complexity and multiple complexes in this chapter we introduce the notion of the complexity of a module and explore some related ideas complexity was first defined by jon alperin in the late 1970 s and it helped to motivate much of the development of the homological properties of modules in fact the complexity of a finitely generated kg module m is a rather crude invariant in that it only measures the polynomial rate of growth of a minimal projective resolution of the module the original theorem of alperin and evens which extended quillen s dimension theorem to modules was proved only in terms of complexity later it was extended to support varieties as well the theorem computes the complexity of a kg module m as the maximum of the complexities of the restrictions of m to the elementary abelian subgroups of the group g

stratification for module categories of finite group schemes the second author was partly supported by nsf grant dms 1503044 the fourth author was partly supported by nsf grant dms 0953011

representation dimension and finitely generated cohomology abstract we consider selfinjective artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators for such an algebra we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules this provides a unified approach to computing lower bounds for the representation dimension of group algebras exterior algebras and artin complete intersections we also obtain new examples of classes of algebras with arbitrarily large representation dimension

a general class of shift splitting preconditioners for non hermitian saddle point problems with applications to time harmonic eddy current models abstract based on a general splitting of the 1 1 leading block matrix we first construct a general class of shift splitting gcss preconditioners for non hermitian saddle point problems convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the gcss preconditioned saddle point matrices are analyzed then the gcss preconditioner is specifically applied to the non hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time harmonic eddy current models with suitable choices of the splittings the new gcss preconditioners are easier to implement and have faster convergence rates than the existing shift splitting preconditioner and its modified variant two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

dynamical analysis in a bioeconomic phytoplankton zooplankton system with double time delays and environmental stochasticity abstract this paper presents a double delayed bioeconomic phytoplankton zooplankton system with commercial harvesting on zooplankton and environmental stochasticity maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered environmental stochasticity is incorporated into the proposed system in form of gaussian white noises some sufficient conditions are derived to show that the proposed system has a unique global positive solution in absence of double time delays stochastic stability and existence of stochastic hopf bifurcation are studied based on invariant measure theory and singular boundary theory of diffusion process for the proposed system in presence of double time delays asymptotic behaviors of the interior equilibrium are discussed by constructing some appropriate lyapunov functions

representations and cohomology of a family of finite supergroup schemes abstract we examine the cohomology and representation theory of a family of finite supergroup schemes of the form g a g a g a r z p s in particular we show that a certain relation holds in the cohomology ring and deduce that for finite supergroup schemes having this as a quotient both cohomology mod nilpotents and projectivity of modules is detected on proper sub supergroup schemes this special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes in a separate work of the authors joint with iyengar and krause we also completely determine the cohomology ring in the smallest cases namely g a g a g a 1 and g a g a z p the computation uses the local cohomology spectral sequence for group cohomology which we describe in the context of finite supergroup schemes

dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations abstract in this paper we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered stochastic fluctuations are incorporated into the proposed system in form of gaussian white noises to depict stochastic environmental factors in plankton system for deterministic system without double time delays existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting and state feedback controllers are designed to eliminate singularity induced bifurcation for deterministic system with double time delays positivity and uniform persistence of solutions are studied and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated for stochastic system without double time delays stochastic stability and existence of stochastic hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory for stochastic system with double time delays existence and uniqueness of global positive solution are investigated and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate lyapunov functions numerical simulations are carried out to validate theoretical analysis

on support varieties and tensor products for finite dimensional algebras abstract it has been asked whether there is a version of the tensor product property for support varieties over finite dimensional algebras defined in terms of hochschild cohomology we show that in general no such version can exist in particular we show that for certain quantum complete intersections there are modules and bimodules for which the variety of the tensor product is not even contained in the variety of the one sided module

detecting cohomology for lie superalgebras abstract in this paper we use invariant theory to develop the notion of cohomological detection for type i classical lie superalgebras in particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras these results are used later to affirmatively answer questions which were originally posed in boe et al 2010 5 and bagci et al 2008 2 about realizing support varieties for lie superalgebras via rank varieties constructed for the smaller detecting subalgebras

global stability and hopf bifurcation of a plankton model with time delay abstract a differential delay equation model with a discrete time delay and a distributed time delay is introduced to simulate zooplankton nutrient interaction the differential inequalities methods and standard hopf bifurcation analysis are applied some sufficient conditions are obtained for persistence and for the global stability of the unique positive steady state respectively it was shown that there is a hopf bifurcation in the model by using the discrete time delay as a bifurcation parameter

