radical layers of representable functors i let 0 f a t4 b o c 0 be an almost split sequence then the induced sequence o g a g b g c ys 0 is exact where s gs c r c the maps a have degree 1 and y has degree 0 ii let i t i sot i be a left minimal almost split map where i is an indecomposable injective let k 03 be maximal such that im sot i c rk i if k co then the induced sequence 0 g sot i g i g i sot z is exact where the first map has degree k and the second has degree 1 if k co then the induced map g i t g z sot i is a monomorphism of degree 1 dual results hold for covariant functors x

the analysis of rates using poisson regression models models are considered in which the underlying rate at which events occur can be represented by a regression function that describes the relation between the predictor variables and the unknown parameters estimates of the parameters can be obtained by means of iteratively reweighted least squares irls when the events of interest follow the poisson distribution the irls algorithm is equivalent to using the method of scoring to obtain maximum likelihood ml estimates the general poisson regression models include log linear quasilinear and intrinsically nonlinear models the approach considered enables one to concentrate on describing the relation between the dependent variable and the predictor variables through the regression model standard statistical packages that support irls can then be used to obtain ml estimates their asymptotic covariance matrix and diagnostic measures that can be used to aid the analyst in detecting outlying responses and extreme points in the model space applications of these methods to epidemiologic follow up studies with the data organized into a life table type of format are discussed the method is illustrated by using a nonlinear model derived from the multistage theory of carcinogenesis to analyze lung cancer death rates among british physicians who were regular cigarette smokers

cartan matrices and grothendieck groups of stable categories let a be a finite dimensional algebra over a field k and a its repetitive algebra 4 in 2 happel proved that if gl dim a cc then there is an equivalence between the derived category d a and projectively stable module catagory mod a such that the triangulations of a and a are retained by the equivalence this investigation is motivated by the converse problem of his theorem if there is a triangulated category equivalence g a n mod g induces naturally the isomorphism between their grothedieck groups ko db a n k mod a as ko db a 1 k a it follows that k a 2 k mod a and consequently ko a is isomorphic to the free abelian group zen of rank n where n is the number of non isomorphic simple a modules the main purpose of this paper is to give a characterization of algebras a such that k 2 n k a in section 2 we prove that in order to hold k a a n k a it is necessary and sufficient that the determinant of the cartan matrix of a be equal to f 1 by this result we can easily find some algebras a such that p a

on the cartan matrix of an artin algebra of global dimension two the purpose of this paper is to show that if n is an artin algebra of g obal dimension 81 dim 2 then the determinant of its cartan matrix equals i this generalizes previous results of donovan and freislich 2 igusa and todorov 4 and wilson 5 we recall that if n is an artin algebra for instance a finite dimensional algebra over a field there are finitely many nonisomorphic indecomposable projective rl modules p p p we denote by end p hom p p the length of hom p pi as an end p module then the h x n matrix c n j having knd cpi hom p p i as the i j th entry is called the cartan matrix of rl let d be now an artin algebra of finite representation typethat is having finitely many nonisomorphic indecomposable 4 modules let h u end tr hom m n be the matrix with integral entries where m n range through the set of the nonisomorphic indecomposable ii modules donovan and freislich 2 have shown that if 4 kg is a group algebra of finite representation type then det h 1 then using associated graded algebras wilson showed that det h 1 if a is a finite dimensional algebra of finite representation type over an algebraically closed field 5 io independently the same result was proved by igusa and todorov 4 without making any assumption on the ground field let p be the auslander algebra of n see 1 for further details that is r end m 0 mt m jod where m m are the nonisomorphic indecomposable n modules then it has been shown that h c t where c t denotes the cartan matrix of the auslander algebra r it is known that gl dim r 2 we recall that

comment on critical behavior of the chain generating function of self avoiding walks on the sierpinski gasket family the euclidean limit we refute the claims made by riera and chalub phys rev e 58 4001 1998 by demonstrating that they have not provided enough data requisite in their series expansion method to draw reliable conclusions about criticality of self avoiding walks on the sierpinski gasket family of fractals

