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

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

the ziegler spectrum for derived discrete algebras let be a derived discrete algebra we show that the krull gabriel dimension of the homotopy category of projective modules and therefore the cantor bendixson rank of its ziegler spectrum is 2 thus extending a result of bobinski and krause 8 we also describe all the indecomposable pure injective complexes and hence the ziegler spectrum for derived discrete algebras extending a result of z han 17 using this we are able to prove that all indecomposable complexes in the homotopy category of projective modules are pure injective so obtaining a class of algebras for which every indecomposable complex is pure injective but which are not derived pure semisimple

the spectrum of prime ideals in tensor triangulated categories we define the spectrum of a tensor triangulated category k as the set of so called prime ideals endowed with a suitable topology in this very generality the spectrum is the universal space in which one can define supports for objects of k this construction is functorial with respect to all tensor triangulated functors several elementary properties of schemes hold for such spaces e g the existence of generic points or some quasi compactness locally trivial morphisms are proved to be nilpotent we establish in complete generality a classification of thick ideal subcategories in terms of arbitrary unions of closed subsets with quasi compact complements thomason s theorem for schemes mutatis mutandis we also equip this spectrum with a sheaf of rings turning it into a locally ringed space we compute examples and show that our spectrum unifies the schemes of algebraic geometry and the support varieties of modular representation theory

stepanov like almost automorphic functions and monotone evolution equations abstract in this paper we are concerned with a new class of stepanov like almost automorphic s p a a functions with values in a banach space x this class contains the space aa x of all bochner almost automorphic functions we use the results obtained to prove the existence and uniqueness of a weak s p a a solution to the parabolic equation u t a t u f t in a reflexive banach space assuming some appropriate conditions of monotonicity coercitivity of the operators a t and s p almost automorphy of the forced term f t this result extends a known result in the case of almost periodicity an application is also given

stepanov like pseudo almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay abstract in this paper we prove the invariance of stepanov like pseudo almost periodic functions under bounded linear operators furthermore we obtain existence and uniqueness theorems of pseudo almost periodic mild solutions to evolution equations u t a t u t h t and u t a t u t f t b u t t c t s u s d s f t on r assuming that a t satisfy acquistapace terreni conditions that the evolution family generated by a t has exponential dichotomy that r 0 a is almost periodic that b c t s t s are bounded linear operators that f is lipschitz with respect to the second argument uniformly in the first argument and that h f f are stepanov like pseudo almost periodic for p 1 and continuous to illustrate our abstract result a concrete example is given

on λ pure acyclic complexes in a grothendieck category abstract let a be a grothendieck category we prove that there is an infinite regular cardinal such that a has pure injective envelopes furthermore we show that any pure acyclic complex in a is a directed union of contractible subcomplexes this provides an answer to a question which is conjectured by simson in 14 conjecture 3 4

on pure derived categories abstract we investigate the properties of pure derived categories of module categories and show that pure derived categories share many nice properties of classical derived categories in particular we show that bounded pure derived categories can be realized as certain homotopy categories we introduce the pure projective resp injective dimension of complexes in pure derived categories and give some criteria for computing these dimensions in terms of the properties of pure projective resp injective resolutions and pure derived functors as a consequence we get some equivalent characterizations for the finiteness of the pure global dimension of rings finally pure projective resp injective resolutions of unbounded complexes are considered

flat complexes pure periodicity and pure acyclic complexes abstract the paper can be viewed as an addition and extension of the recent paper of emmanouil 2016 5 among others an alternative functor category approach to emmanouil s results is presented by applying an idea of neeman 2008 16 and the main results obtained there we prove that a chain complex f in a locally finitely presented grothendieck category a is pure acyclic if and only if any chain map f p f from a complex p of pure projective objects in a to f is null homotopic as a consequence we prove that any pure periodic object in a is pure projective moreover we show that a is pure semisimple if and only if a has the pure qf property that is every pure injective object in a is pure projective

