given by systems of algebraic equations as their Hilbert polynomials. In differential and difference algebra, the analogous roles play characteristic sets.In this paper, algorithms for computations in differential and difference modules, which allow for the computation of characteristic sets (Gröbner bases) in differential, difference, and polynomial modules and differential (difference) dimension polynomials, are described. The algorithms are implemented in the algorithmic language REFAL.     

Noether modules over abelian groups of finite free rank

We prove that if M is a Noether JG-module, where G is an abelian group of finite free rank, and either J=, or J=Ft, where F is a finite field and t is an infinite cyclic group, then the module M belongs to a class (J, ) for some finite set in the sense defined by P. Hall.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1042–1048, July–August, 1991.     

The rank of finitely generated modules over group algebras

We show the existence of a rank function on finitely generated modules over group algebras , where is an arbitrary field and is a finitely generated amenable group. This extends a result of W. Lück (1998).

     

Free resolutions for differential modules over differential algebras

A free resolution (R, d + h) → (M, d) for a DG-module (M, d) over a DG-algebra (A, d) is constructed in the sense of a perturbation of the differential in a free bigraded resolution (R, d) → M of the underlying graded module M over an underlying graded algebra A. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Coleman's power series and Wiles' reciprocity for rank 1 Drinfeld modules

We introduce the formalism of Coleman's power series for rank 1 Drinfeld modules and apply it to formulate and prove the analogue of Wiles' explicit reciprocity law in this setting.     

Classification of irreducible weight modules over higher rank Virasoro algebras

Let G be a rank n additive subgroup of C and Vir[G] the corresponding Virasoro algebra of rank n. In the present paper, irreducible weight modules with finite dimensional weight spaces over Vir[G] are completely determined. There are two different classes of them. One class consists of simple modules of intermediate series whose weight spaces are all 1-dimensional. The other is constructed by using intermediate series modules over a Virasoro subalgebra of rank n−1. The classification of such modules over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a completely different approach.     

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relations, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of n particles in the complex plane, hence to a categorification of the Knizhnik–Zamolodchikov connection. We discuss infinitesimal 2-braidings in a certain monoidal 2-category naturally assigned to every differential crossed module, leading to the notion of a symmetric quasi-invariant tensor in a differential crossed module. Finally, we prove that symmetric quasi-invariant tensors exist in the differential crossed module associated to Wagemann's version of the String Lie-2-algebra. As a corollary, we obtain a more conceptual proof of the flatness of a previously constructed categorified Knizhnik–Zamolodchikov connection with values in the String Lie-2-algebra.     

On commuting differential operators of rank 2

We study examples of formally self-adjoint commuting ordinary differential operators of order 4 or 4g + 2 whose coefficients are analytic on ?. We prove that these operators do not commute with the operators of odd order, justifying rigorously that these operators are of rank 2.