differential operators on stanley reisner rings let k be an algebraically closed field of characteristic zero and let r k xl xn be a polynomial ring suppose that i is an ideal in r that may be generated by monomials we investigate the ring of differential operators d r i on the ring r i and ir i the idealiser of i in r we show that d r i and ir i are always right noetherian rings if i is a square free monomial ideal then we also identify all the two sided ideals of ir i to each simplicial complex a on v vl vn there is a corresponding square free monomial ideal ia and the stanley reisner ring associated to a is defined to be k a r ii we find necessary and sufficient conditions on a for d k a to be left noetherian

maximal cohen macaulay modules over the cone of an elliptic curve let r k y1 y2 y3 f f y13 y23 y33 where k is an algebraically closed field with chark 3 using atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded indecomposable reflexive r modules equivalently we describe explicitly the indecomposable bundles over the projective curve v f pk2 using the fact that over the completion r of r every reflexive module is gradable we obtain a description of the maximal cohen macaulay modules over r k y1 y2 y3 f

d split sequences and derived equivalences abstract in this paper we introduce the so called d split sequences and show that each d split sequence gives rise to a derived equivalence via a tilting module in particular we obtain derived equivalences from auslander reiten sequences via bb tilting modules or from n almost split sequences via n bb tilting modules and auslander reiten triangles further we recover n almost split sequences from n bb tilting modules over n auslander algebras

proper resolutions and gorenstein categories abstract let a be an abelian category and c an additive full subcategory of a we provide a method to construct a proper c resolution resp coproper c coresolution of one term in a short exact sequence in a from that of the other two terms by using these constructions we answer affirmatively an open question on the stability of the gorenstein category g c posed by sather wagstaff sharif and white and also prove that g c is closed under direct summands in addition we obtain some criteria for computing the c dimension and the g c dimension of an object in a

generators and defining relations for the ring of differential operators on a smooth affine algebraic variety in prime characteristic abstract for the ring of differential operators d o x on a smooth affine algebraic variety x over a perfect field of characteristic p 0 explicit algebra generators and defining relations are found a finite set of generators and a finite set of defining relations are given explicitly for the module der k o x of derivations on the algebra o x of regular functions on the variety x

a categorification of biclosed sets of strings abstract we consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type palu pilaud and plamondon proved that the collection of all biclosed sets of strings forms a lattice and moreover that this lattice is congruence uniform many interesting examples of finite congruence uniform lattices may be represented as the lattice of torsion classes of an associative algebra we introduce a generalization the lattice of torsion shadows and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick finite congruence uniform lattices admit an alternate partial order known as the core label order in many cases the core label order of a congruence uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra analogous to torsion shadows we introduce wide shadows and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows

the poincare series of the module of derivations of affine monomial curves abstract let a be a graded k algebra and m be a finitely generated a module the poincare series p m a z is the formal power series i dim k tor i a k m z i we study the poincare series of der k k s the module of derivations of a numerical semigroup ring k s and we relate it to the poincare series of k over k s and to the type of s we then use this in order to determine the poincare series of der k k s or at least its rationality for some classes of examples we finally give an example of a non rational p der k k s k s z

finite length and pure injective modules over a ring of differential operators let k be an algebraically closed field of characteristic zero on k x1 xn the ring of formal power series over k and dn the ring of differential operators over on suppose that is a prime ideal of on of height n 1 i e a on is a curve we prove that every indecomposable finite length module over dn with support on is uniserial with isomorphic or alternating composition factors for the ring d a of differential operators over a we also classify indecomposable pure injective modules and show that the cantor bendixson rank of the ziegler spectrum over d a is equal to 2

standardly stratified algebras and tilting abstract the concept of the characteristic tilting module and of the ringel dual for quasi hereditary algebras is generalized for the setting of standardly stratified algebras

phase transitions and symmetry breaking in genetic algorithms with crossover in this paper we consider the role of the crossover operator in genetic algorithms specifically we study optimisation problems that exhibit many local optima and consider how crossover affects the rate at which the population breaks the symmetry of the problem as an example of such a problem we consider the subset sum problem in doing so we demonstrate a previously unobserved phenomenon whereby the genetic algorithm with crossover exhibits a critical mutation rate at which its performance sharply diverges from that of the genetic algorithm without crossover at this critical mutation rate the genetic algorithm with crossover exhibits a rapid increase in population diversity we calculate the details of this phenomenon on a simple instance of the subset sum problem and show that it is a classic phase transition between ordered and disordered populations finally we show that this critical mutation rate corresponds to the transition between the genetic algorithm accelerating or preventing symmetry breaking and that the critical mutation rate represents an optimum in terms of the balance of exploration and exploitation within the algorithm

