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

the homotopy groups of the localized toda smith spectrum 1 at the prime 3 in this paper we try to compute the homotopy groups of the l2 localized toda smith spectrum v 1 at the prime 3 by using the adamsnovikov spectral sequence and have almost done so this computation involves non trivial differentials d5 and dg of the adams novikov spectral sequence different from the case p 3 we also determine the homotopy groups of some l2 localized finite spectra relating to v 1 we further show some of the non trivial differentials on elements relating so called 3 elements in the adams novikov spectral sequence for 7r s0

critical crossover functions for simple fluids towards the crossover modelling uniqueness based on a single non universal temperature scaling factor present in a simple fluid case a detailed analysis of non universal parameters involved in different critical to classical crossover models is given for the infinite limit of the cutoff wave number a set of three scaling parameters is defined for each model such that it shows all the shapes of the theoretical crossover functions overlap on the mean crossover function shapes close to the non trivial fixed point the analysis of corresponding links between their fluid dependent parameters opens a route to define a parametric model of crossover equation of state closely satisfying the universal features calculated from the ising like limit in the massive renormalization scheme

monte carlo investigations of phase transitions status and perspectives abstract using the concept of finite size scaling monte carlo calculations of various models have become a very useful tool for the study of critical phenomena with the system linear dimension as a variable as an example several recent studies of ising models are discussed as well as the extension to models of polymer mixtures and solutions it is shown that using appropriate cluster algorithms even the scaling functions describing the crossover from the ising universality class to the mean field behavior with increasing interaction range can be described additionally the issue of finite size scaling in ising models above the marginal dimension d 4 is discussed

towards the homotopy of the k 2 local moore spectrum at p 2 abstract let v 0 be the mod 2 moore spectrum and let c be the supersingular elliptic curve over f 4 defined by the weierstrass equation y 2 y x 3 let f c be its formal group law and e c be the spectrum classifying the deformations of f c the group of automorphisms of f c which we denote by s c acts on e c further s c admits a surjective homomorphism to z 2 whose kernel we denote by s c 1 the cohomology of s c 1 with coefficients in e c v 0 is the e 2 term of a spectral sequence converging to the homotopy groups of e c h s c 1 v 0 a spectrum closely related to l k 2 v 0 in this paper we use the algebraic duality resolution spectral sequence to compute an associated graded for h s c 1 e c v 0 these computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2

simultaneous analysis of several models in the three dimensional ising universality class we investigate several three dimensional lattice models believed to be in the ising universality class by means of monte carlo methods and finite size scaling these models include spin 1 models with nearestneighbor interactions on the simple cubic and on the diamond lattice for the simple cubic lattice we also include models with third neighbor interactions of varying strength and some equivalent neighbor models also included are a spin 1 model and a hard core lattice gas separate analyses of the numerical data confirm the ising like critical behavior of these systems on this basis we analyze all these data simultaneously such that the universal parameters occur only once this leads to an improved accuracy the thermal magnetic and irrelevant exponents are determined as y t51 5868 3 y h52 4816 1 and y i520 821 5 respectively the binder ratio is estimated as q5 m 2 2 m 4 50 62 341 4

critical properties of the three dimensional equivalent neighbor model and crossover scaling in finite systems accurate numerical results are presented for the three dimensional equivalent neighbor model on a cubic lattice for 12 different interaction ranges coordination number between 18 and 250 these results allow the determination of the range dependences of the critical temperature and various critical amplitudes which are compared to renormalization group predictions in addition the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin 1 2 model and confirms earlier conclusions that the leading wegner correction must be negative for the three dimensional nearest neighbor ising model by complementing these results with monte carlo data for systems with coordination numbers as large as 52 514 the full finite size crossover curves between classical and ising like behavior are obtained as a function of a generalized ginzburg parameter also the crossover function for the effective magnetic exponent is determined

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

symmetry in the vanishing of ext over stably symmetric algebras abstract a frobenius algebra over a field k is called symmetric if the nakayama automorphism is an inner automorphism a stably symmetric algebra is defined to be a generalization of a symmetric k algebra in this paper we will study symmetry in the vanishing of ext for such algebras r namely for all finitely generated r modules m and n ext r i m n 0 for all i 0 if and only if ext r i n m 0 for all i 0 we show that a certain class of noetherian stably symmetric gorenstein algebras such as the group algebra of a finite group and the exterior algebra k n when n is odd have this symmetry using serre duality we also show that every exterior algebra k n whether n is even or odd has this symmetry for graded modules using koszul duality

