Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms on algebras and their modules over the algebraic closure of the finite field of elements, we establish a relation between the representation theory of over and that of the -fixed point algebra over . More precisely, we prove that the category    mod- of finite-dimensional -modules is equivalent to the subcategory of finite-dimensional -stable -modules, and, when is finite dimensional, we establish a bijection between the isoclasses of indecomposable -modules and the -orbits of the isoclasses of indecomposable -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over can be interpreted as -stable representations of the corresponding quiver over . We further prove that every finite-dimensional hereditary algebra over is Morita equivalent to some , where is the path algebra of a quiver over and is induced from a certain automorphism of . A close relation between the Auslander-Reiten theories for and is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of is obtained by ``folding" the Auslander-Reiten quiver of . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.

     

Open loci of graded modules

Let be an excellent homogeneous Noetherian graded ring and let be a finitely generated graded -module. We consider as a module over and show that the -loci of are open in . In particular, the Cohen-Macaulay locus    is Cohen-Macaulay is an open subset of . We also show that the -loci on the homogeneous parts of are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module over an excellent ring and for an ideal which is not contained in any minimal prime of , the -loci for the modules are eventually stable.

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

Maximal theorems for the directional Hilbert transform on the plane

For a Schwartz function on the plane and a non-zero define the Hilbert transform of in the direction to be

p.v.

Let be a Schwartz function with frequency support in the annulus , and . We prove that the maximal operator maps into weak , and into for . The estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.

     

The braid index is not additive for the connected sum of 2-knots

Any -dimensional knot can be presented in a braid form, and its braid index, , is defined. For the connected sum of -knots and , it is easily seen that holds. Birman and Menasco proved that the braid index (minus one) is additive for the connected sum of -dimensional knots; the equality holds for -knots. We prove that the equality does not hold for -knots unless or is a trivial -knot. We also prove that the -knot obtained from a granny knot by Artin's spinning is of braid index , and there are infinitely many -knots of braid index .

     

Varieties with small discriminant variety

Let be a smooth complex projective variety, let be an ample and spanned line bundle on , defining a morphism and let be its discriminant locus, the variety parameterizing the singular elements of . We present two bounds on the dimension of and its main component relying on the geometry of . Classification results for triplets reaching the bounds as well as significant examples are provided.

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

Koszul duality and equivalences of categories

Let and be dual Koszul algebras. By Positselski a filtered algebra with gr is Koszul dual to a differential graded algebra . We relate the module categories of this dual pair by a Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

     

Unramified cohomology of classifying varieties for exceptional simply connected groups

Let be a classifying variety for an exceptional simple simply connected algebraic group . We compute the degree 3 unramified Galois cohomology of with values in over an arbitrary field . Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for semisimple simply connected over all fields.

These computations provide another family of examples of simple simply connected groups such that is not stably rational.

     

A generalization of Marshall's equivalence relation

For prime and for a field containing a root of unity of order , we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of of index . The equivalence classes then correspond to free pro- factors of the maximal pro- Galois group of . We generalize to this setting results of Jacob on the maximal pro- Galois group of a Pythagorean field.

     

The spectrum of twisted Dirac operators on compact flat manifolds

Let be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the -series, in terms of values of Hurwitz zeta functions, and the -invariant. We give the dimension of the space of harmonic spinors and characterize all -manifolds having asymmetric Dirac spectrum.

Furthermore, we exhibit many examples of Dirac isospectral pairs of -manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat -manifolds, pairwise nonhomeomorphic to each other of the order of .

     

Topological obstructions to certain Lie group actions on manifolds

Given a smooth closed -manifold , this article studies the extent to which certain numbers of the form are determined by the fixed-point set , where classifies the universal cover of , , is a polynomial in the Pontrjagin classes of , and is in the subalgebra of generated by . When , various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free -action extends to an action by some semisimple compact Lie group , then . Similar vanishing results are obtained for spin manifolds admitting certain -actions.

     

Stationary isothermic surfaces and uniformly dense domains

We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain in the -dimensional Euclidean space is said to be uniformly dense in a surface of codimension if, for every small the volume of the intersection of with a ball of radius and center does not depend on for

We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary , and we show that the principal curvatures of satisfy certain identities.

The case in which the surface coincides with is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if , it must be either a circle or a straight line and (ii) if it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

     

The fourth power moment of automorphic -functions for over a short interval

In this paper we will prove bounds for the fourth power moment in the aspect over a short interval of automorphic -functions for on the central critical line Re. Here is a fixed holomorphic or Maass Hecke eigenform for the modular group , or in certain cases, for the Hecke congruence subgroup with . The short interval is from a large to . The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg -function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).

     

Average size of -Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.

     

Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds

The purpose of this article is to study conformal harmonic maps , where is a closed Riemann surface and is a compact Riemannian manifold of dimension at least four. Such maps define parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of automorphisms of . They are Morse nondegenerate in the usual sense if has genus at least two, lie on two-dimensional nondegenerate critical submanifolds if has genus one, and on six-dimensional nondegenerate critical submanifolds if has genus zero.

     

-kernels of Lie groups

We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of and completely. We introduce two natural invariants and defined by the filtration, where is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are -regular or quasi -regular. We apply our results to the groups of self homotopy equivalences.

     

Maximal families of Gorenstein algebras

The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring . Let be the scheme parametrizing graded quotients of with Hilbert function . We prove there is a close relationship between the irreducible components of , whose general member is a Gorenstein codimension quotient, and the irreducible components of , whose general member is a codimension Cohen-Macaulay algebra of Hilbert function related to . If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of , this relationship actually determines a well-defined injective mapping from such ``Cohen-Macaulay' components of to ``Gorenstein' components of , in which generically smooth components correspond. Moreover the dimension of the ``Gorenstein' components is computed in terms of the dimension of the corresponding ``Cohen-Macaulay' component and a sum of two invariants of . Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.

     

Approximation and regularization of Lipschitz functions: Convergence of the gradients

We examine the possible extensions to the Lipschitzian setting of the classical result on -convergence: first (approximation), if a sequence of functions of class from to converges uniformly to a function of class , then the gradient of is a limit of gradients of in the sense that ; second (regularization), the functions can be chosen to be of class and -converging to in the sense that . In other words, the space of functions is dense in the space of functions endowed with the pseudo-norm.

We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials.

We then show that the regularization part can be extended to the Lipschitzian setting, namely if is a locally Lipschitz function, we build a sequence of smooth functions such that

In other words, the space of functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality , which cannot provide the exactness of our result.

As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.

     

Equivalence of domains arising from duality of orbits on flag manifolds

S. Gindikin and the author defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open -orbit on an ``arbitrary' flag manifold generalizing the result of Barchini. This conjecture for closed was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.