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

on certain homological invariants of groups recently the notions of sfli the supremum of the flat lengths of injective modules and silf the supremum of the injective lengths of flat modules have been studied by some authors these homological invariants are based on spli and silp invariants of gedrich and gruenberg and it is shown that they have enough potential to play an important role in studying homological conjectures in cohomology of groups in this paper we will study these invariants it turns out that for any group the finiteness of silf implies the finiteness of sfli but the converse is not known we investigate the situation in which sfli implies silf the statement holds for example for groups with the property that flat modules have finite projective dimension moreover we show that the gorenstein flat dimension of the trivial z module z that will be called gorenstein homological dimension of denoted ghd is completely related to these invariants

a shifted complex global lanczos method and the quasi minimal residual variant for the stein conjugate matrix equation x ax b c abstract in this paper based on a complex global lanczos process a shifted complex global lanczos method is proposed to solve the stein conjugate matrix equation x a x b c the new method adopts the shifted technique and is implemented by the original coefficient matrices the direct version of the proposed method is derived by exploiting the special structure of the l u factorization moreover the quasi minimal residual variant is developed to stabilize the convergence behavior finally some numerical examples are given to illustrate the effectiveness with comparison to some existing methods

pseudo almost periodic solutions to some differential equations abstract this paper is concerned with the pseudo almost periodic solutions to the abstract differential equations of the form d d s w s aw s bw s f s where a b are densely defined closed linear operators acting in a hilbert space h and f r h is a h valued pseudo almost periodic function using the theory of the invariant subspaces for unbounded linear operators we show that every bounded solution to is a pseudo almost periodic function

composition of pseudo almost periodic functions and semilinear differential equations under the assumption of uniformly continuity we establish a composition theorem of pseudo almost periodic functions in general banach spaces as an application we give some existence theorems of pseudo almost periodic solutions and mild pseudo almost periodic solutions for some differential equations

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

homotopy equivalences induced by balanced pairs abstract we introduce the notion of balanced pair of additive subcategories in an abelian category we give sufficient conditions under which a balanced pair of subcategories gives rise to a triangle equivalence between two homotopy categories of complexes as an application we prove that for a left gorenstein ring there exists a triangle equivalence between the homotopy category of its gorenstein projective modules and the homotopy category of its gorenstein injective modules which restricts to a triangle equivalence between the homotopy category of projective modules and the homotopy category of injective modules in the case of commutative gorenstein rings we prove that up to a natural isomorphism our equivalence extends iyengar krause s equivalence

existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional differential equations and applications the paper considers the existence and uniqueness of pseudo almost periodic mild solutions to some classes of first order partial neutral functional differential equations upon making some suitable assumptions existence and uniqueness results are obtained applications include both a partial integro differential equation arising in control systems and the scalar reaction diffusion equation with delay

cohomology theories based on flats abstract let a be an associative ring with identity k flat a the homotopy category of flat modules and k p flat a the full subcategory of pure complexes the quotient category k flat a k p flat a called here the pure derived category of flats was introduced by neeman in this category flat resolutions are unique up to homotopy and so can be used to compute cohomology we develop theories of tate and complete cohomology in the pure derived category of flats these theories extend naturally to sheaves over semi separated noetherian schemes where there are not always enough projectives but we do have enough flats as applications we characterize rings with finite sfli and schemes which are locally gorenstein

the homotopy category of complexes of projective modules abstract the homotopy category of complexes of projective left modules over any reasonably nice ring is proved to be a compactly generated triangulated category and a duality is given between its subcategory of compact objects and the finite derived category of right modules

transfinite adams representability abstract we consider the following problems in a well generated triangulated category t let be a regular cardinal and t t the full subcategory of compact objects is every functor h t op ab that preserves products of h t x t for some x in t is every natural transformation t x t t y t of the form t f t for some f x y in t if the answer to both questions is positive we say that t satisfies adams representability a classical result going back to brown and adams shows that the stable homotopy category satisfies 0 adams representability the case 0 is well understood thanks to the work of christensen keller and neeman in this paper we develop an obstruction theory to decide whether t satisfies adams representability we derive necessary and sufficient conditions of homological nature and we compute several examples in particular we show that there are rings satisfying adams representability for all 0 and rings which do not satisfy adams representability for any 0 moreover we exhibit rings for which the answer to both questions is no for all 2

