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.

     

A general theory of almost convex functions

Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -almost convex iff for all and the inequality

holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function is constructed with the properties that it vanishes on the vertices of and if is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

     

Length, multiplicity, and multiplier ideals

Let be an -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an -primary ideal of , the relationship between the singularities of the scheme defined by and those defined by the multiplier ideals , with varying in , are quantified in this paper by showing that the Samuel multiplicity of satisfies whenever . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

     

Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Let be a bigraded ideal in the bigraded polynomial ring . Assume that has codimension 2. Then is a finite set of points. We prove that if is a local complete intersection, then any syzygy of the vanishing at , and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

     

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.

     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of . In this paper we define and study a sequence of metric invariants , , associated to a compact convex body with base point in the interior of . The invariant measures how lopsided is in dimension with respect to . The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate for convex polytopes.

     

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.

     

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.

     

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.

     

On higher syzygies of ruled surfaces

We study higher syzygies of a ruled surface over a curve of genus with the numerical invariant . Let    Pic be a line bundle in the numerical class of . We prove that for , satisfies property if and , and for , satisfies property if and . By using these facts, we obtain Mukai-type results. For ample line bundles , we show that satisfies property when and or when and . Therefore we prove Mukai's conjecture for ruled surface with . We also prove that when is an elliptic ruled surface with , satisfies property if and only if and .

     

Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

In the first section of this paper we revisit the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra over an arbitrary field . Our main observation is that , the minimal polynomial of , may depend not only on , but also on the underlying algebra. More precisely, if is a subalgebra of , and if is the minimal polynomial of in , then may differ from , in which case we have .

In the second section we restrict attention to the case where is either the real or the complex numbers, and define , the radius of an element in , to be the largest root in absolute value of the minimal polynomial of . We show that possesses some of the familiar properties of the classical spectral radius. In particular, we prove that is a continuous function on .

In the third and last section, we deal with stability of subnorms acting on subsets of finite-dimensional power-associative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings of the previous section. Our main new result states that if , a subset of an algebra , satisfies certain assumptions, and is a continuous subnorm on , then is stable on if and only if majorizes the radius defined above.

     

On initial segment complexity and degrees of randomness

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define to mean that . The equivalence classes under this relation are the -degrees. We prove that if is -random, then and have no upper bound in the -degrees (hence, no join). We also prove that -randomness is closed upward in the -degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the -degrees. Unlike the -degrees, many basic properties of the -degrees are easy to prove. We show that implies , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for , the analogue of for plain Kolmogorov complexity.

Two other interesting results are included. First, we prove that for any , a -random real computable from a --random real is automatically --random. Second, we give a plain Kolmogorov complexity characterization of -randomness. This characterization is related to our proof that implies .

     

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 .

     

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.

     

Effective cones of quotients of moduli spaces of stable -pointed curves of genus zero

Let , the moduli space of -pointed stable genus zero curves, and let be the quotient of by the action of on the last marked points. The cones of effective divisors , , are calculated. Using this, upper bounds for the cones generated by divisors with moving linear systems are calculated, , along with the induced bounds on the cones of ample divisors of and . As an application, the cone is analyzed in detail.

     

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.

     

A tensor norm preserving unconditionality for -spaces

We show that, for each , there is an -tensor norm (in the sense of Grothendieck) with the surprising property that the -tensor product has local unconditional structure for each choice of arbitrary -spaces . In fact, is the tensor norm associated to the ideal of multiple -summing -linear forms on Banach spaces.

     

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.

     

-theorems for fusion systems

For an odd prime, we generalise the Glauberman-Thompson -nilpotency theorem (Gorenstein, 1980) to arbitrary fusion systems. We define a notion of -free fusion systems and show that if is a -free fusion system on some finite -group , then is controlled by for any Glauberman functor , generalising Glauberman's -theorem (Glauberman, 1968) to arbitrary fusion systems.