kaplansky classes and derived categories we put a monoidal model category structure on the category of chain complexes of quasi coherent sheaves over a quasi compact and semi separated scheme x the approach generalizes and simplifies the method used by the author in trans am math soc 356 8 3369 3390 2004 and trans am math soc 358 7 2855 2874 2006 to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space indeed much of the paper is dedicated to showing that in any grothendieck category mathcal g any nice enough class of objects mathcal f induces a model structure on the category ch mathcal g of chain complexes the main technical requirement on mathcal f is the existence of a regular cardinal such that every object f in mathcal f satisfies the following property each generated subobject of f is contained in another generated subobject s for which s f s in mathcal f such a class mathcal f is called a kaplansky class kaplansky classes first appeared in enochs and lopez ramos rend sem mat univ padova 107 67 79 2002 in the context of modules over a ring r we study in detail the connection between kaplansky classes and model categories we also find simple conditions to put on mathcal f which will guarantee that our model structure is monoidal we will see that in several categories the class of flat objects form such kaplansky classes and hence induce monoidal model structures on the associated chain complex categories we will also see that in any grothendieck category mathcal g the class of all objects is a kaplansky class which induces the usual non monoidal injective model structure on ch mathcal g

orientable homotopy modules we prove a conjecture of morel identifying voevodsky s homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy t structure and have a trivial action of the hopf map this is done by relating these two kind of objects to rost s cycle modules applications to algebraic cobordism and construction of cycle classes are given

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

the additivity of traces in triangulated categories abstract we explain a fundamental additivity theorem for euler characteristics and generalized trace maps in triangulated categories the proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation the refinement consists of several new axioms relating products and distinguished triangles the axioms hold in the examples and shed light on generalized homology and cohomology theories

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

the derived category of complex periodic k theory localized at an odd prime abstract we prove that for an odd prime p the derived category d k u p of the p local complex periodic k theory spectrum k u p is triangulated equivalent to the derived category of its homotopy ring k u p this implies that if p is an odd prime the triangulated category d k u p is algebraic

achieving the rate distortion bound with low density generator matrix codes it is shown that binary low density generator matrix codes can achieve the rate distortion bound of discrete memoryless sources with general distortion measure via multilevel quantization a practical encoding scheme based on the survey propagation algorithm is proposed the effectiveness of the proposed scheme is verified through simulation

simplicial structures on model categories and functors we produce a highly structured way of associating a simplicial category to a model category which improves on work of dwyer and kan and answers a question of hovey we show that model categories satisfying a certain axiom are quillen equivalent to simplicial model categories a simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits we also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial or continuous functors this is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial quillen equivalence

smashing localizations of rings of weak global dimension at most one abstract we show for a ring r of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in r if moreover r is commutative we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms we also classify smashing localizations of the derived category of any valuation domain and provide an easy criterion for the telescope conjecture tc for any commutative ring of weak global dimension at most one as a consequence we show that the tc holds for any commutative von neumann regular ring r and it holds precisely for those prufer domains which are strongly discrete

integrable systems on so 4 related to xxx spin chains with boundaries we consider two site xxx heisenberg magnets with different boundary conditions which are integrable systems on so 4 possessing additional cubic and quartic integrals of motion the separated variables for these models are constructed using the sklyanin method

derived representation schemes and cyclic homology we describe the derived functor drepv a of the affine representation scheme repv a parametrizing the representations of an associative k algebra a on a finite dimensional vector space v we construct the characteristic maps trv a n hcn a hn drepv a extending the canonical trace trv a hc0 a k repv a to the higher cyclic homology of the algebra a and describe a related derived version of the representation functor introduced recently by m van den bergh we study various operations on the homology of drepv a induced by known operations on cyclic and hochschild homology of a

grothendieck categories of enriched functors abstract it is shown that the category of enriched functors c v is grothendieck whenever v is a closed symmetric monoidal grothendieck category and c is a category enriched over v localizations in c v associated to collections of objects of c are studied also the category of chain complexes of generalized modules ch c r is shown to be identified with the grothendieck category of enriched functors mod r ch mod r over a commutative ring r where the category of finitely presented r modules mod r is enriched over the closed symmetric monoidal grothendieck category ch mod r as complexes concentrated in zeroth degree as an application it is proved that ch c r is a closed symmetric monoidal grothendieck model category with explicit formulas for tensor product and internal hom objects furthermore the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of hovey palmieri strickland 14 is introduced it is shown that the derived category of generalized modules d c r over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category

