the representation dimension of a selfinjective algebra of euclidean type abstract we prove that the representation dimension of a selfinjective algebra of euclidean type is equal to three and give an explicit construction of the auslander generator of its module category

parametric galois extensions abstract given a field k and a finite group h an h parametric extension over k is a finite galois extension of k t of galois group containing h which is regular over k and has all the galois extensions of k of group h among its specializations we are mainly interested in producing non h parametric extensions which relates to classical questions in inverse galois theory like the beckmann black problem and the existence of one parameter generic polynomials we develop a general approach started in a preceding paper and provide new non parametricity criteria and new examples

algebras with separating almost cyclic coherent auslander reiten components we prove that the class of artin algebras whose auslander reiten quiver admits a separating family of almost cyclic coherent components coincides with the class of generalized multicoil enlargements of concealed canonical algebras moreover the module category homological properties and the representation type of artin algebras with separating families of almost cyclic coherent auslander reiten components are described

periodicity of self injective algebras of polynomial growth abstract let a be an indecomposable representation infinite tame finite dimensional algebra of polynomial growth over an algebraically closed field we prove that a is a periodic algebra with respect to action of the bimodule syzygy operator if and only if a is morita equivalent to a socle deformation of an orbit algebra b g where b is the repetitive category of a tubular algebra b and g is an admissible infinite cyclic automorphism group of b the main contribution in the paper is to prove the sufficiency part of this equivalence it is known that every orbit algebra b g of a tubular algebra b admits a presentation as an orbit algebra t b r h of an r fold trivial extension algebra t b r of b with respect to free action of a finite cyclic automorphism group h of t b r a significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self injective algebras of polynomial growth and showing that all of them are periodic algebras then we deduce the periodicity of all algebras socle equivalent to the orbit algebras b g t b r h of tubular algebras b from the periodicity of these ten exceptional algebras using invariance of periodicity for finite galois coverings and derived equivalences of algebras

quadratic forms and iterated tilted algebras given a finite quiver a without oriented cycles an algebra a is called an iterated tilted algebra of type a 2 if there exists a sequence of algebras a a a 1 a where a is the path algebra of a and a sequence of tilting modules t 0 and every indecomposable a module satisfies either hom j t m 0 or exta t m 0 if m 1 a is called a tilted algebra of type a 25 the representation theory of iterated tilted algebras was proved to be closely related to that of a class of symmetric algebras namely the trivial extension algebras see 3 4 18 27 291 recently they were also shown to arise naturally in the study of the derived category of a finite dimensional algebra see 23 24 71 iterated tilted algebras of type a where the underlying graph of a is a dynkin diagram were studied in 1 2 8 223 and the iterated tilted algebras of euclidean type a m 3 1 were classified in s 55 0021 8693 90 3 00

on krull gabriel dimension and galois coverings abstract assume that k is an algebraically closed field r a locally support finite locally bounded k category g a torsion free admissible group of k linear automorphisms of r and a r g we show that the krull gabriel dimension kg r of r equals the krull gabriel dimension kg a of a furthermore we determine the krull gabriel dimension of locally support finite repetitive k categories these results are applied to determine the krull gabriel dimension of standard selfinjective algebras of polynomial growth finally we show that there are no super decomposable pure injective modules over standard selfinjective algebras of domestic type

rational rigidity and the sporadic groups the question of which finite groups can or do occur as the galois group of a galois extension of the rational numbers is one which has been studied since early in the twentieth century it is conjectured that all finite groups do occur as galois groups over the rationals this is known to be true for all solvable groups see jacobson 73 for brief historical notes in a breakthrough paper thompson 9 has defined the new concept of rigidity of a finite group he then proves that if a group g with trivial centre is rigid then an associated group g is a galois group over the rationals in this paper it is shown that 14 of the 26 sporadic simple groups are galois groups over the rationals the method used involves the complex character table and knowledge of maximal subgroups of the group in some cases where the maximal subgroups are not known the classification of finite simple groups is used section 1 contains the results section 2 describes the notation used and section 3 details the proof for each of the 14 groups separately

tame generalized canonical algebras abstract recently a new wide class of finite dimensional algebras whose auslander reiten quivers admit infinite families of faithful standard stable tubes called generalized canonical algebras has been introduced in a skowronski colloq math 90 2001 77 93 we give a complete description of all tame generalized canonical algebras and prove that all these algebras have tame derived categories of bounded complexes

tame algebras and tits quadratic forms abstract we solve a long standing problem concerning the connection between the tameness of simply connected algebras and the weak nonnegativity of the associated tits integral quadratic forms and derive some consequences

minimal realisation of bilinear and quadratic input output difference equations in state space form this article studies the realisability property of discrete time bilinear and quadratic input output i o equations in the classical state space form constraints on the parameters of the i o model are suggested that lead to realisable models using new formulae for computing basis vectors of certain vector spaces of differential one forms we present in this article the complete list of the third and fourth order realisable i o bilinear models and a new realisable subclass of an arbitrary order is suggested moreover we provide the sufficient conditions of the second and third order realisable i o quadratic models respectively all the developed theory and algorithms are illustrated by means of several examples

two sided gluings of tilted algebras we study the class of algebras a satisfying the property all but at most finitely many non isomorphic indecomposable a modules are such that all their predecessors have projective dimension at most one or all their successors have injective dimension at most one such a class includes the tilted algebras d happel c ringel trans amer math soc 274 1982 399 443 the quasi tilted algebras d happel i reiten s smalo mem am math soc 120 1996 575 the shod algebras f u coelho m lanzilotta manuscripta mathematica 100 1999 1 11 the weakly shod f u coelho m lanzilotta preprint 2001 and the left and right glued algebras i assem f u coelho j pure appl algebra 96 3 1994 225 243

on the support algebras of indecomposable modules over tame algebras of polynomial growth we investigate the support algebras of finite dimensional indecomposable modules over representation infinite tame algebras of polynomial growth over an algebraically closed field

strongly simply connected schurian algebras and multiplicative bases abstract in this paper we define concepts of crowns and quasi crowns valid in an arbitrary schurian algebra and which generalise the corresponding concepts in an incidence algebra we show first that a triangular schurian algebra is strongly simply connected if and only if it is simply connected and contains no quasi crown we then prove that the absence of quasi crowns in a triangular schurian algebra implies the existence of a multiplicative basis

classical state space realizability of input output bilinear models this paper studies the realizability property of bilinear input output i o models in the classical state space form constraints on the parameters of the bilinear i o model are suggested that lead to realizable models the complete list of 2nd and 3rd order realizable input output bilinear models together with the corresponding state equations is given in the general case some subclasses of realizable bilinear models together with their state space realizations are presented including the diagonal bilinear model and the special subclass of the superdiagonal bilinear model

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

on classical state space realisability of quadratic input output differential equations this article studies the realisability property of continuous time quadratic input output i o equations in the classical state space form constraints on the parameters of the quadratic i o model are suggested that lead to realisable models the complete list of second and third order realisable i o quadratic models is given and two subclasses of the n th order realisable i o quadratic systems are suggested our conditions rely basically upon the property that certain combinations of coefficients of the i o equations are zero or not zero we provide explicit state equations for realisable second order quadratic i o equations and for one realisable subclass of quadratic i o equations of arbitrary order

the universal cover of an algebra without double bypass let a be a basic and connected finite dimensional algebra over a field k of characteristic zero we show that if the quiver of a has no double bypass then the fundamental group as defined in r martinez villa j a de la pena the universal cover of a quiver with relations j pure appl algebra 30 1983 277 292 of any presentation of a by quiver and relations is the quotient of the fundamental group of a privileged presentation of a then we show that the galois covering of a associated with this privileged presentation satisfies a universal property with respect to the connected galois coverings of a in a similar fashion to the universal cover of a topological space

some realizations of algorithms for nonlinear input output systems this paper develops an algorithm which given an input output differential equation produces under appropriate circumstances a parametrization of the observation space of the underlying state space system the paper extends previous results by the authors to a more general setting so that state space systems in which the controls enter nonlinearly can be accommodated and most importantly the paper deals with implicitly defined systems at both the input output and state space level fliess and hasler have demonstrated the importance of implicit input output models which yield only local state space descriptions various realizations are obtained depending on the properties of the derived candidate observation space examples of the algorithm are described with the aid of an implementation using the computer algebra system axiom

on auslander reiten components of blocks and self injective biserial algebras we investigate the existence of auslander reiten components of euclidean type for special biserial self injective algebras and for blocks of group algebras in particular we obtain a complete description of stable auslander reiten quivers for the tame self injective algebras considered here

state space realization of bilinear continuous time input output equations this paper studies the realizability property of continuous time bilinear input output i o equations in the classical state space form constraints on the parameters of the bilinear i o model are suggested that lead to realizable models the paper proves that the 2nd order bilinear i o differential equation unlike the discrete time case is always realizable in the classical state space form the complete list of 3rd and 4th order realizable i o bilinear models is given and two subclasses of realizable i o bilinear systems are suggested our conditions rely basically upon the property that certain combinations of coefficients of the i o equations are zero or not zero we provide explicit state equations for all realizable 2nd and 3rd order bilinear i o equations and for one realizable subclass of bilinear i o equations of arbitrary order

the simple connectedness of a tame tilted algebra let k be an algebraically closed field by algebra a we mean a finite dimensional associative k algebra which we assume moreover to be basic and connected for such an algebra a there exists a uniquely deter mined connected quiver q and at least a surjective algebra morphism a from the path algebra kq of q into a whose kernel is denoted by i a a see 6 the algebra a is called triangular if q contains no oriented a

on galois coverings and tilting modules abstract let a be a basic connected finite dimensional algebra over an algebraically closed field k let t be a basic tilting a module with arbitrary finite projective dimension for a fixed group g we compare the set of isoclasses of connected galois coverings of a with group g and the set of isoclasses of connected galois coverings of end a t with group g using the hasse diagram k a see d happel l unger on a partial order of tilting modules algebr represent theory 8 2 2005 147 156 and c riedtmann a schofield on a simplicial complex associated with tilting modules comment math helv 66 1 1991 70 78 of basic tilting a modules we give sufficient conditions on t under which there is a bijection between these two sets these conditions are always verified when a is of finite representation type then we apply these results to study when the simple connectedness of a implies the one of end a t see i assem e n marcos j a de la pena the simple connectedness of a tame tilted algebra j algebra 237 2 2001 647 656 finally using an argument due to w crawley boevey we prove that the type of any simply connected tilted algebra is a tree and that its first hochschild cohomology group vanishes

irreducibility of moduli spaces of cyclic unramified covers of genus g curves let c1 cr g c g be an r tuple consisting of a transitive subgroup g of sm and r conjugacy classes c1 cr of g we consider the concept of the moduli space w c g of compact riemann surface covers of the riemann sphere of nielsen class c g the irreducibility of c g is equivalent to the transitivity of a specific permutation representation of the hurwitz monodromy group 1 but there are few general tools to decide questions about this representation theorem 2 gives a class of examples of c g for which c g is irreducible as an immediate corollary this gives an elementary proof and generalization of the irreduciblity of the moduli space of cyclic unramified covers of genus g curves for which deligne and mumford dm theorem 5 15 applied teichmuller theory and dehn s theorem this contrasts with the examples of c g in bfr for which c g is reducible these kinds of questions combined with the study of the existence of rational subvarieties of w c g have application to the realization of a group g as the galois group of a regular extension of 12 t fr3 4j

algebras with cycle finite galois coverings we prove that the finite dimensional algebras over an algebraically closed field which admit cycle finite galois coverings with torsion free galois groups are of tame representation type and derive some consequences

characterizations of algebras with small homological dimensions abstract we introduce the class of double tilted algebras containing the class of tilted algebras and prove various characterizations in particular we show that the class of double tilted algebras is the class of all artin algebras whose ar quiver admits a faithful double section with a natural property moreover we prove that the class of double tilted algebras coincides with the class of all artin algebras of global dimension three for which every indecomposable finitely generated module has projective or injective dimension at most one we also describe the structure of the category of finitely generated modules as well as the ar quiver of double tilted algebras

on the additive categories of generalized standard almost cyclic coherent auslander reiten components abstract we prove geometric and homological properties of modules in the additive categories of generalized standard almost cyclic coherent components in the auslander reiten quivers of finite dimensional algebras

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p