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

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

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

symbolic dynamics for the renormalization map of a quasiperiodic schrodinger equation a rigorous analysis is given of the dynamics of the renormalization map associated to a discrete schrodinger operatorh onl 2 defined byh n n 1 n 1 vf n n wherev is a real parameter f is a certain discontinuous period 1 function and sigma left 1 sqrt 5 right mathord left vphantom left 1 sqrt 5 right 2 right kern nulldelimiterspace 2 is the golden mean the renormalization map forh is a diffeomorphism t of 3 preserving a cubic surfaces v forv 8 we prove that the non wandering set of the restriction oft tos v is a hyperbolic set on whicht is conjugate to a subshift on six symbols it follows from results in dynamical systems theory that the optimally approximating periodic operators toh have spectra which obey a global scaling law we also define a set which we call the pseudospectrum of the operatorh we prove it to be a cantor set of measure zero and obtain bounds on its hausdorff dimension it is an open question whether the pseudospectrum coincides with the spectrum ofh

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

intermittency in random maps abstract the escape probability for a random walk on a one dimensional lattice is discussed in terms of random maps the global dynamics is found to be intermittent the laminar regions with long sojourn near the upper fixed point are broken by irregular bursts intermittency emerges even if the sinai condition is not satisfied it is shown that if p is the probability of choosing the upper map the average laminar length diverges as 1 p

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

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

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

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

exact wave functions of an electron on a quasiperiodic lattice definition of an infinite dimensional riemann theta function a scheme for obtaining the exact wave functions of an electron on a quasiperiodic lattice is presented it is shown that the trace map plays a very important role for construction of the infinite dimensional riemann theta function in terms of which the wave functions can be represented

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

first passage probability of a random walk on a disordered one dimensional lattice a method is presented which yields the exact solution of the generating function for the first passage probability and related observables for a one dimensional random walk with random hopping rates on each site a geometrical and statistical mechanical interpretation of the problem is suggested and is used for obtaining the asymptotic properties of the occupation and first passage probabilities the class of values with the property of self averaging is found

energy spectra for one dimensional quasiperiodic potentials bandwidth scaling mapping and relation with local isomorphism energy spectra for one dimensional tight binding models with two types of quasiperiodic potentials are studied for which the incommensurability is characterised by quadratic irrationals one is the step potential model for which the structure is a generalised fibonacci chain for special structures scaling properties of the spectrum are found numerically a critical index delta for the total bandwidth is determined after deriving recursion relations for the trace of transfer matrices it is shown that generalised fibonacci chains have the same energy spectrum if and only if they are locally isomorphic the other potential is sinusoidal for which the critical index delta is determined at the critical point

sinai disorder intermittency for random maps abstract the properties of the first passage probability for a random walk with sinai disorder are shown to be similar to those observed for deterministic intermittent chaos

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

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

endofiniteness in stable homotopy theory we study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects our results apply in particular to the stable homotopy category this leads for example to a new interpretation of stable splittings for classifying spaces of finite groups

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

the unstable slice filtration the main goal of this paper is to construct an analogue of vo evodsky s slice filtration in the motivic unstable homotopy category the construction is done via birational invariants this is motivated by the exis tence of an equivalence of categories between the orthogonal components for voevodsky s slice filtration and the birational motivic stable homotopy cat egories constructed in 9 another advantage of this approach is that the slices appear naturally as homotopy fibres and not as in the stable setting where they are defined as homotopy cofibres which behave much better in the unstable setting

random walk in a one dimensional random medium abstract the random walk of a particle in a one dimensional random medium is examined by means of the equivalent transfer rates technique in the discrete as well as in the continuous version of the model the probability distributions of the energy dependent equivalent transfer rates are found analytically either by a matching procedure in the discrete case or exactly in the continuous model both discrete and continuous models are shown to belong to the same universality class the average probability of presence of the particle at its initial point is then computed for a non zero global bias it decreases at large times according to a negative power law with an exponent depending on disorder and bias when there is no global bias sinai s model the decay at large times follows a logarithmic law

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

exact results and self averaging properties for random random walks on a one dimensional infinite lattice we present new exact results for a one dimensional asymmetric disordered hopping model the lattice is taken infinite from the start and we do not resort to the periodization scheme used by derrida an explicit resummation allows for the calculation of the velocityv and the diffusion constantd which are found to coincide with those given by derrida and for demonstrating thatv is indeed a self averaging quantity the same property is established ford in the limiting case of a directed walk