homological properties of the perfect and absolute integral closures of noetherian domains for a noetherian local domain r let r be the absolute integral closure of r and let r be the perfect closure of r when r has prime characteristic in this paper we investigate the projective dimension of residue rings of certain ideals of r and r in particular we show that any prime ideal of r has a bounded free resolution of countably generated free r modules also we show that the analogue of this result is true for the maximal ideals of r when r has residue prime characteristic we compute global dimensions of r and r in some cases some applications of these results are given

a note on k theory and triangulated categories we provide an example of two closed model categories having equivalent homotopy categories but different waldhausen k theories we also show that there cannot exist a functor from small triangulated categories to spaces which recovers quillen s k theory for exact categories and which satisfies localization

on the conservativity of the functor assigning to a motivic spectrum its motive given a 0 connective motivic spectrum e is an element of sh k over a perfect field k we determine h under bar 0 of the associated motive me is an element of dm k in terms of pi under bar 0 e using this we show that if k has finite 2 etale cohomological dimension then the functor m w sh k dm k is conservative when restricted to the subcategory of compact spectra and induces an injection on picard groups we extend the conservativity result to fields of finite virtual 2 etale cohomological dimension by considering what we call real motives

higher traces noncommutative motives and the categorified chern character abstract we propose a categorification of the chern character that refines earlier work of toen and vezzosi and of ganter and kapranov if x is an algebraic stack our categorified chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over x which we introduce to s 1 equivariant perfect complexes on the derived free loop stack l x as an application of the theory we show that toen and vezzosi s secondary chern character factors through secondary k theory our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal n categories which is of independent interest

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

the six operations in equivariant motivic homotopy theory abstract we introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks x g where g is a linearly reductive linear algebraic group we extend to this equivariant setting the main foundational results of motivic homotopy theory the unstable purity and gluing theorems of morel voevodsky and the stable ambidexterity theorem of ayoub our proof of the latter is different than ayoub s and is of interest even when g is trivial using these results we construct a formalism of six operations for equivariant motivic spectra and we deduce that any cohomology theory for g schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology

derived azumaya algebras and generators for twisted derived categories we introduce a notion of derived azumaya algebras over ring and schemes generalizing the notion of azumaya algebras of grothendieck le groupe de brauer i algebres d azumaya et interpretations diverses dix exposes sur la cohomologie des schemas pp 46 66 north holland amsterdam 1968 we prove that any such algebra b on a scheme x provides a class b in h 1 et x mathbb z times h 2 et x mathbb g m we prove that for x a quasi compact and quasi separated scheme defines a bijective correspondence and in particular that any class in h 2 et x mathbb g m torsion or not can be represented by a derived azumaya algebra on x our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories with coefficients in any local system of reasonable dg categories generalizing the well known existence of compact generators in derived categories of quasi coherent sheaves of bondal and van den bergh mosc math j 3 1 1 36 2003 a huge part of this paper concerns the treatment of twisted derived categories as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator we present explicit examples of derived azumaya algebras that are not represented by classical azumaya algebras as well as applications of our main result to the localization for twisted algebraic k theory and to the stability of saturated dg categories by direct push forwards along smooth and proper maps

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

s matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices from isolated to overlapping resonances we study complex eigenvalues of large symmetric random matrices of the form where both and are real symmetric is a random gaussian and is such that when when the model can be used to describe the universal statistics of s matrix poles resonances in the complex energy plane we derive the ensuing distribution of the resonance widths which generalizes the well known distribution to the case of overlapping resonances we also consider a different class of almost real matrices when is random and uncorrelated with

gluing semi orthogonal decompositions abstract we introduce preordered semi orthogonal decompositions psod s of dg categories we show that homotopy limits of dg categories equipped with compatible psod s carry a natural psod this gives a way to glue semi orthogonal decompositions along faithfully flat covers extending the main result of 4 as applications we will construct semi orthogonal decompositions for root stacks of log pairs x d where d is a not necessarily simple normal crossing divisor generalizing results from 17 and 3 further we will compute the kummer flat k theory of general log pairs x d generalizing earlier results of hagihara and niziol in the simple normal crossing case 15 23

basepoint freeness for nef and big line bundles in positive characteristic a necessary and sufficient condition is given for semi ampleness of a numerically effective nef and big line bundle in positive characteristic one application is to the geometry of the universal stable curve over mg specifically the semi ampleness of the relative dualizing sheaf in positive characteristic an example is given which shows this and the semi ampleness criterion fail in characteristic zero a second application is to mori s program for minimal models of 3 folds in positive characteristic namely to the existence of birational extremal contractions

