An algorithm for finding all preprojective components of the Auslander-Reiten quiver

The Auslander-Reiten quiver of a finite-dimensional associative algebra encodes information about the indecomposable finite-dimensional representations of and their homomorphisms. A component of the Auslander-Reiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the Auslander-Reiten translation. Preprojective components play an important role in the present research on algebras of finite and tame representation type. We present an algorithm which detects all preprojective components of a given algebra.

     

Classification of abelian hereditary directed categories satisfying Serre duality

In an ongoing project to classify all hereditary abelian categories, we provide a classification of -finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence.

In order to prove the classification, we will study the shapes of Auslander-Reiten components extensively and use appropriate generalizations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples.

     

The relationship between homological properties and representation theoretic realization of artin algebras

We will study the relationship of quite different objects in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, -categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers.

     

Noetherian hereditary abelian categories satisfying Serre duality

In this paper we classify -finite noetherian hereditary abelian categories over an algebraically closed field satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories.

As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.

     

A note on Selmer groups of abelian varieties over the trivializing extensions

We prove that for any abelian variety defined over a number field that is not isogenous to a product of CM elliptic curves, the pontrjagin dual of the Selmer group of the abelian variety over the trivializing extension has no nonzero pseudo-null submodules.

     

On the length of the spectral sequence of a Lie algebra extension

The length of the spectral sequence of a Lie algebra extension is at most the dimension of the quotient algebra. We show that this bound can be attained for arbitrarily large quotient algebras even when the algebra is nilpotent and the extension splits.

     

Almost split sequences in categories of representations of quivers

Let be a field and a connected quiver. In this note it is proved that the category of finite dimensional representations of over has almost split sequences if and only if either is without oriented cycles or consists of a single oriented cycle.

     

Classification of quasifinite modules over Lie algebras of matrix differential operators on the circle

We prove that an irreducible quasifinite module over the central extension of the Lie algebra of -matrix differential operators on the circle is either a highest or lowest weight module or else a module of the intermediate series. Furthermore, we give a complete classification of indecomposable uniformly bounded modules.

     

A smooth counterexample to Nori's conjecture on the fundamental group scheme

We show that Nori's fundamental group scheme does not base change correctly under extension of the base field for certain smooth projective ordinary curves of genus defined over a field of characteristic .

     

A Hopf algebra having a separable Galois extension is finite dimensional

It is shown that a Hopf algebra over a field admitting a Galois extension separable over its subalgebra of coinvariants is of finite dimension. This answers in the affirmative a question posed by Beattie et al. in [Proc. Amer. Math. Soc. 128, No. 11 (2000), 3201-3203]. It is also proven that this result holds true if has bijective antipode and the extension is Frobenius.

     

Existence of Auslander-Reiten sequences in subcategories

This paper studies the existence of Auslander-Reiten sequences in subcategories of mod(Λ), where Λ is a finite dimensional algebra over a field. The two main theorems give necessary and sufficient conditions for the existence of Auslander-Reiten sequences in subcategories.

Theorem. LetMbe a subcategory ofmod(Λ)closed under extensions and direct summands, and letMbe an indecomposable module inMsuch thatfor someinM. Then the following are equivalent.

(i)

has an-precover in the stable category,

(ii)

There is an Auslander-Reiten sequence0→X→Y→M→0inM.

     

Quadratic extensions of totally real quintic fields

In this work, we establish lists for each signature of tenth degree number fields containing a totally real quintic subfield and of discriminant less than in absolute value. For each field in the list we give its discriminant, the discriminant of its subfield, a relative polynomial generating the field over one of its subfields, the corresponding polynomial over , and the Galois group of its Galois closure.

We have examined the existence of several non-isomorphic fields with the same discriminants, and also the existence of unramified extensions and cyclic extensions.

     

An arithmetic obstruction to division algebra decomposability

This paper presents an indecomposable finite-dimensional division algebra of -power index that decomposes over a prime-to- degree field extension, obtained by adjoining -th roots of unity to the base. This shows that the theory of decomposability has an arithmetic aspect.

     

Sequences of consecutive smooth polynomials over a finite field

Given , we show that there are infinitely many sequences of consecutive -smooth polynomials over a finite field. The number of polynomials in each sequence is approximately .

     

Quadratic division algebras revisited (Remarks on an article by J. M. Osborn)

In his remarkable article ``Quadratic division algebras' (Trans. Amer. Math. Soc. 105 (1962), 202-221), J. M. Osborn claims to solve `the problem of determining all quadratic division algebras of order 4 over an arbitrary field of characteristic not two modulo the theory of quadratic forms over ' (cf. p. 206). While we shall explain in which respect he has not achieved this goal, we shall on the other hand complete Osborn's basic results (by a reasoning which is finer than his) to derive in the real ground field case a classification of all 4-dimensional quadratic division algebras and the construction of a 49-parameter family of pairwise nonisomorphic 8-dimensional quadratic division algebras.

To make these points clear, we begin by reformulating Osborn's fundamental observations on quadratic algebras in categorical terms.

     

Isomorphisms between Artin-Schreier towers

We give a method for efficiently computing isomorphisms between towers of Artin-Schreier extensions over a finite field. We find that isomorphisms between towers of degree over a fixed field can be computed, composed, and inverted in time essentially linear in . The method relies on an approximation process.

     

Value groups, residue fields, and bad places of rational function fields

We classify all possible extensions of a valuation from a ground field to a rational function field in one or several variables over . We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field in one variable, we consider the relative algebraic closure of in the henselization of with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of . In the ``tame case', we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the -adics.

     

MacNeille completions and canonical extensions

Let be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety is generated by an elementary class of relational structures.

Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.