Injective modules and linear growth of primary decompositions

The purposes of this paper are to generalize, and to provide a short proof of, I. Swanson's Theorem that each proper ideal in a commutative Noetherian ring has linear growth of primary decompositions, that is, there exists a positive integer such that, for every positive integer , there exists a minimal primary decomposition with for all . The generalization involves a finitely generated -module and several ideals; the short proof is based on the theory of injective -modules.

     

Lipschitz continuity of oblique projections

Let and be complementary spaces of a finite dimensional unitary space and let denote the projection of on parallel to . Estimates for the norm of are derived which involve the norm of the restriction of to or the gap between and .

     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

Completely continuous multilinear operators on spaces

Given a -linear operator from a product of spaces into a Banach space , our main result proves the equivalence between being completely continuous, having an -valued separately continuous extension to the product of the biduals and having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to being weakly compact, and that, for , being weakly compact implies the conditions above but the converse fails.

     

Conjugate -connections and holonomy groups

In this article we show that when the structure group of the reducible principal bundle is and is an -subbundle of , the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to , and use the estimate to show that for all odd prime , if the holonomy group of the irreducible connection as above is simple and is not isomorphic to , , or , then it is isomorphic to .

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

On Diophantine sets over polynomial rings

We prove that the recursively enumerable relations over a polynomial ring , where is the ring of integers in a totally real number field, are exactly the Diophantine relations over .

     

On the existence of maximal Cohen-Macaulay modules over root extensions

Let be an unramified regular local ring having mixed characteristic and the integral closure of in a th root extension of its quotient field. We show that admits a finite, birational module such that . In other words, admits a maximal Cohen-Macaulay module.

     

Injective resolutions of Noetherian rings and cogenerators

We give new construction of injective resolutions of complexes and bimodules. Applying this construction to an injective resolution of a Noetherian ring, we construct a -embedding cogenerator for the category of modules of projective dimension . Moreover, for a Noetherian projective -algebra , we show that satisfies the Auslander condition if and only if the flat dimension of every -module is equal to or larger than the one of the injective hull .

     

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 .

     

Asymptotic behaviour of Castelnuovo-Mumford regularity

Let be a polynomial ring over a field. For a graded -module generated in degree at most , the Castelnuovo-Mumford regularity of each of (i) its symmetric power, (ii) its torsion-free symmetric power and (iii) the integral closure of its torsion-free symmetric power is bounded above by a linear function in with leading coefficient at most . For a graded ideal of , the regularity of is given by a linear function of for all sufficiently large . The leading coefficient of this function is identified.

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Yang index of the deleted product

For any a -dimensional polyhedron is constructed such that the Yang index of its deleted product equals . This answers a question of Izydorek and Jaworowski (1995). For any a -dimensional closed manifold with involution is constructed such that , but can be mapped into a -dimensional polyhedron without antipodal coincidence.

     

Generalized Matlis duality

Let be a commutative noetherian ring and let be the minimal injective cogenerator of the category of -modules. A module is said to be reflexive with respect to if the natural evaluation map from to is an isomorphism. We give a classification of modules which are reflexive with respect to . A module is reflexive with respect to if and only if has a finitely generated submodule such that is artinian and is a complete semi-local ring.

     

The spectral properties of certain linear operators and their extensions

Let be a Hilbert space with inner-product , and let be a bounded positive operator on which determines an inner-product, . Denote by the completion of with respect to the norm . In this paper, operators having certain relationships with are studied. In particular, if where , then has an extension , and and have essentially the same spectral and Fredholm properties.

     

Every -regular

We prove the following: Theorem A. If is a -regular ultrafilter, then either

(a)

is -regular, or

(b)

the cofinality of the linear order is , and is -regular for all .

Corollary B. Suppose that is singular, and either is regular, or . Then every -regular ultrafilter is -regular.

We also discuss some consequences and variations.

     

Abstract blowing down

Assume that is a surface over an algebraically closed field . Let be obtained from by blowing up a smooth point and let be the exceptional curve. Let be the category of coherent sheaves on . In this note we show how to recover from , if we know the object .

     

A note on -bases of rings

Let be a tower of rings of characteristic . Suppose that is a finitely presented -module. We give necessary and sufficient conditions for the existence of -bases of over . Next, let be a polynomial ring where is a perfect field of characteristic , and let be a regular noetherian subring of containing such that . Suppose that is a free -module. Then, applying the above result to a tower of rings, we shall show that a polynomial of minimal degree in is a -basis of over .

     

Lindelöf property and absolute embeddings

It is proved that a Tychonoff space is Lindelöf if and only if whenever a Tychonoff space contains two disjoint closed copies and of , then these copies can be separated in by open sets. We also show that a Tychonoff space is weakly -embedded (relatively normal) in every larger Tychonoff space if and only if is either almost compact or Lindelöf (normal almost compact or Lindelöf).

     

Noncontinuity of spectrum for the adjoint of an operator

This paper deals with the connection between continuity of spectrum at an element of the Banach algebra of all bounded linear operators on a Banach space and at the adjoint of . In particular, we show that, if is not reflexive, the spectrum function may be continuous at and discontinuous at .