differential operators on monomial curves abstract let k be an algebraically closed field of characteristic 0 let n 0 be a numerical semigroup and let a k be the corresponding semigroup algebra we give an explicit description of the ring d d a of k linear differential operators on a the associated graded ring grd a and the module of derivations derk a we also classify all graded d modules which are finitely generated torsion free a modules of rank 1 considered as modules over the sub ring a d a

phase transitions in sexual populations subject to stabilizing selection we show that a simple model of an evolving sexual population which dates back to some of the earliest work in theoretical population genetics exhibits an unexpected and previously unobserved phase transition between ordered and disordered states this behavior is not present in populations evolving asexually without recombination and is thus important in any comparison of sexual and asexual populations in order to calculate the details of the phase transition we use techniques from statistical physics we introduce the correlation of the population as the order parameter of the system and use maximum entropy inference to find the state of the population at any time

lie rinehart cohomology and integrable connections on modules of rank one abstract let k be an algebraically closed field of characteristic 0 let r be a commutative k algebra and let m be a torsion free r module of rank one with a connection we consider the lie rinehart cohomology with values in end r m with its induced connection and give an interpretation of this cohomology in terms of the integrable connections on m when r is an isolated singularity of dimension d 2 we relate the lie rinehart cohomology to the topological cohomology of the link of the singularity and when r is a quasi homogenous hypersurface of dimension two we give a complete computation of the cohomology

differential operators on an affine curve let x denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero denote by 3 x the sheaf of differential operators on x and 2 x f x 3 x the ring of global differential operators on x the following is established theorem 3 x is a finitely generated k algebra and a noetherian ring furthermore 2 x has a unique minimal non zero ideal j and 3 x j is a finite dimensional k algebra let x denoted the normalisation of x and n x x the projection map the main technique is to compare 2 x with 3 x

mad subalgebras of rings of differential operators on curves we study the maximal abelian ad nilpotent mad subalgebras of the domains morita equivalent to the first weyl algebra we give a complete description both of the individual mad subalgebras and of the space of all such a surprising consequence is that this last space is independent of our results generalize some classic theorems of dixmier about the weyl algebra

d module structure of local cohomology modules of toric algebras let s be a toric algebra over a field k of characteristic 0 and let i be a monomial ideal of s we show that the local cohomology modules h i i s are of finite length over the ring of differential operators d s k generalizing the classical case of a polynomial algebra s as an application we compute the characteristic cycles of some local cohomology modules

algebras with irreducible module varieties i abstract we call a finite dimensional k algebra a geometrically irreducible if for all d 0 all connected components of the affine scheme of d dimensional a modules are irreducible we show that the geometrically irreducible algebras without loops this includes all algebras of finite global dimension are the hereditary algebras we also show that truncated polynomial rings are the only geometrically irreducible local algebras finally we formulate a conjectural classification of all geometrically irreducible algebras

partial tilting modules over m replicated algebras abstract let a be a hereditary algebra over an algebraically closed field k and a m be the m replicated algebra of a given an a m module t we denote by t the number of non isomorphic indecomposable summands of t in this paper we prove that a partial tilting a m module t is a tilting a m module if and only if t a m and that every partial tilting a m module has complements as an application we deduce that the tilting quiver k a m of a m is connected moreover we investigate the number of complements to almost tilting modules over duplicated algebras

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

on the structure of free resolutions of length 3 the sources of structure theory of finite free resolutions go back to hilbert in his famous paper on invariant theory he proved that every ideal of projective dimension 1 over a polynomial ring is determinantal this theorem was then extended by burch to arbitrary commutative rings there are various motivations for extending this results one of them is a variety of applications of the hilbert burch theorem in deformation theory and algebraic geometry cf for example s another reason comes from serre s conjecture one would like to understand in terms of resolutions why the lifting property of grothendieck does not hold in 1974 buchsbaum and eisenbud in bel introduced so called structure theorems various factorisations which occur for acyclic complexes of arbitrary length then hochster in h generalized the hilbert burch theorem to arbitrary free complexes of length 2 his results were then improved in br pw still the structure of complexes of length bigger than 2 is not understood the first steps in this direction are the results of b and the construction of a generic ring for complexes of length 3 of type 1 rz 1 1 given in pw in this paper i construct a ring which i believe is the generic ring for complexes of length 3 the construction involves new structure theorems and a new idea the notion of defects of structure theorems the paper is organized as follows section 1 is introductory i give there the background from representation theory and the cohomology groups of a lie algebra section 2 contains the proof of the new structure theorems we need the construction of the generic ring r and of the defect lie algebra l in section 3 i show that genericity of r follows from vanising of the homology of a certain family of complexes over u l the enveloping algebra of l in section 4 i apply the results of section 2 to study multi