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

jordan holder theorems for derived categories of derived discrete algebras abstract for any positive integer n n derived simple derived discrete algebras are classified up to derived equivalence furthermore the jordan holder theorems for all kinds of derived categories of derived discrete algebras are obtained

artin rings of global dimension two in this paper we show that a left artin ring a of global dimension at most two has loewy length at most 2 1 where n is the number of simple components of a rad a we assume that the reader is familiar with the trends but not necessarily the details of contemporary representation theory of artin rings let us record a few conventions the rings discussed here will be left artinian unless otherwise specified and will sometimes be finite dimensional algebras over an algebraically closed field henceforth denoted by k modules will be finitely generated left modules unless otherwise specified the radical of a will usually be denoted by r the loewy

negative k theory of derived categories we define negative k groups for exact categories and for derived categories in the framework of frobenius pairs generalizing definitions of bass karoubi carter pedersen weibel and thomason we prove localization and vanishing theorems for these groups devissage for noetherian abelian categories additivity and resolution hold we show that the first negative k group of an abelian category vanishes and that in general negative k groups of a noetherian abelian category vanish our methods yield an explicit non connective delooping of the k theory of exact categories and chain complexes generalizing constructions of wagoner and pedersen weibel extending a theorem of auslander and sherman we discuss the k theory homotopy fiber of e e and its implications for negative k groups in the appendix we replace waldhausen s cylinder functor by a slightly weaker form of non functorial factorization which is still sufficient to prove his approximation and fibration theorems

the graded centers of derived discrete algebras abstract we describe in the paper the graded centers of the bounded derived categories of the derived discrete algebras in particular we prove that if a is a derived discrete algebra then the reduced part of the graded center of the bounded derived category of a is nontrivial if and only if a has infinite global dimension moreover we show that the nilpotent part of the graded center is controlled by the objects for which the auslander reiten translation coincides with a power of the suspension functor

statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three dimensional space abstract we study the adsorption problem of linear polymers immersed in a good solvent when the container of the polymer solvent system is taken to be a member of the sierpinski gasket sg family of fractals embedded in the three dimensional euclidean space members of the sg family are enumerated by an integer b 2 b and it is assumed that one side of each sg fractal is impenetrable adsorbing boundary we calculate the surface critical exponents 11 1 and s which within the self avoiding walk model saw of polymer chain are associated with the numbers of all possible saws with both one and no ends grafted to the adsorbing surface adsorbing boundary respectively by applying the exact renormalization group method for 2 b 4 we have obtained specific values for these exponents for various types of polymer conformations to extend the obtained sequences of exact values for surface critical exponents we have applied the monte carlo renormalization group method for fractals with 2 b 40 the obtained results show that all studied exponents are monotonically increasing functions of the parameter b for all possible polymer states we discuss mutual relations between the studied critical exponents and compare their values with those found for other types of lattices in order to attain a unified picture of the attacked problem

on the cyclic homology of monomial relation algebras abstract a finite dimensional monomial relation algebra has finite global dimension if and only if the relative cyclic homology of its radical is equal to zero

self avoiding walks on self similar structures finite versus infinite ramification self avoiding walks saws generated by exact enumeration techniques are studied on the sierpinski carpet d 2 and on the sierpinski sponge d 3 also called sierpinski square lattices a detailed comparison of the results for saws on these infinitely ramified fractals to saws on finitely ramified sierpinski gaskets sierpinski triangular lattices on regular lattices and on the incipient percolation cluster is done providing insight into the behaviour of saws on ordered and disordered structures the saws on sierpinski square lattices are found to display a kind of intermediate behaviour sharing aspects of both saws on ordered and on fractal structures as a consequence a des cloizeaux relation does not seem to hold for this structure as opposed to its validity for saws on regular lattices on sierpinski triangular lattices and on the incipient percolation cluster

the algebras with discrete derived category abstract we classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non isomorphic indecomposable complexes

the cartan map on categories of graded modules this paper presents a technique based on elementary k theory it relies on the fact that k for a category of graded modules can be viewed as a z t t l module for some variable t in this context we will deduce a great deal of information from an explicit computation of the coordinates of the matrix of the cartan homomorphism in section 1 we begin by recalling some definitions and results that we will use throughout this paper from these we then describe our basic machinery section 2 will provide some applications of this machine we note in particular the following uses nakayama 131 proposed that if a is a finite dimensional algebra of infinite dominant dimension then a is self injective auslander and reiten 4 noticed that the following stronger statement would imply the truth of nakayama s conjecture given a finite dimensional algebra a with minimal injective resolution 0 a e e se every indecomposable injective module is isomorphic to a summand of some ei here we prove this generalized nakayama conjecture for graded algebras another application of this technique shows that the determinant of the cartan map is one for graded algebras of finite global dimension this in turn is used to show that if b is any algebra not necessarily graded with only finitely many nonisomorphic indecomposable modules m 1 m then the matrix whose ijth coordinate is dim hom mi mj has determinant one finally we give an application to the rationality of power series let f be a field and let a be a finite dimensional f algebra we will assume everywhere that a is basic that is for every simple a module s end s f this assumption is primarily to simplify notation and the arguments given could be modified to handle algebras that are not basic we will use the notation mod a for the category of finitely generated left a modules let g be a full subcategory of mod 4 and let x 55 be the free 390 0021 8693 83 3 00

