Pure-injective hulls of modules over valuation rings

If is the pure-injective hull of a valuation ring R, it is proved that is the pure-injective hull of M, for every finitely generated R-module M. Moreover , where (A_k)_1≤k≤n is the annihilator sequence of M. The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module are isomorphic.     

A Morita type equivalence for dual operator algebras

We generalize the main theorem of Rieffel for Morita equivalence of W^∗-algebras to the case of unital dual operator algebras: two unital dual operator algebras A,B have completely isometric normal representations α,β such that α(A)=[M^∗β(B)M]^−w∗ and β(B)=[Mα(A)M^∗]^−w∗ for a ternary ring of operators M (i.e. a linear space M such that MM^∗M⊂M) if and only if there exists an equivalence functor which “extends” to a ∗-functor implementing an equivalence between the categories and . By we denote the category of normal representations of A and by the category with the same objects as and Δ(A)-module maps as morphisms (Δ(A)=A∩A^∗). We prove that this functor is equivalent to a functor “generated” by a B,A bimodule, and that it is normal and completely isometric.     

Reynolds operator on functors

Let be an affine R-monoid scheme. We prove that the category of dual functors (over the category of commutative R-algebras) of G-modules is equivalent to the category of dual functors of A^∗-modules. We prove that G is invariant exact if and only if A^∗=R×B^∗ as R-algebras and the first projection A^∗→R is the unit of A. If M is a dual functor of G-modules and w_G?(1,0)∈R×B^∗=A^∗, we prove that M^G=w_G⋅M and M=w_G⋅M⊕(1−w_G)⋅M; hence, the Reynolds operator can be defined on M.     

A complement of the Ando-Hiai inequality

In this paper, we present a complement of a generalized Ando-Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541-545]. Let A and B be positive operators on a Hilbert space H such that 0<m₁?A?M₁ and 0<m₂?B?M₂ for some scalars m_i?M_i (i=1,2), and let α∈[0,1]. Put for i=1,2. Then for each 0<r?1 and s?1

     

Coactions on Hochschild homology of Hopf-Galois extensions and their coinvariants

Let B⊆A be an H-Galois extension, where H is a Hopf algebra over a field K. If M is a Hopf bimodule then , the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C_H=H/[H,H]. Given an injective left C_H-comodule V, our aim is to understand the relationship between and . The roots of this problem can be found in Lorenz (1994) [15], where and are shown to be isomorphic for any centrally G-Galois extension. To approach the above mentioned problem, in the case when A is a faithfully flat B-module and H satisfies some technical conditions, we construct a spectral sequence

     

Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

Norm equality for a basic elementary operator

Let be the algebra of bounded linear operators on a Hilbert space H. For , define the elementary operator M_A,B by M_A,B(X)=AXB (). We give necessary and sufficient conditions for any pair of operators A and B to satisfy the equation ‖I+M_A,B‖=1+‖A‖‖B‖, where I is the identity operator on H.     

On the telescope conjecture for module categories

In [H. Krause, O. Solberg, Applications of cotorsion pairs, J. London Math. Soc. 68 (2003) 631-650], the Telescope Conjecture was formulated for the module category of an artin algebra R as follows: “If C=(A,B) is a complete hereditary cotorsion pair in with A and B closed under direct limits, then ”. We extend this conjecture to arbitrary rings R, and show that it holds true if and only if the cotorsion pair C is of finite type. Then we prove the conjecture in the case when R is right noetherian and B has bounded injective dimension (thus, in particular, when C is any cotilting cotorsion pair). We also focus on the assumptions that A and B are closed under direct limits and on related closure properties, and detect several asymmetries in the properties of A and B.     

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:

(i)

AM is relatively weakly compact;

(ii)

B is a strict contraction;

(iii)

.

Then A+B has at least one fixed point in M.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.     