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

the quasispecies regime for the simple genetic algorithm with ranking selection we study the simple genetic algorithm with a ranking selection mechanism linear ranking or tournament we denote by the length of the chromosomes by m the population size by pc the crossover probability and by pm the mutation probability we introduce a parameter called the selection drift which measures the selection intensity of the fittest chromosome we show that the dynamics of the genetic algorithm depend in a critical way on the parameter 1 pc 1 pm if 1 then the genetic algorithm operates in a quasispecies regime an advantageous mutant invades a positive fraction of the population with probability larger than a constant p which does not depend on m we estimate next the probability of the occurrence of a catastrophe the whole population falls below a fitness level which was previously reached by a positive fraction of the population the asymptotic results suggest the following rules 1 pc 1 pm should be slightly larger than 1 pm should be of order 1 m should be larger than ln the running time should be of exponential order in m the first condition requires that pm pc ln these conclusions must be taken with great care they come from an asymptotic regime and it is a formidable task to understand the relevance of this regime for a real world problem at least we hope that these conclusions provide interesting guidelines for the practical implementation of the simple genetic algorithm

endomorphism algebras of tilting modules over m replicated algebras abstract let a be a finite dimensional hereditary algebra over an algebraically closed field k and let a m be the m replicated algebra of a in this paper we investigate the structure properties of the endomorphism algebras of tilting modules of a m and prove that all the endomorphism algebras of tilting modules of a m can be realized as the iterated endomorphism algebras of bb tilting modules that is for each pair of basic tilting a m modules t 1 and t 2 there exists a series of finite dimensional algebras 0 1 s which are the endomorphism algebras of some basic tilting a m modules and for each i there is a bb tilting i module m i such that 0 end a m t 1 i end i 1 m i 1 for 1 i s and end a m t 2 end s m s

gorenstein projective modules and symmetric recollements we introduce compatible bimodules if m is a compatible a b bimodule then the gorenstein projective modules over algebra am0b are explicitly described and if is gorenstein then this description implies that m is compatible as an application if m is compatible then there is a left recollement of the stable category gp and if is gorenstein and ma is projective then there is a symmetric recollement of the singularity category dsgb

the transpose of modules relative to subcategories abstract let r be a left noetherian ring and s a right noetherian ring and let x be a subcategory of finitely generated left r modules and b s r a finite r s bimodule as a generalization of the auslander transpose replacing a projective presentation by an x presentation and the functor hom r r by the functor hom r b we introduce the x b transpose of an r module m admitting an x presentation for a suitable subcategory x some useful properties of x b transposes are obtained it is shown that the x b transpose shares many nice properties with the auslander transpose some known results are obtained as corollaries

on arkhipov s and enright s functors we give a description of arkhipov s and joseph s and deodhar mathieu s versions of enright s endofunctors on the category open image in new window associated with a fixed triangular decomposition of a complex finite dimensional semi simple lie algebra in terms of co approximation functors with respect to suitably chosen injective resp projective modules we establish some new connections between these functors for example we show that arkhipov s and joseph s functors are adjoint to each other we also give several proofs of braid relations for arkhipov s and enright s functors

finite dimensional representations of invariant differential operators let k be an algebraically closed field of characteristic 0 y k r k s and let g be an algebraic torus acting diagonally on the ring of algebraic differential operators v y we give necessary and sufficient conditions for d v g to have enough simple finite dimensional representations in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero as an application we show that if k gl v is a representation of a reductive group k and if zero is not a weight of a maximal torus of k on v then d v k has enough finite dimensional representations we also construct examples of fcr algebras with any integer gk dimension 3

