change of rings and singularity categories abstract we investigate the behavior of singularity categories and stable categories of gorenstein projective modules along a morphism of rings the natural context to approach the problem is via change of rings that is the classical adjoint triple between the module categories in particular we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of gorenstein projective modules moreover we study this problem at the level of big singularity categories in the sense of krause 30 along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories this is achieved by introducing the notion of 0 cocompact objects in triangulated categories and proving a dual version of bousfield s localization lemma we provide applications and examples illustrating our main results

smooth and proper noncommutative schemes and gluing of dg categories abstract in this paper we discuss different properties of noncommutative schemes over a field we define a noncommutative scheme as a differential graded category of a special type we study regularity smoothness and properness for noncommutative schemes admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes in this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules as a consequence of the main theorem we obtain that for any finite dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme moreover if the algebra has finite global dimension then the full subcategory is admissible we also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance as another application of the main theorem we obtain in characteristic 0 an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

stable homotopy as a triangulated functor let j i be the homotopy category of all finite spectra the homotopy category of all spectra and y 89 be the homotopy category of all prime to 2 spectra let h y abelian groups be a homological functor i e a generalized homology theory a natural question is whether h can be lifted to a triangulated functor is there a functor t y d abelian groups so that composed with the usual homology functor on d abelian groups is h here d abelian groups d z is the derived category of z modules if h is ordinary homology the answer is clearly yes after all the ordinary homology of a space is given by the singular chain complex almost by definition one has a lifting by the brown representability theorem the general statement follows from the special case h h the stable homotopy functor let us be a little more precise

morita theory and singularity categories abstract we propose an analogue of the bounded derived category for an augmented ring spectrum defined in terms of a notion of noether normalization in many cases we show this category is independent of the chosen normalization based on this we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are koszul dual in the style of the bgg correspondence examples of interest include koszul algebras and ginzburg dg algebras c b g for finite groups or for compact lie groups with orientable adjoint representation cochains in rational homotopy theory and various examples from chromatic homotopy theory

locally contractive iterated function systems an iterated function system on mathscr x subset mathbb r d is defined by successively applying an i i d sequence of random lipschitz functions from to mathscr x to mathscr x this paper shows how f n f 1 circ ldots circ f n may converge even in the absence of the strong contraction conditions for instance lipschitz constant smaller than 1 on average which earlier work has required instead it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions applications to queues to self modifying random walks and to random logistic maps are given

variational inequalities with generalized monotone operators we investigate the variational inequality with pseudomonotone operators in the sense of karamardian in banach spaces new existence results which extend many known results in infinite dimensional spaces are derived under rather weak assumptions new uniqueness results which also seem to be new even in finite dimensional spaces are also derived in particular new existence and uniqueness results for the complementarity problem with pseudomonotone operators in banach spaces are obtained also existence and uniqueness results for minimization problems with pseudoconvex functions in banach spaces are obtained

the dual of brown representability for homotopy categories of complexes abstract we call product generator of an additive category a fixed object satisfying the property that every other object is a direct factor of a product of copies of it in this paper we start with an additive category with products and images e g a module category and we are concerned with the homotopy category of complexes with entries in that additive category we prove that the brown representability theorem is valid for the dual of the homotopy category if and only if the initial additive category has a product generator

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

approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions the purpose of this paper is to investigate the convergence of general approximate proximal algorithm resp general bregman function based approximate proximal algorithm for solving the generalized variational inequality problem for short gvi t w where t is a multifunction the general approximate proximal algorithm resp general bregman function based approximate proximal algorithm is to define new approximating subproblems on the domains w email protected email protected n 1 2 which form a general approximate proximate point scheme resp a general bregman function based approximate proximate point scheme for solving gvi t w it is shown that if t is either relaxed a pseudomonotone or pseudomonotone then the general approximate proximal point scheme resp general bregman function based approximate proximal point scheme generates a sequence which converges weakly to a solution of gvi t w under quite mild conditions