annihilators of cohomology modules let g be a finite group k a field of characteristic p 0 and u a kgmodule recently alperin and evens l proved that the complexity of u a numerical invariant defined in terms of the minimal projective resolution of u is determined by the restrictions of u to elementary abelian p subgroups this result has also been obtained by carlson 2 as a consequence of a theorem concerning the dimension of the sum of the nonprojective components of such a restriction in this paper we focus on annihilators of modules for the cohomology ring h g k and show how to obtain the alperin evens theorem and some of its corollaries from a decomposition theorem for certain ideals

iterative methods for the saddle point problem arising from the hc ei formulation of the eddy current problem this paper is concerned with the solution of the linear system arising from a finite element approximation of the time harmonic eddy current problem we consider the bf h c bf e i formulation introduced and analyzed in a m alonso rodriguez r hiptmair and a valli numer methods partial differential equations 21 2005 pp 742 763 where an optimal error estimate for the finite element approximation using edge elements of the first order is proved now we propose and analyze iterative procedures for the solution of the resulting linear system based on the physical decomposition of the computational domain in an insulating region and a conducting one if the insulator does not contain any nonbounding cycle we prove that the dirichlet neumann iteration converges with a rate that is independent of the mesh size in the case of a connected conductor with general topology we propose to use either a modified version of the dirichlet neumann iteration or an uzawa like method we compare the per

finite generation of the cohomology of quotients of pbw algebras abstract in this article we prove finite generation of the cohomology of quotients of a pbw algebra a by relating it to the cohomology of quotients of a quantum symmetric algebra s which is isomorphic to the associated graded algebra of a the proof uses a spectral sequence argument and a finite generation lemma adapted from friedlander and suslin

support varieties an axiomatic approach we provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category where the tensor product is not necessarily symmetric this is illustrated by examples taken in particular from the representation theory of finite dimensional algebras

support varieties and hochschild cohomology rings we define a support variety for a finitely generated module over an artin algebra lambda over a commutative artinian ring k with lambda flat as a module over k in terms of the maximal ideal spectrum of the algebra rm hh lambda of lambda this is modelled on what is done in modular representation theory and the varieties defined in this way are shown to have many of the same properties as those for group rings in fact the notions of a variety in our sense and those for principal and non principal blocks are related by a finite surjective map of varieties for a finite dimensional self injective algebra over a field the variety is shown to be an invariant of the stable component of the auslander reiten quiver moreover we give information on nilpotent elements in rm hh lambda give a thorough discussion of the ring rm hh lambda on a class of nakayama algebras give a brief discussion of a possible notion of complexity and make a comparison with support varieties for complete intersections

quillen stratification for the steenrod algebra let a be the mod 2 steenrod algebra its cohomology h a f2 exta f2 f2 is the e2 term of the adams spectral sequence converging to the 2 component of ir ss the stable homotopy groups of spheres as such this cohomology algebra has been studied extensively for over thirty five years there have been a number of nice structural results and degree by degree computations many of which have implications for rxs0 information about the ring structure of h a f2 though has been harder to come by analogously in the study of the stable homotopy groups of spheres there were many stem by stem calculations before nishida proved in 14 that every element in e3i o i7r so was nilpotent since 7r8s0 z and rs0 o when i 0 in his landmark 1971 paper 18 quillen gave a description of the cohomology algebra h g k modulo nilpotent elements as an inverse limit of the cohomology algebras of the elementary abelian p subgroups of g this has led for instance to the work of benson carlson et al on the theory of varieties for kg modules and in general to deep structural information about modular representations of finite groups in this paper we describe the cohomology of the steenrod algebra modulo nilpotent elements according to the recipe in quillen s result as the inverse limit of the cohomology rings of the elementary abelian sub hopf algebras of a we hope that this leads to further study in several directions first of all according to the point of view of axiomatic stable homotopy theory in 7 the representation theory of any cocommutative hopf algebra has formal similarities to stable homotopy theory viewed this way our main result is an