the dynamics of a genetic algorithm for simple random ising systems abstract a formalism is presented for analysing genetic algorithms it is used to study a simple genetic algorithm consisting of selection mutation and crossover which is searching for the ground states of simple random ising spin systems a random field ideal paramagnet and a spin glass chain the formalism can also be applied to other population based search techniques and to biological models of micro evolution to make the problem tractable it is assumed that the population dynamics can be described by a few macroscopic order parameters and that the remaining microscopic degrees of freedom can be averaged out the macroscopic quantities that are used are the cumulants of the distribution of fitnesses or energies in the population a statistical mechanics model is presented which describes the population configuration in terms of the cumulants this is used to derive equations of motion for the cumulants predictions of the theory are compared with experiments and are shown to predict the average time to convergence and the average fitness of the final population accurately a simplified version of the equations is produced by keeping only leading nonlinear terms and truncating the cumulant expansion this is shown to give a novel description of the role of genetic operators in search e g it is argued that an important role of crossover is to reduce the skewness of the population

the fundamental module of a normal local domain of dimension 2 the fundamental module e of a normal local domain r m of dimension 2 is defined by the nonsplit exact sequence 0 a k a e a m a 0 where k is the canonical module of r we prove that if r is complete with r m c then e is decomposable if and only if r is a cyclic quotient singularity various other properties of fundamental modules will be discussed 0 introduction let r m be a normal local domain of dimension 2 which possesses the canonical module k let c r denote the category of finitely generated reflexive r modules note that a finitely generated r module is an object in c r if and only if it is a maximal cohen macaulay module over r by definition k is a reflexive module of rank 1 and it satisfies extr r m k r m see herzog and kunz 8 for the details we denote the duality with respect to r resp k by resp that is homr r and homr k remark that a finitely generated r module m lies in c r if and only if m m or equivalently m m we make the following definition 0 1 since ext m k ext2r r m k r m there uniquely exists the nonsplit exact sequence 0 k e m 0 which is called the fundamental sequence in 3 in particular the module e appearing in the middle term of this sequence is also unique up to isomorphism we call e the fundamental module of r e is said to be decomposable if there is an isomorphism e a e b where a and b are nontrivial ideals of r otherwise e is indecomposable in this paper we are interested in the properties of the fundamental modules in particular we are mostly concerned with its decomposability in fact we can prove that if r is complete with the residue field c then r has the decomposable fundamental module if and only if r is a cyclic quotient singularity see theorem 2 1 and if this is the case then the ar quiver of r will be easily described in 1 we summarize some elementary properties of fundamental modules in particular we can prove that if r is a hypersurface then the fundamental module of r is isomorphic to the third syzygy of k in 2 our main theorem 2 1 stated above will be proved the reader will notice that the keys of the proof are the first author s theorem 11 theorem 1 1 and the theorem of herzog auslander auslander 3 theorem 4 9 herzog 71 received by the editors june 26 1986 and in revised form july 29 1987 1980 mathematics subject classification 1985 revision primary 13h10 13c13 secondary 14b05 14j17

maximal rigid subcategories and cluster tilting subcategories in 2 cy categories abstract let c be a connected hom finite 2 cy triangulated category we prove that if t is a functorially finite maximal rigid subcategory of c without loops in its quiver then t is cluster tilting in particular this gives a positive answer to a conjecture proposed in 5

three results on frobenius categories this paper consists of three results on frobenius categories 1 we give sufficient conditions on when a factor category of a frobenius category is still a frobenius category 2 we show that any frobenius category is equivalent to an extension closed exact subcategory of the frobenius category formed by cohen macaulay modules over some additive category this is an analogue of gabriel quillen s embedding theorem for frobenius categories 3 we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure with which the given category becomes a frobenius category several applications of the results are discussed

double centralizer properties dominant dimension and tilting modules double centralizer properties play a central role in many parts of algebraic lie theory soergel s double centralizer theorem relates the principal block of the bernstein gelfand gelfand category o of a semisimple complex lie algebra with the coinvariant algebra i e the cohomology algebra of the corresponding flag manifold schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic or via the quantized version in nondefining characteristic in this paper we exhibit algebraic structures behind these double centralizer properties we show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective injective or tilting modules general arguments which combine methods from ring theory qf 3 rings and dominant dimension with tools from representation theory approximations tilting modules then imply the validity of these double centralizer properties as well as new ones in contrast to the traditional proofs e g by the fundamental theorems of invariant theory no computations are necessary