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

totally acyclic complexes over noetherian schemes abstract we define a notion of total acyclicity for complexes of flat quasi coherent sheaves on a semi separated noetherian scheme and study these complexes using the pure derived category of flat quasi coherent sheaves we prove that a scheme is gorenstein if and only if every acyclic complex of flat quasi coherent sheaves is totally acyclic our formalism also removes the need for a dualising complex in several known results for rings including jorgensen s proof of the existence of gorenstein projective precovers

the twisted inverse image pseudofunctor over commutative dg rings and perfect base change abstract let k be a gorenstein noetherian ring of finite krull dimension and consider the category of cohomologically noetherian commutative differential graded rings a over k such that h 0 a is essentially of finite type over k and a has finite flat dimension over k we extend grothendieck s twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup we prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps and prove a perfect base change result for f in this setting as application we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings our results generalize and solve some recent conjectures of yekutieli

homological dimensions and regular rings abstract a question of avramov and foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings a key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules analogous results for flat dimension and projective dimension are also established

rigid complexes via dg algebras let a be a commutative ring b a commutative a algebra and m a complex of b modules we begin by constructing the square sq b a m which is also a complex of b modules the squaring operation is a quadratic functor and its construction requires differential graded dg algebras if there exists an isomorphism p m sq b a m then the pair m p is called a rigid complex over b relative to a there are some finiteness conditions there is an obvious notion of rigid morphism between rigid complexes we establish several properties of rigid complexes including their uniqueness existence under some extra hypothesis and formation of pullbacks f b m p resp f m p along a finite resp essentially smooth ring homomorphism f b c in the subsequent paper rigid dualizing complexes over commutative rings we consider rigid dualizing complexes over commutative rings building on the results of the present paper the project culminates in our paper rigid dualizing complexes and perverse sheaves on schemes where we give a comprehensive version of grothendieck duality for schemes the idea of rigid complexes originates in noncommutative algebraic geometry and is due to van den bergh 1997

brown representability for the dual let t be the homotopy category of all spectra brown proved that a homological functor h t o p ab is representable if it takes coproducts to products that is the functors h may be characterised as the homological functors taking coproducts to products in this article we will prove the dual a covariant functor h t ab which takes products to products is representable it is of the form h

approximate proximal algorithms for generalized variational inequalities with paramonotonicity and pseudomonotonicity we propose an approximate proximal algorithm for solving generalized variational inequalities in hilbert space extension to bregman function based approximate proximal algorithm is also discussed weak convergence of these two algorithms are established under the paramonotonicity and pseudomonotonicity assumptions of the operators

perverse t structure on artin stacks in this paper we develop the theory of perverse sheaves on artin stacks continuing our earlier study of lisse etale sheaves on stacks in laszlo and olsson the six operations for sheaves on artin stacks i finite coefficients publ math ihes 2008 the six operations for sheaves on artin stacks ii adic coefficients publ math ihes 2008

flops and equivalences of derived categories for threefolds with only terminal gorenstein singularities the main purpose of this paper is to show that bridgeland s moduli space of perverse point sheaves for certain flopping contractions gives the flops and the fourier mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories

on the derived category of a graded commutative noetherian ring abstract for any graded commutative noetherian ring where the grading group is abelian and where commutativity is allowed to hold in a quite general sense we establish an inclusion preserving bijection between on the one hand the twist closed localizing subcategories of the derived category and on the other hand subsets of the homogeneous spectrum of prime ideals of the ring we provide an application to weighted projective schemes

monoidal infinity category of complexes from tannakian viewpoint in this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infty categories of quasi coherent complexes it can be viewed as a derived analogue of tannaka duality as a consequence we deduce that an algebraic stack satisfying a certain condition can be recovered from the symmetric monoidal stable infty category of quasi coherent complexes with tensor operation

hybrid shrinking projection method for a generalized equilibrium problem a maximal monotone operator and a countable family of relatively nonexpansive mappings the purpose of this paper is to introduce and consider a hybrid shrinking projection method for finding a common element of the set ep of solutions of a generalized equilibrium problem the set n 0 f s n of common fixed points of a countable family of relatively nonexpansive mappings s n n 0 and the set t 10 of zeros of a maximal monotone operator t in a uniformly smooth and uniformly convex banach space it is proven that under appropriate conditions the sequence generated by the hybrid shrinking projection method converges strongly to some point in ep t 10 n 0 f s n this new result represents the improvement complement and development of the previously known ones in the literature

zeno s walk a random walk with refinements a self modifying random walk on bbb q is derived from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed this process converges almost surely to a random variable which is totally singular with respect to lebesgue measure and which is supported on a subset of bbb r having hausdorff dimension less than 1 which we calculate by a theorem of billingsley by generating function techniques we then calculate the exponential rate of convergence of the process to its limit point which may be taken as a bound for the convergence of the measure in the wasserstein metric we describe how the process may viewed as a random walk on the space of monotone piecewise linear functions where moves are taken by successive compositions with a randomly chosen such function

variational and generalized variational inequalities with discontinuous mappings abstract in this paper we derive several existence results for the variational and the generalized variational inequality problems where the mappings under consideration need not have any continuity properties uniqueness of the solution of the generalized variational inequality problem is also investigated as a by product we obtain several existence and uniqueness results for the generalized complementarity problem applications to minimization problems with generalized convex functions are considered

gorenstein homological dimensions for triangulated categories abstract let c be a triangulated category with a proper class e of triangles asadollahi and salarian introduced and studied e g projective and e g injective objects and developed a relative homological algebra in c in this paper we further study gorenstein homological dimensions for triangulated categories first we discuss the finiteness of e g projective and e g injective dimensions then we define e gorenstein derived functors ge xt e i and characterize e g projective and e g injective dimensions of objects in c by vanishing of such functors consequently we can prove the equality sup e g pd m for any m c sup e g id m for any m c which is used to define the global e gorenstein dimension of c finally e g phantom tower and e g cellular tower for objects in c with finite e g projective dimension are constructed

an outer approximation method for the variational inequality problem we study two outer approximation schemes applied to the variational inequality problem in reflexive banach spaces first we propose a generic outer approximation scheme and its convergence analysis unifies a wide class of outer approximation methods applied to the constrained optimization problem as is standard in this setting boundedness and optimality of weak limit points are proved to hold under two alternative conditions i boundedness of the feasible set or ii coerciveness of the operator to develop a convergence analysis where i and ii do not hold we consider a second scheme in which the approximated subproblems use a coercive approximation of the original operator under conditions alternative to both i and ii we obtain standard convergence results furthermore when the space is uniformly convex we establish full strong convergence of the second scheme to a solution

the popescu gabriel theorem for triangulated categories abstract the popescu gabriel theorem states that each grothendieck abelian category is a localization of a module category in this paper we prove an analogue where grothendieck abelian categories are replaced by triangulated categories which are well generated in the sense of neeman and algebraic in the sense of keller the role of module categories is played by derived categories of small differential graded categories an analogous result for topological triangulated categories has recently been obtained by a heider

generic vanishing in characteristic p 0 and the characterization of ordinary abelian varieties we prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of kawamata s theorems a characterization of smooth varieties birational to ordinary abelian varieties and the surjectivity of the albanese map when the frobenius stable kodaira dimension is zero

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

phase transition in reinforced random walk and rwre on trees a random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future using exchangeability theory the process is shown to be equivalent to a random walk in a random environment rwre that is to say a mixture of markov chains criteria are given to determine whether a rwre is transient or recurrent these criteria apply to show that the reinforced random walk can vary from transient to recurrent depending on the value of an adjustable parameter measuring the strength of the feedback the value of the parameter at the phase transition is calculated

on deformations of triangulated models abstract this paper is the first part of a project aimed at understanding deformations of triangulated categories and more precisely their dg and a models and applying the resulting theory to the models occurring in the homological mirror symmetry setup in this first paper we focus on models of derived and related categories based upon the classical construction of twisted objects over a dg or a algebra for a hochschild 2 cocycle on such a model we describe a corresponding curvature compensating deformation which can be entirely understood within the framework of twisted objects we unravel the construction in the specific cases of derived a and abelian categories homotopy categories and categories of graded free qdg modules we identify a purity condition on our models which ensures that the structure of the model is preserved under deformation this condition is typically fulfilled for homotopy categories but not for unbounded derived categories