classical wave experiments on chaotic scattering we review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems scattering theory has to be applied for a quantitative interpretation of the measurements most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation in all studied examples a quantitative agreement between experiment and theory is achieved to this end it is necessary however to take absorption and imperfect coupling into account concepts that were ignored in most previous theoretical investigations classical phase space signatures of scattering are being examined in a small number of experiments

on the efficient calculation of a linear combination of chi square random variables with an application in counting string vacua linear combinations of chi square random variables occur in a wide range of fields unfortunately a closed analytic expression for the probability density function is not yet known starting out from an analytic expression for the density of the sum of two gamma variables a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed an explicit expression for the error bound is obtained the proposed technique is shown to be computationally efficient i e only polynomial in growth in the number of terms compared to the exponential growth of most other methods it provides a vast improvement in accuracy and shows only logarithmic growth in the required precision in addition it is applicable to a much greater number of terms and currently the only way of computing the distribution for hundreds of terms as an application the exponential dependence of the eigenvalue fluctuation probability of a random matrix model for 4d supergravity with n scalar fields is found to be of the asymptotic form exp 0 35n

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

a model category structure on the category of simplicial categories in this paper we put a cofibrantly generated model category struc ture on the category of small simplicial categories the weak equivalences are a simplicial analogue of the notion of equivalence of categories

the homotopy theory of dg categories and derived morita theory the main purpose of this work is to study the homotopy theory of dg categories up to quasi equivalences our main result is a description of the mapping spaces between two dg categories c and d in terms of the nerve of a certain category of c d bimodules we also prove that the homotopy category ho dg cat possesses internal hom s relative to the derived tensor product of dg categories we use these two results in order to prove a derived version of morita theory describing the morphisms between dg categories of modules over two dg categories c and d as the dg category of c d bi modules finally we give three applications of our results the first one expresses hochschild cohomology as endomorphisms of the identity functor as well as higher homotopy groups of the classifying space of dg categories i e the nerve of the category of dg categories and quasi equivalences between them the second application is the existence of a good theory of localization for dg categories defined in terms of a natural universal property our last application states that the dg category of continuous morphisms between the dg categories of quasi coherent resp perfect complexes on two schemes resp smooth and proper schemes is quasi equivalent to the dg category of quasi coherent resp perfect complexes on their product

the beilinson regulator is a map of ring spectra abstract we prove that the beilinson regulator which is a map from k theory to absolute hodge cohomology of a smooth variety admits a refinement to a map of e ring spectra in the sense of algebraic topology to this end we exhibit absolute hodge cohomology as the cohomology of a commutative differential graded algebra over r the associated spectrum to this cdga is the target of the refinement of the regulator and the usual k theory spectrum is the source to prove this result we compute the space of maps from the motivic k theory spectrum to the motivic spectrum that represents absolute hodge cohomology using the motivic snaith theorem we identify those maps which admit an e refinement and prove a uniqueness result for these refinements

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

de rham factors we construct a theory of de rham epsilon factors given a smooth projective variety x over a field of characteristic 0 and a open image in new window module open image in new window we may consider the determinant of de rham cohomology open image in new window as a graded super line let s be the singular support of open image in new window u an open in x y x u and a 1 form on u x such that u s then we show the existence of a canonically defined graded super line open image in new window canonically isomorphic to open image in new window the key property is the local nature of the epsilon factor it only depends on the restriction of open image in new window and to any open neighborhood of y restricting to the case of curves gives a theory of epsilon factors for curves previously defined by beilinson bloch deligne and esnault

combinatorial model categories have presentations abstract we show that every combinatorial model category is quillen equivalent to a localization of a diagram category where diagram category means diagrams of simplicial sets this says that every combinatorial model category can be built from a category of generators and a set of relations

universal homotopy theories abstract begin with a small category c the goal of this short note is to point out that there is such a thing as a universal model category built from c we describe applications of this to the study of homotopy colimits the dwyer kan theory of framings sheaf theory and the homotopy theory of schemes

the homotopy limit problem for hermitian k theory equivariant motivic homotopy theory and motivic real cobordism abstract the homotopy limit problem for karoubi s hermitian k theory karoubi 1980 26 was posed by thomason 1983 44 there is a canonical map from algebraic hermitian k theory to the z 2 homotopy fixed points of algebraic k theory the problem asks roughly how close this map is to being an isomorphism specifically after completion at 2 in this paper we solve this problem completely for fields of characteristic 0 theorems 16 20 we show that the 2 completed map is an isomorphism for fields f of characteristic 0 which satisfy cd 2 f i but not in general