representation theoretic support spaces for finite group schemes we introduce the space p g of abelian p points of a finite group scheme over an algebraically closed field of characteristic p 0 we construct a homeomorphism g p g proj g from p g to the projectivization of the cohomology variety for any finite group g for an elementary abelian p group respectively an infinitesimal group scheme p g can be identified with the projectivization of the variety of cyclic shifted subgroups resp variety of 1 parameter subgroups for a finite dimensional g module m g restricts to a homeomorphism p g m proj g m thereby giving a representation theoretic interpretation of the cohomological support variety

some conjectures and their consequences for tensor products of modules over a finite p group abstract we make some conjectures about tensor products of modular representations for a finite p group in characteristic p for p 2 and p 3 the easiest to state is that if m is a module of odd dimension for a 2 group over an algebraically closed field of characteristic two then k is the only direct summand of m m of odd dimension and the remaining indecomposable summands have dimension divisible by four as a consequence the odd dimensional modules form an abelian group where the composition law is to take the unique indecomposable summand of odd dimension of the tensor product and where the inverse of an odd dimensional module is its dual in characteristic three the conjectures are more complicated and in larger characteristics it is not at all clear what the corresponding conjectures should be we prove a number of theorems relating to these conjectures and describe examples illustrating them the examples were computed using the computer algebra package magma

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 cohomology of saturated fusion systems and support varieties abstract in this short note we study the cohomology algebra of saturated fusion systems using finite groups which realize saturated fusion systems and hochschild cohomology of group algebras a similar result to a theorem of alperin 1 is proved for varieties of cohomology algebras of fusion systems associated to block algebras of finite groups

support varieties and representation types for basic classical lie superalgebras abstract let be an algebraically closed field of characteristic p 3 and g a restricted lie superalgebra over we introduce the definition of restricted cohomology for g and show its cohomology ring is finitely generated provided g is a basic classical lie superalgebra as a consequence we show that the restricted enveloping algebra of a basic classical lie superalgebra g is always wild except g sl 2 or g osp 1 2 or g c 2

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

representations of elementary abelian p groups and bundles on grassmannians abstract we initiate the study of representations of elementary abelian p groups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements k t 1 t r t 1 p t r p we introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras we also introduce modules of constant radical and socle type generalizing modules of constant jordan type and provide several general constructions of modules with these properties we show that modules of constant radical and socle type lead to families of algebraic vector bundles on grassmannians and illustrate our theory with numerous examples

on support varieties for lie superalgebras and finite supergroup schemes abstract we study the spectrum of the cohomology rings of cocommutative hopf superalgebras restricted and non restricted lie superalgebras and finite supergroup schemes we also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory we completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero while for non restricted lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by duflo and serganova in characteristic zero but which have no known cohomological interpretation our computations for restricted lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of friedlander and parshall and of bendel friedlander and suslin in the classical setting

modules of constant jordan type we introduce the class of modules of constant jordan type for a finite group scheme g over a field k of characteristic p 0 this class is closed under taking direct sums tensor products duals heller shifts and direct summands and includes endotrivial modules it contains all modules in an auslander reiten component which has at least one module in the class highly non trivial examples are constructed using cohomological techniques we offer conjectures suggesting that there are strong conditions on a partition to be the jordan type associated to a module of constant jordan type

the varieties for some specht modules abstract j carlson introduced the cohomological and rank variety for a module over a finite group algebra we give a general form for the largest component of the variety for the specht module for the partition p p of p 2 restricted to a maximal elementary abelian p subgroup of rank p we determine the varieties of a large class of specht modules corresponding to p regular partitions to any partition of np of not more than p parts with empty p core we associate a unique partition of np where the rank variety of the restricted specht module s e n to a maximal elementary abelian p subgroup e n of rank n is v e n k if and only if v e n s v e n k in some cases where is a 2 part partition we show that the rank variety v e n s is v e n k in particular the complexity of the specht module s is n

generic module theory abstract the process of restricting modules to cyclic shifted subgroups is a fundamental technique in the modular representation theory of elementary abelianp groups ifeis elementary abelian ofp rankrandkis an algebraically closed field of characteristicp then each point inkr 0 determines a cyclic shifted subgroup because the restriction of ake module to this shifted subgroup depends only upon the corresponding point in projective space it is often convenient to work with2kr 1instead ofkr 0 roughly speaking this paper shows that ifvis an irreducible subvariety of3kr 1andmis ake module then for almost all points invthe direct sum decomposition ofmis the same upon restriction moreover this decomposition is completely determined by the behavior ofmupon restriction to the cyclic shifted subgroup corresponding to the generic point ofv a similar idea provides a stratification of the rank variety ofminto a disjoint union of locally closed subspaces the closures of these subspaces are then described in terms of deformations of modules over a group of orderp