recollement for differential graded algebras abstract a recollement of triangulated categories describes one such category as being glued together from two others this paper gives a precise criterion for the existence of a recollement of the derived category of a differential graded algebra in terms of two other such categories

a multiplication in cyclic homology we define a multiplication on the cyclic homology of a commutative cocommutative bialgebra h with superproduct in the case when h is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator the loday quillen multiplication is trivial in that case in this paper we define an associative multiplication on the cyclic homology of certain commutative algebras h the definition is based on the cartesian product of sets instead of the shuffle product as in the case of lq and ga when h contains q our product is compatible with the hodge decomposition of hc h so it has the form hc n h hc j m h hc n m h the existence of this multiplication in the case h k is classical since it is just given by the tensor product of complex line bundle however its significance came to our attention in joint work with john klein as a consequence of our study of fiberwise morse theory on circle bundles ik and in other joint works with r penner and r hain by doing morse theory on circles john klein and i realized that there was a rationally nontrivial associative multiplication on the cyclic homology of the integral group ring on any cyclic group and that it had important consequences for pseudoisotopy theory this will be explained in a later paper in this paper we show that this multiplication in cyclic homotopy is defined first in the case when the algebra is h km k m where m is the underlying additive group of a ring then in the case when h is a hopf algebra with superproduct in the sense of gaucher ga and finally in the more general case of any algebra example 12 4 in fact we do not need additive inverses in m nor do we need an antipode for h so we can take m to be a monoid and h to be a bialgebra when h k i e m 1 the cyclic homology becomes a divided power algebra which is of course isomorphic to a polynomial algebra in one generator if k is a field of characteristic zero we assume throughout the paper that k is a commutative ring with 1 and that all tensor products are taken over k this includes those tensor products which appear in the definition of cyclic homology we also use connes original definition that a cyclic object in a category c is a covariant functor c where is connes cyclic category since this is the same as a contravariant functor c an outline of the paper is as follows in the first section we give a topological definition of the multiplication in hc km the construction is so elementary that received by the editors april 14 1994 1991 mathematics subject classification primary 18g60 secondary 16w30 this research was supported by nsf grant no dms 90 02512 c 1999 american mathematical society

representation theory of graded artin algebras having initiated the study of graded artin algebras a in 91 here we initiate the study of their representation theory our point of view is to study graded i modules which we believe to be somewhat more tractable than ungraded ones in order to obtain information about all i modules this point of view leads to the introduction in section 1 of the full subcategory mod of mod i consisting of the gradable objects of mod i that is the finitely generated i modules which support a gradation in these terms one of the major results of the paper asserts that mod has finite representation type if and only if mod i has finite representation type thus in section 3 we introduce a number g g 4 designed to measure the size of mod i if g co we show that i has infinite representation type if g co we show that there is a graded artin algebra q of a certain specified form such that i has infinite representation type precisely when 0 has infinite representation type the artin algebra fl has in particular desirable diagrammatic properties and these we will exploit elsewhere we speculate that when g is finite every finitely generated i module is gradable now in case n has finite representation type it is obvious that g is finite we show indeed that for every gradation of a given artin algebra of finite representation type every module is gradable this result and the others cited are chiefly consequences of a result proved in section 4 if a component of the auslander reiten graph of a graded artin algebra contains a gradable module then the component

the gorenstein projective modules for the nakayama algebras i abstract the aim of this paper is to outline the structure of the category of the gorenstein projective modules where is a nakayama algebra in addition we are going to introduce the resolution quiver of it provides a fast algorithm in order to obtain the gorenstein projective modules and to decide whether is a gorenstein algebra or not and whether it is cm free or not

