On the finiteness properties of extension and torsion functors of local cohomology modules

Let be a commutative Noetherian ring with non-zero identity, and ideals of with , and a finitely generated -module. In this paper, for fixed integers and , we study the finiteness of and in several cases.

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .

     

Character degree graphs that are complete graphs

Let be a finite group, and write for the set of degrees of irreducible characters of . We define to be the graph whose vertex set is , and there is an edge between and if . We prove that if is a complete graph, then is a solvable group.

     

Lightface -indescribable cardinals

-absoluteness for forcing means that for any forcing , . `` inaccessible to reals' means that for any real , . To measure the exact consistency strength of `` -absoluteness for forcing and is inaccessible to reals', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) -indescribable cardinal; has this property exactly if it is inaccessible and .

     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

The Laplacian subalgebra of is a strongly singular masa

In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the -folded tensor product of the free group factor with itself (). The notions of strongly singular masas in type factors and that of asymptotic homomorphism were introduced by A. Sinclair and R. Smith. One of their first examples was the Laplacian subalgebra of the free group factor, generated by the sum of words of length 1 in . This subalgebra was known to be a singular masa. Using the results of A. Sinclair and R. Smith, we show that the unique trace-preserving conditional expectation onto the Laplacian subalgebra of is an asymptotic homomorphism, and hence the Laplacian subalgebra is a strongly singular masa for every .

     

Restrictions of bounded linear operators: Closed range

Let be a bounded linear operator on a Banach space and let be a subspace of which is a Banach space and invariant. Denote by the restriction of to This paper explores the questions:

If the range of is closed, under what conditions is the range of closed?

If the range of is closed, under what conditions is the range of closed?

     

The degree of the bicanonical map of a surface with

In this note it is shown that, given a smooth minimal complex surface of general type with , , for which the bicanonical map is a morphism, the degree of is not 3. This completes our earlier results, showing that if is a minimal surface of general type with , such that is free, then the bicanonical map of can have degree 1, 2 or 4.

     

On the degree of Hilbert polynomials associated to the torsion functor

Let be a local, Noetherian ring and an ideal. A question of Kodiyalam asks whether for fixed , the polynomial giving the th Betti number of has degree equal to the analytic spread of minus one. Under mild conditions on , we show that the answer is positive in a number of cases, including when is divisible by or is an integrally closed -primary ideal.

     

Extensions of orthosymmetric lattice bimorphisms

Let be an Archimedean vector lattice, let be its Dedekind completion and let be a Dedekind complete vector lattice. If is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism that not just extends but also has to be orthosymmetric. As an application, we prove the following: Let be an Archimedean -algebra. Then the multiplication in can be extended to a multiplication in , the Dedekind completion of , in such a fashion that is again a -algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

Dimension distortion of hyperbolically convex maps

In this note, we provide an answer to a question of D. Mejia and Chr. Pommerenke, by constructing a hyperbolically convex subdomain of the unit disc so that the conformal map from to maps a set of dimension 0 on to a set of dimension

     

A Hilbert -module not anti-isomorphic to itself

We study the complexification of real Hilbert -modules over real -algebras. We give an example of a Hilbert -module that is not the complexification of any Hilbert -module, where is a real -algebra.

     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

Asymptotically harmonic spaces in dimension 3

Let be a Hadamard manifold of dimension whose sectional curvature satisfies and whose curvature tensor satisfies for suitable constants and . We show that is of constant sectional curvature provided is asymptotically harmonic. This was previously only known if admits a compact quotient.

     

A class of differentiable toral maps which are topologically mixing

We show that on the 2-torus there exists a open set of regular maps such that every map belonging to is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of toral diffeomorphisms, but that the property does hold for the class of diffeomorphisms on the 3-torus . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of diffeomorphisms on the -torus ().

     

A linear function associated to asymptotic prime divisors

Let be a Noetherian standard -graded ring and finitely generated, -graded -modules. Let be finitely many homogeneous ideals of . We show that there exist linear functions such that the associated primes over of and are stable whenever satisfies and , respectively.

     

Invariants of semisimple Lie algebras acting on associative algebras

If is a Lie algebra of derivations of an associative algebra , then the subalgebra of invariants is the set In this paper, we study the relationship between the structure of and the structure of , where is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on , when is semiprime.

     

Minimal rank and reflexivity of operator spaces

Let be an -dimensional space of linear operators between the linear spaces and over an algebraically closed field . Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let be a basis of . Assume that every nonzero operator in has rank larger than . Then a linear operator belongs to if and only if for every , is a linear combination of .