nilpotence and descent in equivariant stable homotopy theory abstract let g be a finite group and let f be a family of subgroups of g we introduce a class of g equivariant spectra that we call f nilpotent this definition fits into the general theory of torsion complete and nilpotent objects in a symmetric monoidal stable category with which we begin we then develop some of the basic properties of f nilpotent g spectra which are explored further in the sequel to this paper in the rest of the paper we prove several general structure theorems for categories of module spectra over objects such as equivariant real and complex k theory and borel equivariant mu using these structure theorems and a technique with the flag variety dating back to quillen we then show that large classes of equivariant cohomology theories for which a type of complex orientability holds are nilpotent for the family of abelian subgroups in particular we prove that equivariant real and complex k theory as well as the borel equivariant versions of complex oriented theories have this property

higher homotopy excision and blakers massey theorems for structured ring spectra abstract working in the context of symmetric spectra we prove higher homotopy excision and higher blakers massey theorems and their duals for algebras and left modules over operads in the category of modules over a commutative ring spectrum e g structured ring spectra

the grothendieck construction for model categories the grothendieck construction is a classical correspondence between diagrams of categories and cocartesian fibrations over the indexing category in this paper we consider the analogous correspondence in the setting of model categories as a main result we establish an equivalence between suitable diagrams of model categories indexed by m and a new notion of model fibrations over m when m is a model category our construction endows the grothendieck construction with a model structure which gives a presentation of lurie s categorical grothendieck construction and enjoys several good formal properties we apply our construction to various examples yielding model structures on strict and weak group actions and on modules over algebra objects in suitable monoidal model categories

abstract representation theory of dynkin quivers of type a abstract we study the representation theory of dynkin quivers of type a in abstract stable homotopy theories including those associated to fields rings schemes differential graded algebras and ring spectra reflection functors partial coxeter functors and serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules certain explicitly constructed spectral bimodules in fact these universal tilting modules are spectral refinements of classical tilting complexes as a consequence we obtain split epimorphisms from the spectral picard groupoid to derived picard groupoids over arbitrary fields these results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators as further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field this includes yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral serre duality result finally using abstract representation theory of linearly oriented a n quivers we construct canonical higher triangulations in stable derivators and hence a posteriori in stable model categories and stable categories

the homotopy theory of coalgebras over a comonad let k be a comonad on a model category m we provide conditions under which the associated category of k coalgebras admits a model category structure such that the forgetful functor to m creates both cofibrations and weak equivalences we provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of hopf galois extensions these examples are specific instances of the following categories of comodules over a coring for any semihereditary commutative ring r let a be a dg r algebra that is homologically simply connected let v be an a coring that is semifree as a left a module on a degreewise r free homologically simply connected graded module of finite type we show that there is a model category structure on the category of right a modules satisfying the conditions of our existence theorem with respect to the comonad given by tensoring over a with v and conclude that the category of v comodules in the category of right a modules admits a model category structure of the desired type finally under extra conditions on r a and v we describe fibrant replacements in this category of comodules in terms of a generalized cobar construction

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

modules over motivic cohomology abstract for fields of characteristic zero we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to voevodsky s big category of motives the proof makes use of some highly structured models for motivic stable homotopy theory motivic spanier whitehead duality the homotopy theories of motivic functors and of motivic spaces with transfers as introduced from ground up in this paper working with rational coefficients we extend the equivalence for fields of characteristic zero to all perfect fields by employing the techniques of alterations and homotopy purity in motivic homotopy theory

an algebraic model for free rational g spectra for connected compact lie groups g we show that the category of free rational g spectra for a connected compact lie group g is quillen equivalent to the category of torsion differential graded modules over the polynomial ring h bg the ingredients are the enriched morita equivalences of schwede and shipley topology 42 1 103 153 2003 the functors of shipley am j math 129 351 379 2007 making rational spectra algebraic koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring

s modules and symmetric spectra we study a symmetric monoidal adjoint functor pair between the category of s modules and the category of symmetric spectra the functors induce equivalences between the respective homotopy categories of spectra module spectra and ring spectra then at approximately the same time two categories of spectra with nice smash products were discov ered elmendorf kriz mandell and may constructed the category of s modules ekmm and je smith introduced symmetric spectra hss both categories are quillen model categories and have associated notions of ring and module spectra however these two categories arise in completely dierent ways and even though the homotopy categories are equivalent it is not a priori clear if both frameworks give rise to the same homotopy theory of rings and modules both categories have their merits described in detail in the introductions of ekmm and hss and it is desirable to be able to translate results obtained in one category into conclusions valid in the other the present paper describes an easy mechanism which facilitates such comparisons below we dene a lax symmetric monoidal functor ms sp from the category of s modules to the category of symmetric spectra the functor preserves homotopy groups and has a strong symmetric monoidal left adjoint we show that the two functors induce inverse equivalences of the homotopy categories of spectra ring spectra commutative ring spectra and module spectra main theorem the functor from the category ms of s modules to the category sp of symmetric spectra passes to a symmetric monoidal equivalence of homotopy categories

k theory and derived equivalences we show that if two rings have equivalent derived categories then they have the same algebraic k theory similar results are given for g theory and for a large class of abelian categories

monoidal uniqueness of stable homotopy theory we show that the monoidal product on the stable homotopy category of spectra is essentially unique this strengthens work of this author with schwede on the uniqueness of models of the stable homotopy theory of spectra also the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra s modules w spaces orthogonal spectra simplicial functors with symmetric spectra as an application we show that with an added assumption about underlying model structures margolis axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence

the stable homotopy category has a unique model at the prime 2 the stable homotopy category has been extensively studied by algebraic topologists for a long time for many applications it is convenient or even necessary to work with point set level models of spectra as opposed to working up to homotopy and the outcome of a calculation can depend on the choice of model in recent years many new models for the stable homotopy category have been constructed it is especially useful to have the structure of a closed model category in the sense of quillen qui and many examples of spectra categories t into this context bf rob87 ekmm hss lyd mmss moreover all known examples capture the same homotopy theory in technical terms one speaks of quillen equivalent model categories hov def 1 3 12 hence not only the homotopy categories but also higher order information such as toda brackets homotopy colimits and homotopy types of function spaces coincide in two quillen equivalent model categories the answer to every homotopy theoretic question comes out the same in a model category one can pass to the associated homotopy category by formally inverting the class of weak equivalences however passage to the homotopy category loses information and in general the homotopy theory can not be recovered from the homotopy category see 2 1 and 2 2 for two examples in this paper we show that in contrast to the general case the stable homotopy category completely determines the stable homotopy theory 2 locally we prove a uniqueness theorem which says that there is essentially only one model category structure underlying the stable homotopy category of 2 local spectra the stable homotopy category has no exotic models at the prime 2

quadratic forms and clifford algebras on derived stacks abstract in this paper we present an approach to quadratic structures in derived algebraic geometry we define derived n shifted quadratic complexes over derived affine stacks and over general derived stacks and give several examples of those we define the associated notion of derived clifford algebra in all these contexts and compare it with its classical version when they both apply finally we prove three main existence results for derived shifted quadratic forms over derived stacks define a derived version of the grothendieck witt group of a derived stack and compare it to the classical one

localization of algebras over coloured operads we give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories our approach encompasses a number of previous results about preservation of structures under localizations such as loop spaces or infinite loop spaces and provides new results of the same kind for instance under suitable assumptions homotopical localizations preserve ring spectra in the strict sense not only up to homotopy modules over ring spectra and algebras over commutative ring spectra as well as maps between ring spectra module maps and algebra maps it is principally the treatment of module spectra and their maps that led us to the use of coloured operads also called enriched multicategories in this context

tilting theory via stable homotopy theory we show that certain tilting results for quivers are formal consequences of stability and as such are part of a formal calculus available in any abstract stable homotopy theory thus these results are for example valid over arbitrary ground rings for quasi coherent modules on schemes in the differential graded context in stable homotopy theory and also in the equivariant motivic or parametrized variant thereof in further work we will continue developing this calculus and obtain additional abstract tilting results here we also deduce an additional characterization of stability based on goodwillie s strongly co cartesian n cubes as applications we construct abstract auslander reiten translations and abstract serre functors for the trivalent source and verify the relative fractionally calabi yau property this is used to offer a new perspective on may s axioms for monoidal triangulated categories

on the nonexistence of elements of kervaire invariant one we show that the kervaire invariant one elements j 2j 1 2s exist only for j 6 by browder s theorem this means that smooth framed manifolds of kervaire invariant one exist only in dimensions 2 6 14 30 62 and possibly 126 except for dimension 126 this resolves a longstanding problem in algebraic topology