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

spatio temporal delays in a nutrient plankton model on a finite domain linear stability and bifurcations the issue of how to incorporate time delays into a mathematical model in which individuals are moving around requires careful consideration any time delay term must also involve a weighted spatial averaging to account for movement of individuals during the time delay period most of the current literature on this subject is on reaction diffusion equations and concentrates on the simplest case when the spatial domain is infinite in this paper we consider what changes arise when the domain is finite spatial averaging kernels are computed explicitly for the case of a finite one dimensional domain to illustrate the ideas we concentrate on a diffusive nutrient plankton model the model is analysed in terms of the local stability of the steady states and bifurcations the results of some numerical simulations are also presented

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

on cocommutative hopf algebras of finite representation type abstract let g be a finite algebraic group defined over an algebraically closed field k of characteristic p 0 such a group decomposes into a semidirect product g g 0 g red with a constant group g red and a normal infinitesimal subgroup g 0 if the principal block b 0 g of the group algebra h g has finite representation type then both constituents have the same property with at least one of them being semisimple we determine the structure of the infinitesimal constituent g 0 up to the classification of v uniserial groups

substructuring preconditioners for saddle point problems arising from maxwell s equations in three dimensions this paper is concerned with the saddle point problems arising from edge element discretizations of maxwell s equations in a general three dimensional nonconvex polyhedral domain a new augmented technique is first introduced to transform the problems into equivalent augmented saddle point systems so that they can be solved by some existing preconditioned iterative methods then some substructuring preconditioners are proposed with very simple coarse solvers for the augmented saddle point systems with the preconditioners the condition numbers of the preconditioned systems are nearly optimal namely they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

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

weil restriction and support varieties this paper extends the comparison results for cohomological support varieties for chevalley groups of the form g fp established by j carlson z lin and d nakano to chevalley groups of the form g fpd other topics considered are complexity and non maximal support varieties for modules techniques developed by j pevtsova and the author together with the use of weil restriction functors are employed

permutation complexes and modular representation theory hc g o iti has two basic functors a rg 2 di rh and a rh 2 rg given by restriction and induction which play an essential role in representation theory an important and elementary class of rg representations is permutation modules which are direct sums of modules indz r obtained by induction from the trivial rh module r for various hs g in another extreme one has rg modules which arise by induction from rh projective modules leading to the concept of relative projectivity and green s theory of vertices and sources cr gr the value of these subcategories of modules in representation theory and related areas is well known in a different direction influenced by algebraic geometry and topology one considers not only module categories but various categories of chain complexes of modules and their cohomologies this culminates in the more recent approaches to representation theory through the theory of derived categories see sc cps and their many references

the avrunin and scott theorem and a truncated polynomial algebra we prove an analogue of a theorem of avrunin and scott for truncated polynomial algebras m k x 1 x m x 2 i over an algebraically closed field of arbitrary characteristic the avrunin and scott theorem relates the support variety for a finite dimensional ke module to its rank variety where char k p and e is an elementary abelian p group the analogue of the avrunin and scott theorem relates the support variety for a finite dimensional m module using hochschild cohomology to its rank variety developed in k erdmann m holloway rank varieties and projectivity for a class of local algebras math z 247 2004 441 460 using clifford algebras along the way to proving our main result we provide a new proof of the avrunin and scott theorem for elementary abelian p group algebras which we are then able to generalise to the setting of m algebras

varieties in finite transformation groups the equivariant cohomology ring of a g space x defines a homogeneous affine variety quillen q and w y hsiang hs have determined the relation between such varieties and the family of isotropy subgroups as well as their fixed point sets when dim x oo in modular representation theory j carlson cj introduced cohomological support varieties and rank varieties the latter depending on the group algebra and explored their relationship we define rank and support varieties for g spaces and g chain complexes and apply them to cohomological problems in transformation groups as a corollary a useful criterion for zg projectivity of the reduced total homology of certain g spaces is obtained which improves the projectivity criteria of rim r chouinard ch and dade d