rotating fluids with self gravitation in bounded domains in this paper we study the steady solutions of euler poisson equations in bounded domains with prescribed angular velocity this models a rotating newtonian star consisting of a compressible perfect fluid with given equation of state p es when the domain is a ball and the angular velocity is constant we obtain both existence and non existence theorems depending on the adiabatic gas constant in addition we obtain some interesting properties of the solutions e g monotonicity of the radius of the star with both angular velocity and central density we also prove that the radius of a rotating spherically symmetric star with given constant angular velocity and constant entropy is uniformly bounded independent of the central density this is physically striking and in sharp contrast to the case of the non rotating star for general domains and variable angular velocities both an existence result for the isentropic equations of state and non existence result for the non isentropic equation of state are also obtained

lossy chaotic electromagnetic reverberation chambers universal statistical behavior of the vectorial field the effective hamiltonian formalism is extended to vectorial electromagnetic waves in order to describe statistical properties of the field in reverberation chambers the latter are commonly used in electromagnetic compatibility tests as a first step the distribution of wave intensities in chaotic systems with varying opening in the weak coupling limit for scalar quantum waves is derived by means of random matrix theory in this limit the only parameters are the modal overlap and the number of open channels using the extended effective hamiltonian we describe the intensity statistics of the vectorial electromagnetic eigenmodes of lossy reverberation chambers finally the typical quantity of interest in such chambers namely the distribution of the electromagnetic response is discussed by determining the distribution of the phase rigidity describing the coupling to the environment using random matrix numerical data we find good agreement between the theoretical prediction and numerical calculations of the response

on rotating star solutions to the non isentropic euler poisson equations abstract this paper investigates rotating star solutions to the euler poisson equations with a non isentropic equation of state as a first step the equation for gas density with a prescribed entropy and angular velocity distribution is studied the resulting elliptic equation is solved either by the method of sub and supersolutions or by a variational method depending on the value of the adiabatic index the reverse problem of determining angular velocity from gas density is also considered

projectivity of the witt vector affine grassmannian we prove that the witt vector affine grassmannian which parametrizes w k lattices in w k frac 1 p n for a perfect field k of characteristic p is representable by an ind perfect scheme over k this improves on previous results of zhu by constructing a natural ample line bundle along the way we establish various foundational results on perfect schemes notably h descent results for vector bundles

global solutions to physical vacuum problem of non isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions abstract this paper is concerned with spherically symmetric motions of non isentropic viscous gaseous stars with self gravitation when the stationary entropy s x is spherically symmetric and satisfies a suitable smallness condition the existence and properties of the stationary solutions are obtained for 6 5 2 with weaker constraints upon s x compared with the one in 26 where is the adiabatic exponent the global existence of strong solutions capturing the physical vacuum singularity that the sound speed is c 1 2 holder continuous across the vacuum boundary to a simplified system for non isentropic viscous flow with self gravitation and the nonlinear asymptotic stability of the stationary solution are proved when 4 3 2 with the detailed convergence rates motivated by the results and analysis of the nonlinear asymptotic stability of lane emden solutions for isentropic flows in 29 30

local duality in algebra and topology abstract the first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts for any stable category c together with a collection of compact objects k c we construct local cohomology and local homology functors satisfying an abstract version of local duality when specialized to the derived category of a commutative ring a and a suitable ideal in a we recover the classical local duality due to grothendieck as well as generalizations by greenlees and may more generally applying our result to the derived category of quasi coherent sheaves on a quasi compact and separated scheme x implies the local duality theorem of alonso tarrio jeremias lopez and lipman as a second objective we establish local duality for quasi coherent sheaves over many algebraic stacks in particular those arising naturally in stable homotopy theory after constructing an appropriate model of the derived category in terms of comodules over a hopf algebroid we show that in familiar cases the resulting local cohomology and local homology theories coincide with functors previously studied by hovey and strickland furthermore our framework applies to global and local stable homotopy theory in a way which is compatible with the algebraic avatars of these theories in order to aid computability we provide spectral sequences relating the algebraic and topological local duality contexts

k theory and the bridge from motives to noncommutative motives abstract in this work we present a new approach to the theory of noncommutative motives and use it to explain the different flavors of algebraic k theory of schemes and dg categories the work is divided into three main parts in the first part we use the techniques of higher algebra developed in 63 to provide a universal characterization for the symmetric monoidal 1 category underlying the motivic stable a 1 homotopy theory of morel voevodsky 107 67 more precisely given a symmetric monoidal model category v together with an object x v we characterize the underlying symmetric monoidal 1 category of the symmetric monoidal model category sp v x introduced by hovey in 43 by means of a universal property amongst symmetric monoidal 1 categories this characterization trivializes the problem of finding motivic monoidal realizations in the second part we introduce a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of kontsevich 56 55 54 the key ingredient is a notion of nisnevich topology in the noncommutative setting compatible with the classical notion this compatibility together with the universal property proved in the first part ensures the existence of a canonical monoidal map from the stable motivic theory of morel voevodsky towards these new noncommutative motives that allow us to compare the two theories in the last part of this paper we explain how this bridge can be used to explain the various flavors of algebraic k theory of dg categories more precisely we prove that the non connective k theory of dg categories introduced by schlichting 82 is the non commutative nisnevich sheafification of connective algebraic k theory then we prove that its further non commutative a 1 localization is a tensor unit in our noncommutative motives as a corollary we obtain a precise proof for an original conjecture of kontsevich claiming that k theory gives the correct mapping spaces in noncommutative motives our major application is the discovery of a canonical factorization of our motivic bridge through the 1 category of modules over the commutative algebra object representing homotopy invariant algebraic k theory of schemes the results in 77 imply that this bridge is fully faithful over a field k with resolutions of singularities so that at the motivic level no information below k theory is lost by passing to the noncommutative world

cdh descent cdarc descent and milnor excision we give necessary and sufficient conditions for a cdh sheaf to satisfy milnor excision following ideas of bhatt and mathew along the way we show that the cdh infty topos of a quasi compact quasi separated scheme of finite valuative dimension is hypercomplete extending a theorem of voevodsky to nonnoetherian schemes as an application we show that if e is a motivic spectrum over a field k which is n torsion for some n invertible in k then the cohomology theory on k schemes defined by e satisfies milnor excision

existence of positive solutions for semilinear elliptic equations in general domains the main purpose of this paper is to prove some new existence theorems for positive solutions to the dirichlet problem elta u x f u x 0 x n mega 1 1 u x 0 x n artial mega 1 2

on the expanding configurations of viscous radiation gaseous stars thermodynamic model abstract in this work we study the stability of the expanding configurations of radiation gaseous stars such expanding configurations exist for a thermodynamic model given as a class of self similar solutions to the associated dynamic system with viscosity coefficients satisfying 2 3 0 for the monatomic gas that is the bulk viscosity is vanishing with respect to small perturbations this work shows that the linearly expanding homogeneous solutions are stable for a large expanding rate this is an extensive study of the result in 14 by had ic and jang

dg quotients of dg categories abstract keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with verdier s notion of quotient of a triangulated category modulo a triangulated subcategory this work is an attempt to further develop his theory

an isomorphism of motivic galois groups abstract in characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives one due to nori and the other one due to independently hanamura levine and voevodsky although these approaches are a priori quite different it is expected that ultimately they can be reduced to one another in this article we provide some evidence for this belief by proving that their associated motivic galois groups are canonically isomorphic

derived azumaya algebras and twisted k theory abstract we construct a relative version of topological k theory in the sense of 11 over an arbitrary quasi compact quasi separated c scheme x this has as input a perf x linear stable category and output a sheaf of spectra on x c the space of complex points of x we then characterize the values of this functor on inputs of the form perf x a for a a derived azumaya algebra over x in such cases we show that this coincides with the twisted topological k theory of x c for some appropriately defined twist of k theory we use this to provide a topological analogue of a classical result of quillen s on the algebraic k theory of severi brauer varieties

homotopical algebraic geometry i topos theory this is the rst of a series of papers devoted to lay the foundations of algebraic geometry in homotopical and higher categorical contexts in this rst part we investigate a notion of higher topos for this we use s categories i e simplicially enriched categories as models for certain kind of 1 categories and we develop the notions of s topologies s sites and stacks over them we prove in particular that for an s category t endowed with an s topology there exists a model category of stacks over t generalizing the model category structure on simplicial presheaves over a grothendieck site of a joyal and r jardine we also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves by proving that there exists a one to one correspodence between s topologies on an s category t and certain left exact bouseld localizations of the model category of pre stacks on t based on the above results we study the notion of model topos introduced by c rezk and we relate it to our model categories of stacks over s sites in the second part of the paper we present a parallel theory where s categories s topologies and s sites are replaced by model categories model topologies and model sites we prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over s sites is provided by the simplicial localization construction of dwyer and kan as an example of application we propose a denition of etale k theory of ring spectra extending the etale k theory of commutative rings