graded artin algebras rational series and bounds for homological dimensions is a minimal injective resolution of a then all the i are projective then n is selfinjecrive it is of interest to note that the nakayama conjecture follows in a trivial way from the finitistic dimension conjecture this one says the following let n be an artin algebra then sup proj dim m proj dimm oc is finite where m ranges over the finitely generated amodules the finitistic projective dimension has its origins in the early 1960s and was formulated from results of rosenberg and zelinsky in this paper we study the behavior or certain rational power series associated to every graded module over a graded finite dimensional algebra series which are slightly different from the usual poincare betti series it is interesting to note that rationality is enough to enable us to prove that over a graded finite dimensional algebra the linitistic projective dimension is finite over the class of finitely generated graded modules whose nonprojective syzygies have no projective summands this implies the nakayama conjecture for graded algebras see also wilson 3 also using rationality we find bounds for global dimension in terms of graded length and the number of nonisomorphic simple modules finally let a be an algebra of finite representation type having n nonisomorphic indecomposable modules we prove that every composition of more than 2n 2 nonisomorphisms of amodules is zero this is done by using the result of auslander reiten which states that every nonzero homomorphism which is not an isomorphism is a sum of compositions of irreducible morphisms

resolution quiver and cyclic homology criteria for nakayama algebras abstract if a nakayama algebra is not cyclic it has finite global dimension for a cyclic nakayama algebra there are many characterizations of when it has finite global dimension in 17 shen gave such a characterization using ringel s resolution quiver in 11 the second author with zacharia gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension we show directly that these criteria are equivalent for all nakayama algebras our comparison result also reproves both characterizations in a separate paper we discuss an interesting example that came up in our attempt to generalize this comparison result to arbitrary monomial relation algebras 8

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

approximations with modules having linear resolutions abstract let be a koszul algebra over a field k we study in this paper a class of modules closely related to the koszul modules called weakly koszul modules it turns out that these modules have some special filtrations with modules having linear resolutions and therefore easy to describe minimal projective resolutions we prove that if the koszul dual of a finite dimensional koszul algebra is noetherian then every finitely generated graded module has a weakly koszul syzygy and as a consequence a rational poincare series if is selfinjective koszul we prove that the stable part of each connected component of the graded auslander reiten quiver containing a weakly koszul module is of the form z a and if the koszul dual of is noetherian then every component has its stable part of the form z a

parametrizing recollement data for triangulated categories abstract we give a general parametrization of all the recollement data for a triangulated category with a set of generators from this we deduce a characterization of when a 0 perfectly generated or aisled triangulated category is a recollement of triangulated categories generated by a single compact object also we use homological epimorphisms to give a complete and explicit description of all the recollement data for or smashing subcategories of the derived category of a k flat dg category in the final part we give a bijection between smashing subcategories of compactly generated triangulated categories and certain ideals of the subcategory of compact objects in the spirit of h krause s work henning krause cohomological quotients and smashing localizations amer j math 127 2005 1191 1246 this bijection implies the following weak version of the generalized smashing conjecture in a compactly generated triangulated category every smashing subcategory is generated by a set of milnor colimits of compact objects

correspondence analysis applied to grouped cohort data in this paper we investigate application of generalized correspondence analysis gca to model a two way table of rates consisting of a cross classification of case numerators and person time denominators the gca reconstitution formula is proposed as a correlation model for analysing these data this provides two major advantages over the techniques usually applied the number of interaction parameters to be estimated is reduced and moreover they have useful interpretation the singular values and the scores are measures of row column modification interaction effects in addition the scores allow graphical display of these effects an asymmetrical plot is proposed for optimal representation of interaction parameters as distances finally two examples are presented to illustrate the method of analysing a two way table of covariate by exposure and a two way table derived from a cross classification of two exposure factors

a proof of the strong no loop conjecture abstract the strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non zero self extension the main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field

canonical analysis of contingency tables by maximum likelihood abstract canonical analysis has often been employed instead of log linear models to analyze the relationship of two polytomous random variables however until the last few years analysis has been informal in this article models are examined that place nontrivial restrictions on the values of the canonical parameters so that a parsimonious description of association is obtained maximum likelihood is used to obtain parameter estimates for these restricted models approximate confidence intervals are derived for parameters and chi squared tests are used to check adequacy of models the resulting models may be used to determine the appropriateness of latent class analysis or to determine whether a set of canonical scores has specified patterns results are illustrated through analysis of two tables previously analyzed in the statistical literature comparisons are made with alternate methods of analysis based on a log linear parameterization of cell probabilities it is shown that canonical analysis wh