the representation dimension of k x y x2 yn the representation dimension of an artin algebra was defined by m auslander in 1970 the precise value is not known in general and is very hard to compute even for small examples for group algebras it is known in the case of cyclic sylow subgroups for some group algebras in characteristic 2 of rank at least 3 the precise value of the representation dimension follows from recent work of r rouquier there is a gap for group algebras of rank 2 in this paper we show that for all n 0 and any field k the commutative algebras k x y x2 y2 n have representation dimension 3 for the proof we give an explicit inductive construction of a suitable generator cogenerator as a consequence we obtain that the group algebras in characteristic 2 of the groups c2 c2m have representation dimension 3 note that for m 3 these group algebras have wild representation type

finite element simulation of eddy current problems using magnetic scalar potentials we propose a new implementation of the finite element approximation of eddy current problems using as the principal unknown the magnetic field in the non conducting region a scalar magnetic potential is introduced the method can deal automatically with any topological configuration of the conducting region and being based on the search of a scalar magnetic potential in the non conducting region has the advantage of making use of a reduced number of unknowns several numerical tests are presented for illustrating the performance of the proposed method in particular the numerical simulation of a new type of transformer of complicated topological shape is shown

cohomological symmetry in triangulated categories abstract we give a criterion for cohomological symmetry in a triangulated category as an application we show that such cohomological symmetry holds for all pairs of modules over any exterior algebra

the avrunin scott theorem for quantum complete intersections abstract we prove the avrunin scott theorem for quantum complete intersections relating the rank variety of a module to its support variety

boundedness and periodicity of modules over qf rings be a minimal projective resolution m is called bounded if the lengths of the kernels ker di have a common upper bound and m is called periodic if some ker di is isomorphic to m in 1 p 7761 alperin posed the following question is every bounded kg module periodic alperin answered this question in the affirmative in case k is algebraic over its prime field l theorem 11 without any restriction on k eisenbud settled alperin s question in 6 theorem 9 21 in section 1 of the present paper we look at an arbitrary qf ring r instead of a group algebra and we give a condition which forces a bounded r module m to be periodic before we will formulate our result we make some preliminary remarks let m and x be modules over a ring r with the yoneda product for multiplication the abelian group extx m m oi extk m m becomes a graded ring which we call the extension ring of m and the abelian group ext m x becomes a graded right extr m m module we recall some classical examples of extension rings let g be a finite group and let k be a field on which g acts trivially then the cohomology ring h g k is the extension ring ext k k more generally let m be a finitely generated kg module then the ring hom m m is a g ring and the cohomology ring h g hom m m with cup product is canonically isomorphic to the extension ring extx m m with yoneda product these rings have been intensively studied see carlson 3 for details and for further references other well known exampes of extension rings arise in the homological algebra of local rings namely the extension

rank varieties and projectivity for a class of local algebras we consider algebras open image in new window where k is an algebraically closed fields to any finite dimensional module for this algebra we associate a rank variety when char k 2 we recover carlson s rank variety the main result states that a module is projective if and only if its rank variety vanishes this has applications to other algebras including tensor products of certain brauer tree algebras and certain parabolic hecke algebras in addition the result has implications for the graph structure of the stable auslander reiten quiver mathematics subject classification 2000 16g10 18g05 20c20

complexes and vanishing of cohomology for group schemes abstract completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology in this paper we study these complexes and the vanishing of cohomology for arbitrary finite dimensional cocommutative hopf algebras applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups

modules with small loewy length a module of complexity c for e z p r in characteristic p has loewy length at least p 1 r c 1 we study the case of equality if p is odd the only rank varieties possible are finite unions of linear subspaces of dimension c and every such rank variety occurs if p 2 the variety has to be equidimensional if such a variety is a finite union of set theoretic complete intersections then it occurs for such a module but otherwise the situation is unclear exterior algebras in any characteristic are also treated and follow the same behaviour as the case p 2 above

on module varieties and quotient groups let g be a finite group and k be an algebraically closed field of characteristic p 0 all modules considered here are assumed to be finitely generated let e k 1 p ext k k then e k is a finitely generated commutative k algebra and so it has an associated afline variety v k max eg k which is the set of maximal ideals of e k endowed with the zariski topology let m be a kg module and j m be the annihilator in e k of ext fc m m x to extijm m following carlson 4 we define the variety v m of m as the subvariety of v k associated to jg m which is equal to the support of m in z let h be a subgroup of g then the affine map h g v k v k is induced by the restriction map resg h ef k e k in their paper 1 avrunin and scott showed that i has the property vc m v m see theorem 3 1 in 1 i the main purpose of this paper is to give a similar result as the above for the inflation map let h be a normal subgroup of g and m be a k g h module then we can define the affine map agqgih v k p vgih k induced by the inflation map inf e k e k and under the above notation we shall prove the following

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

the dimensional splitting iteration methods for solving saddle point problems arising from time harmonic eddy current models abstract a dimensional splitting iteration method is proposed for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time harmonic eddy current models which is by making use of the special positive semidefinite splittings of the saddle point matrix it is proved that the proposed iteration method is unconditionally convergent for both cases of simple topology and general topology numerical results show that the corresponding preconditioner is superior to the existing preconditioners when those preconditioners are used to accelerate the convergence rate of krylov subspace methods

support varieties and cohomology over complete intersections quillen s geometric approach to the cohomology of finite groups 30 has revolutionized modular representation theory the ideology and techniques involved have been generalized and extended to representations of various hopf algebras culminating in the recent work of friedlander and suslin 17 on finite group schemes in this paper we develop geometric methods for the study of finite modules over a local complete intersection r if r is such a ring and m is its maximal ideal then the m adic completion r has the form q f where f is a regular sequence and q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring the least number of equations needed to cut out r from a regular local ring equals the codimension of r where codim r r m dim r and r m denotes the minimal number of generators of an r module m in 5 a cone that is a homogeneous algebraic set vr m is attached to each finite r module m and used to study its minimal free resolution here we prove

global stability in chemostat type competition models with nutrient recycling this note corrects some typos and errors in the paper s ruan and x z he siam j appl math 58 1998 pp 170 192

pi supports for modules for finite group schemes we introduce the space g of equivalence classes of points of a finite group scheme g and associate a subspace g m to any g module m our results extend to arbitrary finite group schemes g over arbitrary fields k of positive characteristic and to arbitrarily large g modules the basic results about cohomological support varieties and their interpretation in terms of representation theory in particular we prove that the projectivity of any possibly infinite dimensional g module can be detected by its restriction along points of g unlike the cohomological support variety of a g module m the invariant m 7 g m satisfies good properties for all modules thereby enabling us to determine the thick tensor ideal subcategories of the stable module category of finite dimensional g modules finally using the stable module category of g we provide g with the structure of a ringed space which we show to be isomorphic to the scheme projh g k

representation dimension of exterior algebras we determine the representation dimension of exterior algebras this provides the first known examples of representation dimension 3 we deduce that the loewy length of the group algebra over f2 of a finite group is strictly bounded below by the 2 rank of the group a conjecture of benson a key tool is the use of the concept of dimension of a triangulated category

support varieties for the steenrod algebra in recent years the spectrum of the cohomology ring has been a fundamental object of interest in the study of modular group cohomology quillen q1 2 proved that this variety can be expressed as a disjoint union of locally closed varieties involving elementary abelian subgroups avrunin and scott as later generalized quillen s result for varieties associated to modules defined by carlson c1 for restricted lie algebra cohomology no such stratification theorem exists friedlander and parshall fp1 2 have shown that the cohomological varieties in this context can be realized as subvarieties of the given lie algebra when viewed as an affine space fix a prime p and let a be the mod p steenrod algebra the cohomology of a exta fp fp is an object of much interest in algebraic topology one way to study the cohomology ring is to approximate a by finite dimensional sub hopf algebras and then try to compute their cohomology rings by a theorem of wilkerson w the cohomology of any of these finite dimensional hopf algebras is a finitely generated commutative fp algebra hence one can get a great deal of information as in the case of groups and lie algebras by studying the spectrum of this ring one of the goals of this paper will be to show how the aforementioned results for finite groups and restricted lie algebras can be used to study the cohomological varieties for finite dimensional subhopf algebras of the steenrod algebra given a finite dimensional sub hopf algebra b of a let b fp be the spectrum of extb fp fp for a finite dimensional b module m the

on the structure and cohomology ring of connected hopf algebras abstract let p be a prime and k be a field of characteristic p we investigate the algebra structure and the structure of the cohomology ring for the connected hopf algebras of dimension p 3 which appear in the classification obtained in 20 the list consists of 23 algebras together with two infinite families we identify the morita type of the algebra and in almost all cases this is sufficient to clarify the structure of the cohomology ring