The flat model structure on

Given a cotorsion pair in an abelian category with enough objects and enough objects, we define two cotorsion pairs in the category of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when is hereditary. We then show that both of these induced cotorsion pairs are complete when is the ``flat' cotorsion pair of -modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new ``flat' model category structure on . In the last section we use the theory of model categories to show that we can define using a flat resolution of and a cotorsion coresolution of .

     

Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

Higher homotopy commutativity of -spaces and the permuto-associahedra

In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an -space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected -space has the finitely generated mod cohomology for a prime and the multiplication of it is homotopy commutative of the -th order, then it has the mod homotopy type of a finite product of Eilenberg-Mac Lane spaces s, s and s for .

     

On one-dimensional self-similar tilings and -tiles

Let be an integer base, a digit set and the set of radix expansions. It is well known that if has nonvoid interior, then can tile with some translation set ( is called a tile and a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of ; (ii) for a given , characterize so that is a tile.

We show that for a given pair , there is a unique self-replicating translation set , and it has period for some . This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for when are distinct primes. The only other known characterization is for , due to Lagarias and Wang. The proof for the case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.

     

The -module structure of -modules

Let be a regular ring, essentially of finite type over a perfect field . An -module is called a unit -module if it comes equipped with an isomorphism , where denotes the Frobenius map on , and is the associated pullback functor. It is well known that then carries a natural -module structure. In this paper we investigate the relation between the unit -structure and the induced -structure on . In particular, it is shown that if is algebraically closed and is a simple finitely generated unit -module, then it is also simple as a -module. An example showing the necessity of being algebraically closed is also given.

     

The cyclic and simplicial cohomology of

Let be the unital semigroup algebra of . We show that the cyclic cohomology groups vanish when is odd and are one dimensional when is even (). Using Connes' exact sequence, these results are used to show that the simplicial cohomology groups vanish for . The results obtained are extended to unital algebras for some other semigroups of .

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

Linking numbers in rational homology -spheres, cyclic branched covers and infinite cyclic covers

We study the linking numbers in a rational homology -sphere and in the infinite cyclic cover of the complement of a knot. They take values in and in , respectively, where denotes the quotient field of . It is known that the modulo- linking number in the rational homology -sphere is determined by the linking matrix of the framed link and that the modulo- linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate `modulo  ' and `modulo  '. When the finite cyclic cover of the -sphere branched over a knot is a rational homology -sphere, the linking number of a pair in the preimage of a link in the -sphere is determined by the Goeritz/Seifert matrix of the knot.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

Lusternik-Schnirelmann category of

First we give an upper bound of , the L-S category of a principal -bundle for a connected compact group with a characteristic map . Assume that there is a cone-decomposition of in the sense of Ganea that is compatible with multiplication. Then we have for , if is compressible into with trivial higher Hopf invariant . Second, we introduce a new computable lower bound, for . The two new estimates imply , where is a category weight due to Rudyak and Strom.

     

A sharp weak type inequality for martingale transforms and other subordinate martingales

If is a martingale difference sequence, a sequence of numbers in , and a positive integer, then

Here denotes the best constant. If , then as was shown by Burkholder. We show here that for the case 2$">, and that is also the best constant in the analogous inequality for two martingales and indexed by , right continuous with limits from the left, adapted to the same filtration, and such that is nonnegative and nondecreasing in . In Section 7, we prove a similar inequality for harmonic functions.

     

properties for Gaussian random series

Let be an arbitrary sequence of and let be a random series of the type

where is a sequence of independent Gaussian random variables and an orthonormal basis of (the finite measure space being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and sufficient condition for to belong to , for any almost surely is that . One of the main motivations behind this result is the construction of a nontrivial Gibbs measure invariant under the flow of the cubic defocusing nonlinear Schrödinger equation posed on the open unit disc of .

     

Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.

     

Character sums and congruences with

We estimate character sums with , on average, and individually. These bounds are used to derive new results about various congruences modulo a prime and obtain new information about the spacings between quadratic nonresidues modulo . In particular, we show that there exists a positive integer such that is a primitive root modulo . We also show that every nonzero congruence class can be represented as a product of 7 factorials, , where , and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials with represent ``almost all' residue classes modulo p, and that products of 3 factorials with are uniformly distributed modulo .

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

Involutions fixing , II

This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space with its normal bundle nonbounding and a Dold manifold with a positive even and 0$">. The complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on the codimension of may not be best possible. In particular, we find that there exist such involutions with nonstandard normal bundle to . Together with the results of part I of this title (Trans. Amer. Math. Soc. 354 (2002), 4539-4570), the argument for involutions fixing is finished.

     

Weighted estimates in for Laplace's equation on Lipschitz domains

Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

     

On the representation of integers as linear combinations of consecutive values of a polynomial

Let be a cyclotomic field with ring of integers and let be a polynomial whose values on belong to . If the ideal of generated by the values of on is itself, then every algebraic integer of may be written in the following form:

for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where

     

Simple birational extensions of the polynomial algebra

The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .