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

geometrically reductive hopf algebras and their invariants abstract by analogy with the mumford definition of geometrically reductive algebraic group we introduce the concept of geometrically reductive hopf algebra over a field then we prove that if h is a geometrically reductive hopf algebra and a is a commutative finitely generated and locally finite h module algebra then the algebra of invariants a h is finitely generated we also prove that in characteristic 0 a hopf algebra h is geometrically reductive if and only if every finite dimensional h module is semisimple and that in positive characteristic every finite dimensional hopf algebra is geometrically reductive finally we prove that in positive characteristic the quantum enveloping hopf algebras u q sl n n 2 are geometrically reductive for any parameter q 1

additive group actions on affine t varieties of complexity one in arbitrary characteristic abstract let x be a normal affine t variety of complexity at most one over a perfect field k where t g m n stands for the split algebraic torus our main result is a classification of additive group actions on x that are normalized by the t action this generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero with the assumption that the characteristic of k is positive we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine t varieties normalized up to a frobenius map as a preliminary result we provide a complete description of these g a actions in the toric situation

integrality of module algebras over its invariants abstract let k be a field and h be a finite dimensional hopf algebra over k if h is cocommutative then it is known that all commutative h module algebras are integral over its invariants here we prove this result for those hopf algebras h such that either h is involutionary with char k dim h or h has a cocommutative coradical and k is of positive characteristic examples are constructed to illustrate that the conclusion is not true in general

on finite generation of symbolic rees rings of space monomial curves and existence of negative curves abstract in this paper we shall study finite generation of symbolic rees rings of the defining ideal of the space monomial curves t a t b t c for pairwise coprime integers a b c such that a b c 1 1 1 if such a ring is not finitely generated over a base field then it is a counterexample to the hilbert s fourteenth problem finite generation of such rings is deeply related to existence of negative curves on certain normal projective surfaces we study a sufficient condition definition 3 6 for existence of a negative curve using it we prove that in the case of a b c 2 a b c a negative curve exists using a computer we shall show that there exist examples in which this sufficient condition is not satisfied

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

the separating variety for the basic representations of the additive group abstract for a group g acting on an affine variety x the separating variety is the closed subvariety of x x encoding which points of x are separated by invariants we concentrate on the indecomposable rational linear representations v n of dimension n 1 of the additive group of a field of characteristic zero and decompose the separating variety into the union of irreducible components we show that if n is odd divisible by four or equal to two the closure of the graph of the action which has dimension n 2 is the only component of the separating variety in the remaining cases there is a second irreducible component of dimension n 1 we conclude that in these cases there are no polynomial separating algebras

computation of invariants for reductive groups abstract we will give an algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group the algorithm basically consists of a single grobner basis computation we will also show a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants

locally nilpotent derivations over a ufd and an application to rank two locally nilpotent derivations ofk x1 xn abstract given a ufd r containing the rational numbers we study locally nilpotent r derivations of the polynomial ring r x y in particular we give a generalization of rentschler s theorem and a criterion for the existence of a slice these results are then applied to describe rank two locally nilpotent derivations of k x 1 x n where k is a field of characteristic zero we also give an example of a nontriangulable locally nilpotent derivation of k x y z whose set of fixed points is a line

representation dimension and quasi hereditary algebras we study auslander s representation dimension of artin algebras which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators we show the following statements 1 if an artin algebra a is stably hereditary then the representation dimension of a is at most 3 2 if two artin algebras are stably equivalent of morita type then they have the same representation dimension particularly if two self injective algebras are derived equivalent then they have the same representation dimension 3 any incidence algebra of a finite partially ordered set over a field has finite representation dimension moreover we use results on quasi hereditary algebras to show that 4 the auslander algebra of a nakayama algebra has finite representation dimension

hopf co addition for free magma algebras and the non associative hausdorff series abstract generalizations of the series exp and log to noncommutative non associative and other types of algebras were considered by m lazard and recently by v drensky and l gerritzen there is a unique power series exp x in one non associative variable x such that exp x exp x exp 2 x exp 0 1 we call the unique series h h x y in two non associative variables satisfying exp h exp x exp y the non associative hausdorff series and we show that the homogeneous components h n of h are primitive elements with respect to the co addition for non associative variables we describe the space of primitive elements for the co addition in non associative variables using taylor expansion and a projector onto the algebra a 0 of constants for the partial derivations by a theorem of kurosh a 0 is a free algebra we describe a procedure to construct a free algebra basis consisting of primitive elements

multifractal decompositions of moran fractals we present a rigorous construction and generalization of the multifractal decomposition for moran fractals with infinite product measure the generalization is specified by a system of nonnegative weights in the partition sum all the usual smooth properties of the f theory are recovered for the case that the weights are equal to unity the generalized spectrum f w is invariant to a group of gauge transformations of the weights and in addition need no longer be concave in case the fractal is a cantor set generated by an iterated function system of similarities is the pointwise dimension of the measure we discuss properties of some examples

kernel of locally nilpotent r derivations of r x y in this paper we study the kernel of a non zero locally nilpotent r derivation of the polynomial ring r x y over a noetherian integral domain r containing a field of characteristic zero we show that if r is normal then the kernel has a graded r algebra structure isomorphic to the symbolic rees algebra of an unmixed ideal of height one in r and conversely the symbolic rees algebra of any unmixed height one ideal in r can be embedded in r x y as the kernel of a locally nilpotent r derivation of r x y we also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general

invariant rings through categories abstract we formulate a notion of geometric reductivity in an abstract categorical setting which we refer to as adequacy the main theorem states that the adequacy condition implies that the ring of invariants is finitely generated this result applies to the category of modules over a bialgebra the category of comodules over a bialgebra and the category of quasi coherent sheaves on an algebraic stack of finite type over an affine base

weitzenbock derivations of free metabelian lie algebras abstract a nonzero locally nilpotent linear derivation of the polynomial algebra k x d k x 1 x d in several variables over a field k of characteristic 0 is called a weitzenbock derivation the classical theorem of weitzenbock states that the algebra of constants k x d which coincides with the algebra of invariants of a single unipotent transformation is finitely generated similarly one may consider the algebra of constants of a locally nilpotent linear derivation of a finitely generated not necessarily commutative or associative algebra which is relatively free in a variety of algebras over k now the algebra of constants is usually not finitely generated except for some trivial cases this holds for the algebra of constants l d l d of the free metabelian lie algebra l d l d with d generators we show that the vector space of the constants l d l d in the commutator ideal l d l d is a finitely generated k x d module for small d we calculate the hilbert series of l d l d and find the generators of the k x d module l d l d this gives also an infinite set of generators of the algebra l d l d

representation dimension via tilting theory abstract let be an artin algebra over a commutative artinian ring r and let t be a separating splitting tilting module with endomorphism end t the aim of this paper is to use tilting theory to study the representation dimension of the main result asserts that for an integer n 1 if is n gorenstein of finite cohen macaulay type then r e p d i m n 2 we conclude that if is a n gorenstein artin algebra of finite cohen macaulay type then r e p d i m n 2 this in particular shows that for any artin algebra r e p d i m g l d i m 2

on the representation dimension of tilted and laura algebras abstract we prove that the representation dimension of a tilted or of a strict laura algebra is at most three

finitistic dimension conjecture and relative hereditary algebras abstract we introduce the notion of relative hereditary artin algebras as a generalization of algebras with representation dimension at most 3 we prove the following results 1 the relative hereditariness of an artin algebra is left right symmetric and is inherited by endomorphism algebras of projective modules 2 the finitistic dimensions of a relative hereditary algebra and its opposite algebra are finite as a consequence the finitistic projective dimension conjecture the finitistic injective dimension conjecture the gorenstein symmetry conjecture the wakamatsu tilting conjecture and the generalized nakayama conjecture hold for relative hereditary artin algebras and endomorphism algebras of projective modules over them in particular over algebras with representation dimension at most 3 we also show that the torsionless finiteness of an artin algebra is inherited by endomorphism algebras of projective modules and consequently give a partial answer to the question if the representation dimension of the endomorphism algebra of any projective module over an artin algebra a is bounded by the representation dimension of a

radical embeddings and representation dimension given a representation finite algebra b and a subalgebra a of b such that the jacobson radicals of a and b coincide we prove that the representation dimension of a is at most three by a result of igusa and todorov this implies that the finitistic dimension of a is finite

invariants of finite hopf algebras abstract this paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite hopf algebras on commutative rings topics considered include integrality over the invariant rings properties of the canonical map between the prime spectra orbital and stabilizer algebras projectivity over the invariant rings and descent of cohen macaulayness

a counterexample to the fourteenth problem of hilbert in dimension four abstract this paper presents a family of new counterexamples to hilbert s fourteenth problem they are realized as subfields of the rational function field of four variables over a field of characteristic zero of transcendence degree three it was previously not known whether any counterexample could be found as a subfield of a rational function field of four variables or whether counterexamples with minimal transcendence degree three could exist

linear counterexamples to the fourteenth problem of hilbert abstract following ideas of a campo neuen indag math n s 5 1994 253 257 we construct linear counterexamples to the fourteenth problem of hilbert from non linear counterexamples d daigle g freudenburg j algebra 221 1999 528 535 g freudenburg transform groups 5 2000 61 71

cohomology of twisted tensor products abstract it is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies we look at twisted tensor products and investigate to which extent this is still true we give an explicit description of the ext algebra of the tensor product of two modules and under certain additional conditions describe an essential part of the hochschild cohomology ring of a twisted tensor product as an application we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the hochschild cohomology ring moreover both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra

on the finitistic dimension conjecture iii related to the pair eae a let a be an artin algebra and e an idempotent element in a in this paper we use co homological conditions on a to control the finitistic dimension of eae such a consideration is of particular interest for understanding the finitistic dimension conjecture let us denote the finitistic dimension and global dimension of a by fin dim a and gl dim a respectively suppose gl dim a 4 then fin dim eae if one of the following conditions holds 1 a aea has representation dimension at most 3 2 a 3 a is an a aea module 3 proj dim sa 3 for all simple a aea modules s this result can be considered as a first step to the question of whether gl dim a 4 implies fin dim eae moreover we show the following let a be an arbitrary artin algebra and e an idempotent element of a such that the syzygy dimension or the gorenstein dimension of the eae module ae is finite if fin dim a then fin dim eae

ml invariant of the symmetric algebra of a projective module over a one dimensional affine domain abstract let k be an algebraically closed field and let a be an affine domain of dimension one over k let p be a finitely generated projective a module of rank d and let r sym a p be the symmetric algebra of p over a assume that a is not a polynomial algebra over k in this article we show that under these assumptions the makar limanov invariant ml r a

on the representation dimension of finite dimensional algebras the importance of the representation finite artin algebras for the whole representation theory of artin algebras is well understood and there is much literature on the subject see 1 6 12 these are artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules moreover there is a bijective correspondence between the class of representation finite artin algebras and that of artin algebras with global dimension of at most 2 and with dominant dimension of at least 2 this result was established by auslander in 1 motivated by this correspondence auslander introduced the concept of representation dimension for artin algebras to study the connection of arbitrary artin algebras with representation finite artin algebras it is hoped that this notion gives a reasonable way of measuring how far an artin algebra is from being representation finite type 1 p 134 representation dimension is a morita invariant of artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators unfortunately little seems to be known about representation dimension so there are a lot of essential questions on this invariant dealing with the computation and estimation of the representation dimension of an artin algebra there are a few cases known auslander proved that the representation dimension of an artin algebra is two if and

dynamical system related to quasiperiodic schrodinger equations in one dimension a dynamical map is obtained from a class of quasiperiodic discrete schrodinger equations in one dimension which include the fibonacci system the potentials are constant except for steps at special points

divisors on rational normal scrolls let a be the homogeneous coordinate ring of a rational normal scroll the ring a is equal to the quotient of a polynomial ring s by the ideal generated by the two by two minors of a scroll matrix with two rows and l catalecticant blocks the class group of a is cyclic and is infinite provided l is at least two one generator of the class group is j where j is the ideal of a generated by the entries of the first column of the positive powers of j are well understood if l is at least two then the nth ordinary power the nth symmetric power and the nth symbolic power coincide and therefore all three nth powers are resolved by a generalized eagon northcott complex the inverse of j in the class group of a is k where k is the ideal generated by the entries of the first row of we study the positive powers of k we obtain a minimal generating set and a grobner basis for the preimage in s of the symbolic power k n we describe a filtration of k n in which all of the factors are cohen macaulay s modules resolved by generalized eagon northcott complexes we use this filtration to describe the modules in a finely graded resolution of k n by free s modules we calculate the regularity of the graded s module k n and we show that the symbolic rees ring of k is noetherian

geometrically reductive hopf algebras ii abstract in this paper we prove that every finite dimensional commutative hopf algebra is geometrically reductive in the case of characteristic zero this implies that it is the sum of simple subcoalgebras then we study the case of characteristic p and relate the geometric reductivity of a hopf algebra with the corresponding property of its frobenius kernels along the way we prove some results on finite generation of invariants

on the polynomiality of invariant rings for codimension one ga modules abstract let k be an algebraically closed field of positive characteristic and let g a be its additive group let g a act linearly on a finite dimensional vector space v let k v be the ring of polynomial functions on v and let k v g a be the ring of g a invariants in this article we show that the ring of g a invariants k v g a is a polynomial ring under the assumption that v is a codimension one g a module

multiplicities of schur functions in invariants of two 3 3 matrices abstract we relate with any symmetric function f x y c x y presented as an infinite linear combination of schur functions f x y m 1 2 s 1 2 x y the multiplicity series m f m 1 2 t 1u 2 we study the behavior of m f under natural combinatorial and algebraic constructions in particular we calculate the multiplicity series for the symmetric algebra of the irreducible gl 2 c module corresponding to the complete symmetric function of degree 3 our main result is that we have found the explicit form of the multiplicity series for the hilbert or poincare series of the algebra of invariants of two 3 3 matrices as a consequence we have precised the result of berele on the asymptotics of the multiplicities in the trace cocharacter sequence of two 3 3 matrices

representation dimensions of triangular matrix algebras abstract let a be a finite dimensional hereditary algebra over an algebraically closed field k t 2 a a 0 a a be the triangular matrix algebra we prove that rep dim t 2 a is at most three if a is dynkin type and rep dim t 2 a is at most four if a is not dynkin type let a 1 a 0 da a be the duplicated algebra of a let t be a tilting a module and t t p be a tilting a 1 module we show that end a 1 t is representation finite if and only if the full subcategory x a y a f x a mod a y a 1 f t a add a of mod t 2 a is of finite type where is the auslander reiten translation and f t a is the torsion free class of mod a associated with t moreover we also prove that rep dim end a 1 t is at most three if a is dynkin type

homological theory of recollements of abelian categories abstract we investigate several homological aspects of recollements of abelian categories in particular we study how various homological invariants and dimensions of the categories involved in a recollement situation are related and when recollements of abelian categories induce recollements at the level of the bounded derived categories finally we give applications to global finitistic and representation dimension of rings and artin algebras and to rouquier s dimension of triangulated categories

on the finite generation of additive group invariants in positive characteristic abstract roberts freudenburg and daigle and freudenburg have given the smallest counterexamples to hilbert s fourteenth problem as rings of invariants of algebraic groups each is of an action of the additive group on a finite dimensional vector space over a field of characteristic zero and thus each is the kernel of a locally nilpotent derivation in positive characteristic additive group actions correspond to locally finite iterative higher derivations we set up characteristic free analogs of the three examples and show that contrary to characteristic zero in every positive characteristic the invariants are finitely generated

the representation dimension of hecke algebras and symmetric groups abstract we establish a lower bound for the representation dimension of all the classical hecke algebras of types a b and d for all the type a algebras and most of the algebras of types b and d we also establish upper bounds moreover we establish bounds for the representation dimension of group algebras of some symmetric groups

factorial algebraic group actions and categorical quotients abstract given an action of an affine algebraic group with only trivial characters on a factorial variety we ask for categorical quotients we characterize existence in the category of algebraic varieties moreover allowing constructible sets as quotients we obtain a more general existence result which for example settles the case of a finitely generated algebra of invariants as an application we provide a combinatorial git type construction of categorical quotients for actions of not necessarily reductive groups on e g complete varieties with finitely generated cox ring via lifting to the characteristic space

on finite generation of kernels of locally nilpotent r derivations of r x y z let r be a noetherian domain containing the field of rationals we show that if r is dedekind then the kernel of any locally nilpotent r derivation of r x y z is a finitely generated r algebra conversely we show that if r is neither a field nor a dedekind domain then there exists a locally nilpotent r derivation of r x y z whose kernel is not finitely generated over r

separating invariants for 2 2 matrices abstract a minimal separating set is found for the algebra of matrix invariants of several 2 2 matrices over an infinite field of arbitrary characteristic

triangular derivations of k x1 x2 x3 x4 abstract it is shown that if k is a field of characteristic zero then the kernel of any triangular k derivation of k x 1 x 2 x 3 x 4 is finitely generated as a k algebra this is obtained as a corollary of a more general result concerning triangular r derivations of r x y z for certain rings r

a hilbert nagata theorem in noncommutative invariant theory nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subal gebra of invariants in this paper we consider groups acting on noncommutative algebras over a field of characteristic zero we characterize all the t ideals of the free associative algebra such that the algebra of invariants in the co rresponding relatively free algebra is finitely generated for any group action from the class of nagata in particular in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in lie sense as a consequence we extend the hilbert nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is lie nilpotent we also prove that the hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non matrix polynomial identity is rational if the action satisfies the condition of nagata

multifractal analysis of brownian zero set the multifractal structure of zerosz of brownian motion is considered for different measures onz we find typical characteristics the function and the multifractal spectrumf a dimensional interpretation off is also discussed

on the representation dimension of some classes of algebras abstract we show that the representation dimension of the following classes of algebras is at most 3 a artin algebras a such that the functor hom a d a has finite length or dually hom a a has finite length these algebras coincide with the right left glued algebras as introduced in i assem f u coelho j pure appl algebra 96 3 1994 225 and b trivial extensions of iterated tilted algebras

the nagata automorphism is wild it is proved that the well known nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild that is it can not be decomposed into a product of elementary automorphisms

anomalies in multifractal formalism for local time of brownian motion the renyi function for the logical time measure of brownian motion is found it is shown that this function the legendre transform of the multifractal spectrum of and the function derived by the reciprocal measure formalism are not identical more examples of having similar anomalies are discussed

hilbert series of fixed free algebras and noncommutative classical invariant theory we compute an asymptotic formula for the number of invariants of a given degree which compares nicely with the corresponding result in the classical commutative case in section 6 we show that if g is an infinite cyclic group generated by a unipotent matrix then

invariant properties of representations under excellent extensions abstract in this paper we introduce the notion of weak excellent extensions of rings as a generalization of that of excellent extensions of rings let be a weak excellent extension of an artinian algebra we prove that if is of finite representation type resp cm finite cm free then so is furthermore if is an excellent extension of then the converse also holds true we also study when the representation dimension of an artinian algebra is invariant under excellent extensions

constants of weitzenbock derivations and invariants of unipotent transformations acting on relatively free algebras abstract in commutative algebra a weitzenbock derivation is a nonzero triangular linear derivation of the polynomial algebra k x 1 x m in several variables over a field k of characteristic 0 the classical theorem of weitzenbock states that the algebra of constants is finitely generated this algebra coincides with the algebra of invariants of a single unipotent transformation in this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated not necessarily commutative or associative algebras over k assuming that the algebras are free in some sense in most of the cases relatively free algebras in varieties of associative or lie algebras in this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation the main results are the following 1 we show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants 2 for all varieties of associative algebras which are not nilpotent in lie sense the subalgebras of constants of the relatively free algebras of rank 2 are not finitely generated 3 we describe the generators of the subalgebra of constants for all factor algebras k x y i modulo a gl 2 k invariant ideal i 4 applying known results from commutative algebra we construct classes of automorphisms of the algebra generated by two generic 2 2 matrices we obtain also some partial results on relatively free lie algebras

new stably tame automorphisms of polynomial algebras abstract let k x y k x1 xn y1 ym be the polynomial algebra in m n variables over a field k of characteristic 0 let be a locally nilpotent derivation of k x y such that yi 0 i 1 m and let act as a k y affine transformation over the free k y module freely generated by x1 xn we prove that the automorphism exp w of k x y is stably tame for every polynomial w from the kernel ker of this result is applied to the automorphisms of the polynomial algebra in five variables introduced recently by drensky and gupta and arising from wild automorphisms of generic matrix algebras we also give an algorithm for constructing new stably tame automorphisms in any number of variables and hence new potential candidates for wild automorphisms

some formulas in invariant theory in an earlier paper 121 the author discussed the relationship between invariants of the group g z pz in characteristic p and the classical covariants of sl 2 k in characteristic zero this is more closely studied here in the corresponding representation rings in characteristic zero the only indecomposable 1 irreducible sl 2 k modules are the r si where r is the set of homogeneous forms of degree n of the polynomial ring r k x y in characteristic p there are p indecomposable g modules i vp the corresponding representation rings are

hilbert s fourteenth problem and algebraic extensions let k x be the polynomial ring in n variables over a field k for some n n and k x its field of fractions assume that l is a subfield of k x containing k over which k x is algebraic in spite of being an important issue in hilbert s fourteenth problem relations between finite generation of the k subalgebra l k x of k x and the extension degree k x l of k x over l have not been investigated in the present paper we give an example of l with k x l d such that l k x is not finitely generated for each d n with d 3 for n 3

fields defined by locally nilpotent derivations and monomials abstract we prove a necessary and sufficient condition for certain fields defined by locally nilpotent derivations and monomials to be algebraically closed in a rational function field this implies that a counterexample to the fourteenth problem of hilbert in dimension four which was recently given by the author is obtained as the kernel of a derivation it was previously unknown only in dimension four whether there exists such a counterexample

on the finitistic dimension conjecture ii related to finite global dimension abstract in this paper we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras on the one hand and the finitistic dimensions of their subalgebras on the other hand this is a continuation of the project in j pure appl algebra 193 2004 287 305 for an artin algebra a we denote by gl dim a fin dim a and rep dim a the global dimension finitistic dimension and representation dimension of a respectively the jacobson radical of a is denoted by rad a the main results in the paper are as follows let b be a subalgebra of an artin algebra a such that rad b is a left ideal in a then 1 if gl dim a 4 and rad a rad b a then fin dim b 2 if rep dim a 3 then fin dim b the results are applied to pullbacks of algebras over semi simple algebras moreover we have also the following dual statement 3 let b a be a surjective homomorphism between two algebras b and a suppose that the kernel of is contained in the socle of the right b module b b if gl dim a 4 or rep dim a 3 then fin dim b finally we provide a class of algebras with representation dimension at most three 4 if a is stably hereditary and rad b is an ideal in a then rep dim b 3

hilbert s fourteenth problem and field modifications abstract let k x k x 1 x n be the rational function field and k l k x an intermediate field hilbert s fourteenth problem asks whether the k algebra l k x 1 x n is finitely generated various counterexamples to this problem were already given but there are still many open cases in this paper we study the problem in terms of the field theoretic properties of l our result implies that for any s k x 1 x n 1 with k s k x 1 x n 1 and tr deg k k s 2 there exists aut k k x such that k s x n is a counterexample to hilbert s fourteenth problem this implies the existence of several interesting new counterexamples such as one with k x l 2

lengths of tors determined by killing powers of ideals in a local ring given a local ring r and n ideals whose sum is primary to the maximal ideal of r one may define a function which takes an n tuple of exponents a1 an to the length of tork r i1 r in these functions are shown to have rational generating functions in certain cases

computing invariants of algebraic groups in arbitrary characteristic let g be an affine algebraic group acting on an affine variety x we present an algorithm for computing generators of the invariant ring k x g in the case where g is reductive furthermore we address the case where g is connected and unipotent so the invariant ring need not be finitely generated for this case we develop an algorithm which computes k x g in terms of a so called colon operation from this generators of k x g can be obtained in finite time if it is finitely generated under the additional hypothesis that k x is factorial we present an algorithm that finds a quasi affine variety whose coordinate ring is k x g along the way we develop some techniques for dealing with nonfinitely generated algebras in particular we introduce the finite generation ideal

separating invariants and finite reflection groups abstract a separating algebra is roughly speaking a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants in this paper we introduce a geometric notion of separating algebra this allows us to prove that only groups generated by reflections may have polynomial separating algebras and only groups generated by bireflections may have complete intersection separating algebras

tame and wild projective curves and classification of vector bundles abstract we propose a new method of classifying vector bundles on projective curves especially singular ones according to their representation type in particular we prove that the classification problem of vector bundles respectively of torsion free sheaves on projective curves is always finite tame or wild we completely classify curves which are of finite respectively tame vector bundle type by their dual graph moreover our methods yield a geometric description of all indecomposable vector bundles and torsion free sheaves on finite and tame curves

invariants and separating morphisms for algebraic group actions the first part of this paper is a refinement of winkelmann s work on invariant rings and quotients of algebraic group actions on affine varieties where we take a more geometric point of view we show that the algebraic quotient x g given by the possibly not finitely generated ring of invariants is almost an algebraic variety and that the quotient morphism pi x rightarrow x g has a number of nice properties one of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective these general results are then refined for actions of the additive group mathbb g a where we can say much more we get a rather explicit description of the so called plinth variety and of the separating variety which measures how much orbits are separated by invariants the most complete results are obtained for representations we also give a complete and detailed analysis of roberts famous example of a an action of mathbb g a on 7 dimensional affine space with a non finitely generated ring of invariants

a condition for finite generation of the kernel of a derivation abstract we consider a homogeneous derivation on a finitely generated graded normal domain over a field and give a sufficient condition for finite generation of its kernel as its consequence we show some sufficient conditions for finite generation of the kernel of a derivation on a polynomial ring in particular we prove that the kernel of a locally nilpotent monomial derivation on a polynomial ring in four variables is finitely generated

linear equivalence of ideal topologies it is proved that whenever p is a prime ideal in a commutative noetherian ring such that the p adic and thep symbolic topologies are equivalent then the two topologies are equivalent linearly several explicit examples are calculated in particular for all prime ideals corresponding to non torsion points on nonsingular elliptic cubic curves

new automorphisms of generic matrix algebras and polynomial algebras abstract let t be the generic trace algebra generated by the algebra r of two generic 2 2 matrices and by all traces of the matrices from r over a field k we construct new automorphisms of t and r they induce automorphisms of the polynomial algebra in five variables which fix two of the variables our automorphisms of r are wild we do not know if the new automorphisms of k x 1 x 5 are tame when char k 2 we give the explicit form of all the constructed automorphisms

a counterexample to hilbert s fourteenth problem in dimension 5 abstract the kernel of a certain triangular derivation of the polynomial ring k x 1 x 2 x 3 x 4 x 5 is shown to be non finitely generated over k a field of characteristic zero thus giving a new counterexample to hilbert s fourteenth problem in the lowest dimension to date