On the Deligne-Simpson problem and its weak version

We consider the Deligne-Simpson problem (DSP) (respectively the weak DSP): Give necessary and sufficient conditions upon the choice of the p+1 conjugacy classes or so that there exist irreducible (p+1)-tuples (respectively (p+1)-tuples with trivial centralizers) of matrices A_j∈c_j with zero sum or of matrices M_j∈C_j whose product is I. The matrices A_j (respectively M_j) are interpreted as matrices-residua of Fuchsian linear systems (respectively as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues.     

Almost fully decomposable infinite rank lattices over orders

Let Λ be an order over a Dedekind domain R with quotient field K. An object of , the category of R-projective Λ-modules, is said to be fully decomposable if it admits a decomposition into (finitely generated) Λ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105-128], we give a necessary and sufficient criterion for R-orders Λ in a separable K algebra A with the property that every is fully decomposable. In the present paper, we assume that is separable, but that the p-adic completion A_p is not semisimple for at least one . We show that there exists an , such that KL admits a decomposition KL=M₀⊕M₁ with finitely generated, where L∩M₁ is fully decomposable, but L itself is not fully decomposable.     

Sigma-cotorsion rings

It is proved that a ring R is right perfect if and only if it is Σ-cotorsion as a right module over itself. Several other conditions are shown to be equivalent. For example, that every pure submodule of a free right R-module is strongly pure-essential in a direct summand, or that the countable direct sum of the cotorsion envelope of R_R is cotorsion.If C_R is a flat Σ-cotorsion module, then C_R admits a decomposition into a direct sum of indecomposable modules with a local endomorphism ring. The Jacobson radical J(S) of the endomorphism ring S=End_RC is characterized as the maximum ideal that acts locally T-nilpotently on C_R. If R is semilocal and C=C(R), then the radical consists of those endomorphisms whose image is contained in CJ.     

Relative copure injective and copure flat modules

Let R be a ring, n a fixed nonnegative integer and I_n (F_n) the class of all left (right) R-modules of injective (flat) dimension at most n. A left R-module M (resp., right R-module F) is called n-copure injective (resp., n-copure flat) if (resp., ) for any N∈I_n. It is shown that a left R-module M over any ring R is n-copure injective if and only if M is a kernel of an I_n-precover f:A→B of a left R-module B with A injective. For a left coherent ring R, it is proven that every right R-module has an F_n-preenvelope, and a finitely presented right R-module M is n-copure flat if and only if M is a cokernel of an F_n-preenvelope K→F of a right R-module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.     

Cap pairings and spectral sequences

Let be a small category. For an -diagram X and -diagrams A and B of pointed spaces, each pairing X∧A→B satisfying the projection formula induces a pairing . In this note we show that there is an induced pairing of homotopy spectral sequences compatible with abutments in the sense that

     

On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

The classical Weyl-von Neumann theorem states that for any self-adjoint operator A₀ in a separable Hilbert space H there exists a (non-unique) Hilbert-Schmidt operator C=C^? such that the perturbed operator A₀+C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering the set ExtA of self-adjoint extensions of a given densely defined symmetric operator A in H and some fixed . We show that the ac-parts and of and A₀ are unitarily equivalent provided that the resolvent difference is compact and the Weyl function M(⋅) of the pair {A,A₀} admits weak boundary limits M(t):=w-lim_y→+0M(t+iy) for a.e. t∈R. This result generalizes the classical Kato-Rosenblum theorem. Moreover, it demonstrates that for such pairs {A,A₀} the Weyl-von Neumann theorem is in general not true in the class ExtA.     

Singular fibers and characteristic classes

Let be a proper generic map between smooth manifolds with dimN−dimM=−1. We explicitly calculate the cohomology class dual to the closure of the set of points in N over which lies a specific singular fiber in terms of characteristic classes of M and N.     

The phantom cover of a module

A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor₁(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor₁(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.     

Higher power moments of the Riesz mean error term of symmetric square L-function

Let be the error term of the Riesz mean of the symmetric square L-function. We give the higher power moments of and show that if there exists a real number A₀:=A₀(ρ)>3 such that , then we can derive asymptotic formulas for , 3?h<A₀, h∈N. Particularly, we get asymptotic formulas for , h=3,4,5 unconditionally.