the differential operator ring of an affine curve the purpose of this paper is to investigate the structure of the ring d r of all linear differential operators on the coordinate ring of an affine algebraic variety x possibly reducible over a field k not necessarily algebraically closed of characteristic zero concentrating on the case that dim x 1 in this case it is proved that d r is a left and right noetherian ring with left and right krull dimension equal to dim x that the endomorphism ring of any simple left or right d r module is finite dimensional over k that d r has a unique smallest ideal l essential as a left or right ideal and that d r l is finite dimensional over k the following ring theoretic tool is developed for use in deriving the above results let d be a subalgebra of a left noetherian k algebra e such that e is finitely generated as a left dmodule and all simple left e modules have finite dimensional endomorphism rings over k and assume that d contains a left ideal i of e such that e i has finite length then it is proved that d is left noetherian and that the endomorphism ring of any simple left d module is finite dimensional over k introduction in this paper we will study the ring d r of k linear differential operators on a commutative k algebra r where k is a field of characteristic zero of special interest is the case where r is the coordinate ring of an affine algebraic variety x when x is nonsingular the ring d r has been extensively studied and enjoys many nice properties for example d r is noetherian we will use the term noetherian to indicate that a ring is both left and right noetherian when x is singular d r need not be noetherian as shown by j n bernstein i m gelfand and s i gelfand 3 if x is the normal cubic cone i e the surface in complex 3 space given by x3 y3 z3 0 then d r is neither left nor right noetherian thus a major goal is to discover for which varieties x the ring d r is noetherian the main contribution of this paper is to prove that d r is noetherian when dim x 1 and to develop some of the structure of d r in this case the paper is organized as follows 1 contains a number of basic results about the differential operators on commutative rings in 2 the algebraic tool used in proving d r is noetherian is developed this result overlaps with the independent work of j c robson and l w small 11 3 and 4 contain the main results on the structure of d r when dim x 1 this work was motivated by the calculations of i m musson 10 these results were independently obtained by s p smith and j t stafford 12 in the case that x is an irreducible curve over an algebraically closed field of characteristic zero 5 contains an example of a nonreduced k algebra r with krull dimension one such that d r is right but not left noetherian received by the editors may 22 1986 and in revised form april 13 1987 1980 mathematics subject classification 1985 revision primary 16a33 secondary 13b10 1988 american mathematical society 0002 9947 88 1 00 25 per page

differential operators on rational projective curves fix a singular projective rational curve 3 over an algebraically closed field k of characteristic zero with normalisation map a p 3 and write s 5 r 9 for its ring of global differential operators we prove theorem 1 the ring g a is a noerherian domain finirely generated as a k algebra with a unique minimal non zero ideal j z moreover f y 9 j f is finite dimensional over k in the case when n is injective we give complete description of the structure of 2 x

rings of differential operators and zero divisors throughout k will denote an algebraically closed field of characteristic zero if a is a commutative k algebra we denote by 9 a the ring of all k linear differential operators on 4 if a is a domain it is well known and easy to see that 9 a is also a domain in most studies of 9 a see for example s it is assumed that a is a domain here we are mainly concerned with the case where a has zero divisors and we study annihilator conditions in 9 a for example in section 2 we prove

tilting modules of finite projective dimension tilting modules are well known and are useful in the representation theory of artinian algebras 2 4 6 11 the purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results a unital left a module a t over a ring a is said to be a tilting module of projective dimension 0 is an integer if it satisfies the following three conditions 1 a t has a projective resolution

triple cohomology of lie rinehart algebras and the canonical class of associative algebras abstract we introduce a bicomplex which computes the triple cohomology of lie rinehart algebras we prove that the triple cohomology is isomorphic to the rinehart cohomology provided the lie rinehart algebra is projective over the corresponding commutative algebra as an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra and extend sridharan s result on almost commutative algebras

representation theoretic properties of balanced big cohen macaulay modules let r mathfrak m k be a complete cohen macaulay local ring in this paper we assign a numerical invariant for any balanced big cohen macaulay module called underline mathsf h length among other results it is proved that for a given balanced big cohen macaulay r module m with an mathfrak m primary cohomological annihilator if there is a bound on the underline mathsf h length of all modules appearing in mathsf cm support of m then it is fully decomposable i e it is a direct sum of finitely generated modules while the first brauer thrall conjecture fails in general by a counterexample of dieterich dealing with multiplicities to measure the size of maximal cohen macaulay modules our formalism establishes the validity of the conjecture for complete cohen macaulay local rings in addition the pure semisimplicity of a subcategory of balanced big cohen macaulay modules is settled namely it is shown that r is of finite mathsf cm type if and only if r is an isolated singularity and the category of all fully decomposable balanced big cohen macaulay modules is closed under kernels of epimorphisms finally we examine the mentioned results in the context of cohen macaulay artin algebras admitting a dualizing bimodule omega as defined by auslander and reiten it will turn out that omega gorenstein projective modules with bounded mathsf cm support are fully decomposable in particular a cohen macaulay algebra lambda is of finite mathsf cm type if and only if every omega gorenstein projective module is of finite mathsf cm type which generalizes a result of chen for gorenstein algebras our main tool in the proof of results is gabriel roiter co measure an invariant assigned to modules of finite length and defined by gabriel and ringel this in fact provides an application of the gabriel roiter co measure in the category of maximal cohen macaulay modules

defining an m cluster category we show that a certain orbit category considered by keller encodes the combinatorics of the m clusters of fomin and reading in a fashion similar to the way the cluster category of buan marsh reineke reiten and todorov encodes the combinatorics of the clusters of fomin and zelevinsky this allows us to give type uniform proofs of certain results of fomin and reading in the simply laced cases

the structure of preenvelopes with respect to maximal cohen macaulay modules abstract this paper studies the structure of special preenvelopes and envelopes with respect to maximal cohen macaulay modules we investigate the structure of them in terms of their kernels and cokernels moreover using this result we also study the structure of special proper coresolutions with respect to maximal cohen macaulay modules over a henselian cohen macaulay local ring

rings of differential operators on invariant rings of tori let k be an algebraically closed field of characteristic zero and g a torus acting diagonally on k for a subset 8 of s 1 2 s set up u e k u o if j e 8 then g acts on d up the ring of regular functions on up and we study the ring d d up g of all differential operators on the invariantvring more generally suppose that is a set of subsets of s such that each invariant ring d u g b e has the same quotient field we prove thatn g d d up g is noetherian and finitely generated as a k algebra now g acts on each d d up and there is a natural map 0 n d d ub n d d up g d ya g obtained by restriction of the differential operators we find necessary and sufficient conditions for 0 to be suuective and describe the kernel of the algebras n g d up g and n g a d up g carry a natural filtration given by the order of the differential operators we show that the associated graded rings are finitely generated commutative algebras and are gorensetin rings we also determine the centers of n ga d up5 g and n ga d d u g introduction throughout this paper k will be an algebraically closed field of characteristic zero if k is a commutative k algebra we denote by do k the set of k linear maps k k and if 1 a k linear map f k k belongs to dp k provided the map f r defined by f r s f rs rf s for s e k belongs to dp l k for all r e k the set d k upaodp k forms a subring of endk k called the ring of differential operators on k we consider k as a left d k module where f r f r for all f e d k r e k the ring of differential operators on the invariant ring of a finite group acting on a polynomial ring has been studied in ka l1 and l2 we carry out a similar study for the invariant ring of a torus we assume that the torus g acts on ks as a group of diagonal matrices if jb is a subset of s 1 2 s then g acts on ua u e ksluj o if j e jb also g acts on p u via gf u f g 1u for g e g f e g u 3 u e u 3 more generally suppose that for some set of subsets a of s we have y ue ua where each invariant ring p u c has the same quotient field f then each d p u 6 is a received by the editors june 18 1986 and in revised form december 11 1986 1980 mclthemcltiss subject clclssificcltion 1985 revision primary 16a33 secondary 13n05 14l30 p 1987 american mathematical s clety 0002 9947 87 1 00 25 per page

one dimensional local rings of maximal and almost maximal length let a m k denote a one dimensional cohen macaulay local ring with maximal ideal m and residue class field k we assume a is a reduced excellent local ring which has a canonical module a with no loss of generality we can also assume k is infinite let denote the integral closure of a in its total quotient ring q and let b denote the conductor of a in a has maximal length if la aa la ab t a here t a denotes the cohen macaulay type of a and la denotes the length of the a module a has almost maximal length if la aa la ab t a 1 the two main results of this paper are as follows a has maximal length if and only if either a is gorenstein or there exists an x m and a positive integer p such that m x b andba xpa let e a denote the multiplicity of a then a has almost maximal length and 1 t a e a if and only if there exists a transversal x of m such that m x b and la bxpa 1 here p min i xi b these two theorems generalize more specific results for semigroup rings obtained in brown and curtis numerical semigroups of maximal and almost maximal length semigroup forum vol 42 springer verlag berlin new york 1991 219 235

an auslander buchweitz approximation approach to pre silting subcategories in triangulated categories abstract we apply the auslander buchweitz approximation triangles to study pre silting subcategories in a triangulated catego ry t an auslander reiten type correspondence between the class of silting subcategories of t and that of certain covariantly finite subcategories of t is established we introduce a relation on presilting subcategories of t which can be used to obtain another silting subcategory from a given one we also give a bazzoni s characterization for two presilting subcategories satisfying such a relation the results can be used to improve the auslander reiten type correspondence and bazzoni s characterization for small silting complexes established by wei

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

generalized tilting modules with finite injective dimension let r be a left noetherian ring s be a right noetherian ring and r be a generalized tilting module with s end r we mainly consider the conditions such that l id r 1 in this paper and obtain some characterizations for this inequality further when l id r and r id s are finite we provide the conditions such that r and s are injective as a direct application we get some necessary and sufficient conditions for a gorenstein ring to be a qf ring

a note on abelian quotient categories abstract let c be a triangulated category with a serre functor s and x a non zero contravariantly finite rigid subcategory of c then x is cluster tilting if and only if the quotient category c x is abelian and s x x 2 as an application this result generalizes the work by beligiannis

on modules with trivial self extensions throughout this note all rings will be self basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated homomorphisms will be written on the opposite side of the scalars by mod a resp a mod we denote the category of all right resp left a modules for an artin algebra a the ordinary duality functor is denoted by d mod a 2 a mod i e d hom f with the minimal injective cogenerator i over k we are interested in the bimodules bta with the following properties

a capelli harish chandra homomorphism for a real reductive dual pair the capelli identities define a homomorphism c from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group in terms of the harish chandra isomorphism this map involves a shift we view a dual pair as a lie supergroup and offer a construction of the homomorphism c based solely on the harish chandra s radial component maps thus we provide a geometric interpretation of the shift

approximations and the little finitistic dimension of artinian rings let r be a ring and let simp r be a representative set of all simple right r modules denote by p the class of all modules which are finitely generated and have finite projective dimension the little finitistic dimension of r is defined by fdim r sup proj dim m m p let a b be the complete cotorsion theory cogenerated by p for each s simp r let fs xs s be a special a precover of s we prove that fdim r max proj dim xs s simp r provided that r is right artinian as a corollary we extend to right artinian rings the well known auslander reiten sufficient condition for finiteness of the little finitistic dimension

endomorphisms of right ideals of the weyl algebra let a a k be the first weyl algebra over an infinite field k let p be any noncyclic projective right ideal of a and set s end p we prove that as k algebras s f a in contrast there exists a noncyclic projective right ideal q of s such that s end q thus despite the fact that they are morita equivalent s and a have surprisingly different properties for example under the canonical maps autk a pick a pick s in contrast autk s has infinite index in pick s introduction given a comnmutative domain r the first weyl algebra a r is defined to be the associative r algebra thus r is central subring generated by elements x and y subject to the relation xy yx 1 when no ambiguity is possible we write a for a r let f be a field of characteristic zero then a a f is a simple ring and indeed may be thought of as one of the nicest and most important examples of simple noetherian rings the initial motivation for this paper was the following result of smith if p is the noncyclic right ideal p x2a xy 1 a of a then end p t a as f algebras 11 note that as char f 0 a is a simple hereditary ring and so end p is automatically morita equivalent to a even worse as pedp ada 17 the full matrix rings m2 a and m2 end p are isomorphic thus any proof of smith s result must be fairly subtle and may therefore provide useful invariants for a the first main aim of this paper is to generalize smith s result to an arbitrary projective right ideal of a while the proof of this is harder than that of smith s result it does provide a more informative proof in the sense that unlike smith we do not and cannot require an explicit description of end p for the rest of this introduction k will denote a field of arbitrary characteristic and all isomorphisms of rings will be k algebra isomorphisms theorem a let p be a projective right ideal of a a k then end p a if and only if p is a cyclic right ideal of a see theorem 3 1 this has some easy corollaries corollary b let p and q be projective right ideals of a a k then end p end q if and only if p to q for some t e d k the division ring of fractions of a and a e autk a the group of k automorphisms of a received by the editors february 13 1986 1980 mathematics subject classification 1985 revision primary 16a65 16a72 16a19 supported in part by an nsf grant 1987 american mathematical society 0002 9947 87 1 00 25 per page

totally reflexive extensions and modules abstract we introduce the notion of totally reflexive extension of rings it unifies gorenstein orders and frobenius extensions we prove that for a totally reflexive extension a module over the extension ring is totally reflexive if and only if its underlying module over the base ring is totally reflexive

cusps and d modules 1 1 v modules on singular varieties let y denote a variety over a field and let vy denote the full sheaf of differential operators on y in characteristic zero vy is the familiar sheaf of differential operators and in characteristic p the sheaf vy will contain all divided powers of operators d it is well known that on a general singular variety y the ring of differential operators is badly behaved in particular it need not be noetherian nor if y is affine a simple ring see bgg following grothendieck there is an alternative notion of a sheaf with infinitesi mal parallel transport called a co stratification see be2 be3 bdi or section 2 which agrees with the notion of left respectively right py module when y is smooth but will not agree in general when y is singular our first result presents a simple vanishing condition that guarantees that the notions of x module and co stratification on a singular variety coincide

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

wakamatsu tilting modules abstract we study a generalization of tilting modules to modules of possibly infinite projective dimension introduced by wakamatsu in j algebra 114 1988 106 114 in particular we characterize these modules in terms of suitable subcategories of finitely generated modules and in terms of cotorsion theories

d modules on smooth toric varieties abstract let x be a smooth toric variety cox introduced the homogeneous coordinate ring s of x and its irrelevant ideal b let a denote the ring of differential operators on spec s we show that the category of d modules on x is equivalent to a subcategory of graded a modules modulo b torsion additionally we prove that the characteristic variety of a d module is a geometric quotient of an open subset of the characteristic variety of the associated a module and that holonomic d modules correspond to holonomic a modules

dixmier groups and borel subgroups abstract in this paper we study a family g n n 0 of infinite dimensional ind algebraic groups associated with algebras morita equivalent to the weyl algebra a 1 c we give a geometric presentation of these groups in terms of amalgamated products generalizing classical theorems of dixmier and makar limanov our main result is a classification of borel subgroups of g n for all n we show that the conjugacy classes of non abelian borel subgroups of g n are in bijection with the partitions of n furthermore we prove an infinite dimensional analogue of the classical theorem of steinberg 52 that characterizes borel subgroups in purely group theoretic terms combined together the last two results imply that the g n are pairwise non isomorphic as abstract groups this settles an old question of stafford 51

polynomial solutions to constant coefficient differential equations let d 1 d r c x 1 x n be constant coefficient differnetial operators with zero constant term let s f c x 1 x n d j f 0 for all 1 j r be the space of polynomial solutions to the system of simultaneous differential equations d j f 0 it is proved that s is a module over d v the ring of differential operators on the affine scheme v with coordinate ring c x 1 x n if v is smooth and irreducible then s is a simple d v module s 1 d v and the generators for d v yield an algorithm for obtaining a basis for s

on algebras of finite cohen macaulay type abstract we study artin algebras and commutative noetherian complete local rings r in connection with the following decomposition property of gorenstein projective modules any gorenstein projective module is a direct sum of finitely generated modules we show that the class of algebras enjoying coincides with the class of virtually gorenstein algebras of finite cohen macaulay type introduced in beligiannis and reiten 2007 27 beligiannis 2005 24 thus we solve the problem stated in chen 2008 33 this is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits thus generalizing a classical result of auslander 1976 9 and ringel and tachikawa 1974 63 in the commutative case if r admits a non free finitely generated gorenstein projective module then we show that r is of finite cohen macaulay type iff r is gorenstein and satisfies we also generalize a result of yoshino 2005 68 by characterizing when finitely generated modules without extensions with the ring are gorenstein projective finally we study the stable relative auslander algebra of a virtually gorenstein algebra of finite cohen macaulay type and under the presence of a cluster tilting object we give descriptions of the stable category of gorenstein projective modules in terms of the cluster category associated to the quiver of the stable relative auslander algebra in this setting we show that the cluster category is invariant under derived equivalences

quantifying singularities with differential operators abstract the f signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties for example this invariant detects non singularity and strong f regularity however it is very difficult to compute motivated by different aspects of the f signature we define a numerical invariant for rings of characteristic zero or p 0 that exhibits many of the useful properties of the f signature we also compute many examples of this invariant including cases where the f signature is not known we also obtain a number of results on symbolic powers and bernstein sato polynomials

gelfand kirillov dimension and associated graded modules abstract let s a 0sa be a filtered algebra over a field k and m ma a filtered s module with associated graded module gr m m a m a 1 over gr s s a s a 1 we consider conditions which ensure that gr m and m have the same gelfand kirillov dimension and show that this equality holds in particular if gr s is an affine commutative k algebra