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

a resolution of the k 2 local sphere at the prime 3 we develop a framework for displaying the stable homotopy theory of the sphere at least after localization at the second morava k theory k 2 at the prime 3 we write the spectrum lk 2 s 0 as the inverse limit of a tower of fibrations with four layers the successive fibers are of the form e hf where f is a finite subgroup of the morava stabilizer group and e2 is the second morava or lubin tate homology theory we give explicit calculation of the homotopy groups of these fibers the case n 2a tp 3 represents the edge of our current knowledge n 1 is classical and at n 2 the prime 3 is the largest prime where the morava stabilizer group has a p torsion subgroup so that the homotopy theory is not entirely algebraic the problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics a period of calculation beginning with serre s computation of the cohomology of eilenberg maclane spaces and the advent of the adams spectral sequence culminated in the late 1970s with the work of miller ravenel and wilson on periodic phenomena in the homotopy groups of spheres and ravenel s nilpotence conjectures the solutions to most of these conjectures by devinatz hopkins and smith in the middle 1980s established the primacy of the chromatic point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory computations passed from center stage to some extent although there has been steady work in the wings most notably by shimomura and his coworkers and ravenel and more lately by hopkins and

centralizers of elementary abelian p subgroups and mod p cohomology of profinite groups 1 1 let g be a profinite group and p be a fixed prime in this paper we will be concerned with h c g fp the continuous cohomology of g with coefficients in the trivial module fp we will abbreviate h c g fp by h g fp or simply by h g if p is understood from the context we recall that if g is the inverse limit of finite groups gi then h g colim h gi throughout this paper we will assume that h g is finitely generated as fp algebra by work of lazard la it is known that this holds for many interesting groups for example for profinite p analytic groups like gl n zp the general linear groups over the p adic integers in case h g is finitely generated as fp algebra quillen has shown q1 that there are only finitely many conjugacy classes of elementary abelian p subgroups of g i e groups isomorphic to z p for some natural number n in other words the following category a g is equivalent to a finite category objects of a g are all elementary abelian p subgroups of g and if e1 and e2 are elementary abelian p subgroups of g then the set of morphisms from e1 to e2 in a g consists precisely of those homomorphisms e1 e2 of abelian groups for which there exists an element g g with e geg 1 e e1 the category a g plays an important role both in quillen s results and in the work presented here this category entered into quillen s work as follows the assignment e 7 h e extends to a functor from the opposite category a g op to graded fp algebras and the restriction homomorphisms h g h e for e running through the elementary abelian p subgroups of g induce a canonical map of algebras q h g lima g op h e

the homotopy groups of se 2 at p 5 revisited abstract we present a new technique for analyzing the v 0 bockstein spectral sequence studied by shimomura and yabe employing this technique we derive a conceptually simpler presentation of the homotopy groups of the e 2 local sphere at primes p 5 we identify and correct some errors in the original shimomura yabe calculation we deduce the related k 2 local homotopy groups and discuss their manifestation of gross hopkins duality

induction theorems for finite groups including a common generalization of two classical theorems abstract roquette proved results involving induction about rings related to the character rings of finite groups and consisting of functions that vanish outside a p section kletzing did something similar for q characters this paper gives new generalizations of these results involving k characters for any field k of characteristic 0 as well as corresponding theorems for a set of primes that these facts are useful is illustrated by showing that they imply a common generalization of the witt berman and artin induction theorems as well as results of chen fan hu robinson and sin

self equivalences of stable module categories let p be an abelian p group e a cyclic p group acting freely on p and k an algebraically closed field of characteristic p 0 in this work we prove that every self equivalence of the stable module category of k p rtimes e comes from a self equivalence of the derived category of k p rtimes e work of puig and rickard allows us to deduce that if a block b with defect group p and inertial quotient e is rickard equivalent to k p rtimes e then they are splendidly rickard equivalent that is broue s original conjecture implies rickard s refinement of the conjecture in this case all of this follows from a general result concerning the self equivalences of the thick subcategory generated by the trivial module

crossover critical behavior in the three dimensional ising model the character of critical behavior in physical systems depends on the range of interactions in the limit of infinite range of the interactions systems will exhibit mean field critical behavior i e critical behavior not affected by fluctuations of the order parameter if the interaction range is finite the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class a variety of systems including fluids and anisotropic ferromagnets belongs to the three dimensional ising universality class recent numerical studies of ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation induced critical behavior and mean field type critical behavior in this work we compare these numerical results with a crossover landau model based on renormalization group matching for this purpose we consider an application of the crossover landau model to the three dimensional ising model without fitting to any adjustable parameters the crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range ten orders of the scaled distance to the critical temperature the dependence of the coupling constant on the interaction range governing the crossover critical behavior is discussed

the homotopy of the k 2 local moore spectrum at the prime 3 revisited in this paper we use the approach introduced in goerss et al ann math 162 2 777 822 2005 in order to analyze the homotopy groups of l k 2 v 0 the mod 3 moore spectrum v 0 localized with respect to morava k theory k 2 these homotopy groups have already been calculated by shimomura j math soc japan 52 1 65 90 2000 the results are very complicated so that an independent verification via an alternative approach is of interest in fact we end up with a result which is more precise and also differs in some of its details from that of shimomura j math soc japan 52 1 65 90 2000 an additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the k 2 localization of the spectrum tmf of topological modular forms and related spectra even more the adams novikov differentials for l k 2 v 0 can be read off from those for tmf