Nonlinear Perron–Frobenius theory in finite dimensions

A nonlinear Perron–Frobenius theory in finite dimensions is described which applies to positively homogeneous and increasing maps. Sufficient conditions for the existence and uniqueness of eigenvectors in the interior of a cone are developed even when eigenvectors at the boundary of the cone exist. Several ways to benefit from nonlinearities are pointed out in order to show that the classical, linear, theory truly deserves nonlinear generalization. The main technical novelties are: a link between Collatz–Wielandt numbers and Gâteaux derivatives, and the almost one-dimensional dynamics of the nonlinear map.     

Frobenius pull backs of vector bundles in higher dimensions

We prove that for a smooth projective variety X of arbitrary dimension and for a vector bundle E over X, the Harder?CNarasimhan filtration of a Frobenius pull back of E is a refinement of the Frobenius pull back of the Harder?CNarasimhan filtration of E, provided there is a lower bound on the characteristic p (in terms of rank of E and the slope of the destabilizing sheaf of the cotangent bundle of X). We also recall some examples, due to Raynaud and Monsky, to show that some lower bound on p is necessary. We also give a bound on the instability degree of the Frobenius pull back of E in terms of the instability degree of E and well defined invariants of X.     

Associator-cyclic bimodules

Moscow Pedagogical State University. Translated from Matematicheskie Zametki, Vol. 56, No. 5, pp. 22–26, November, 1994.     

Formally smooth bimodules

The notion of a formally smooth bimodule is introduced and its basic properties are analyzed. In particular it is proven that a B-A bimodule M which is a generator left B-module is formally smooth if and only if the M-Hochschild dimension of B is at most one. It is also shown that modules M which are generators in the category σ[M] of M-subgenerated modules provide natural examples of formally smooth bimodules.     

Yetter-Drinfeld-Long bimodules are modules

Let H be a finite-dimensional bialgebra. In this paper, we prove that the category ?R(H) of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category \({}_{H \otimes H*}^{H \otimes H*}YD\) over the tensor product bialgebra H ? H* as monoidal categories. Moreover if H is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.     

Comatrix corings and invertible bimodules

We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimodules in a non-commutative ring extension and the group of automorphisms of the associated Sweedler's canonical coring, to the class of finite comatrix corings introduced in [6].

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Sunto Estendiamo il Teorema di Masuoka [11] riguardante l'isomorfismo tra il gruppo dei bimoduli invertibili su un'estensione di anelli non commutativa e il gruppo di automorfismi del coanello canonico associato di Sweedler, alla classe dei coanelli di comatrici finiti introdotta in [6].

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Research supported by the grant BFM2001-3141 from the Ministerio de Ciencia y Tecnología of Spain and FEDER.

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Supported by the grant SB2003-0064 from the Ministerio de Educación, Cultura y Deporte of Spain.     

Equivalences induced by graded bimodules

Let k be a commutative ringG a finite groupR and S fully G-graded k-algebras. In this paper we investigate Morita equivalences, derived equivalences and stable equivalences of Morita type between R and S, which are induced by G-graded R, 5-bimodules or complexes of G-graded bimodules. Such equivalences occur naturally in the case of group algebras in certain reduction steps for Broue's conjecture, and we show how they can be lifted from equivalences between R ₁ and S ₁.     

Endomorphism rings of bimodules

Let \(M\) be an \(R\) - \(R\) -bimodule over a semi-prime right and left Goldie ring \(R\) . We investigate how non-singularity conditions on \(M_R\) are related to such conditions on \(_RM\) . In particular, we say an \(R\) - \(R\) -bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular has the right essentiality property if \(IM_R\) is essential in \(M_R\) for all essential right ideals \(I\) of \(R\) , and investigate several questions related to this property.     

Ribet bimodules and the specialization of Heegner points

For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X ₀(D, N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P ∈ CM(R) specializes to a singular point (resp. a irreducible component) of the special fiber [(X)\tilde]\tilde X of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X ₀(D, N) and K. Ribet’s theory of bimodules, we give a construction of a correspondence Φ between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebra that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin-Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points in [(X)\tilde]\tilde X. We also illustrate our results with an explicit example.     

On bimodules determining stable equivalences

In the paper one shows that for two indecomposable non-simple self-injective algebras over a field K we have that if the functor induces a stable equivalence then the bimodule _AN_B is contained in the frame of a connected component in the Auslander-Reiten quiver Γ_A⊗KB^op.     

Left-sided quasi-invertible bimodules over Nakayama algebras

Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence.     

Local Hodge theory of Soergel bimodules

We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.     

Norm-principal bimodules of nest algebras II

Let be a nest in a separably acting factor . We present conditions for a factor to be a bimodule of the nest subalgebraalgß in which is singly generated in the norm topology. Some relevant results about the compact ideal , the quasitriangular algebra and the Calkin algebra are established.This work was supported, in part, by funds provided by the University of North Carolina at Charlotte.