partial silting objects and smashing subcategories we study smashing subcategories of a triangulated category with coproducts via silting theory our main result states that for derived categories of dg modules over a non positive differential graded ring every compactly generated localising subcategory is generated by a partial silting object in particular every such smashing subcategory admits a silting t structure

homotopy category of projective complexes and complexes of gorenstein projective modules abstract let r be a ring with identity and c r denote the category of complexes of r modules in this paper we study the homotopy categories arising from projective resp injective complexes as well as gorenstein projective resp gorenstein injective modules we show that the homotopy category of projective complexes over r denoted k prj c r is compactly generated provided k prj r is so based on this result it will be proved that the class of gorenstein projective complexes is precovering whenever r is a commutative noetherian ring of finite krull dimension furthermore it turns out that over such rings the inclusion functor k gprj r k r has a right adjoint where k gprj r is the homotopy category of gorenstein projective r modules similar or rather dual results for the injective resp gorenstein injective complexes will be provided if r has a dualising complex a triangle equivalence between homotopy categories of projective and of injective complexes will be provided as an application we obtain an equivalence between the triangulated categories k gprj r and k ginj r that restricts to an equivalence between k prj r and k inj r whenever r is commutative noetherian and admits a dualising complex

stepanov like almost automorphic solutions of nonautonomous semilinear evolution equations with delay abstract we consider the existence and uniqueness of a stepanov like almost automorphic solution to the nonautonomous semilinear evolution equations with a constant delay u t a t u t f t u t h t r where a t t r generates an exponentially stable evolution family u t s and f r x x satisfies a lipschitz condition with respect to the second argument

on pure acyclic complexes abstract in this paper we study the pure acyclic complexes of modules we obtain several characterizations of these complexes extending results that are known for the pure acyclic complexes of flat modules in particular we extend neeman s characterization 18 of the pure acyclic complexes of flat modules in terms of complexes of projective modules by considering more generally complexes of pure projective modules as a consequence we obtain simson s result 22 on the pure projectivity of pure periodic modules

on purely generated α smashing weight structures and weight exact localizations abstract this paper is dedicated to new methods of constructing weight structures and weight exact localizations our arguments generalize their bounded versions considered in previous papers of the authors we start from a class of objects p of a triangulated category c that satisfies a certain countable negativity condition there are no c extensions of positive degrees between elements of p we actually need a somewhat stronger condition of this sort to obtain a weight structure both halves of which are closed either with respect to c coproducts of less than objects where is a fixed regular cardinal or with respect to all coproducts provided that c is closed with respect to coproducts of this sort this construction gives all reasonable weight structures satisfying the latter conditions in particular one can obtain certain weight structures on spectra in sh consisting of less than cells and on certain localizations of sh these results are new

exact model structures and recollements abstract we show how to obtain recollements of triangulated categories using the theory of exact model structures from 13 after noting how the theory relates to well known notions in the simplest case of frobenius categories we apply these ideas to recollements of homotopy categories of chain complexes in short we give model categorical explanations for the classical verdier localization recollement as well as several recollement situations due to neeman and murfet

on the relation between k flatness and k projectivity abstract krause 19 has proved that the homotopy category k r pureproj of pure projective modules over an associative ring r is compactly generated and equivalent to the verdier quotient k r t r where t r is the triangulated subcategory of the homotopy category k r consisting of the pure acyclic complexes we provide an alternative proof of this result when restricted to the homotopy category of projective modules this equivalence reduces to that obtained by neeman 26 as an application of this description of k r pureproj we show that the k flat complexes defined by spaltenstein 29 are up to homotopy equivalence the filtered colimits of k projective complexes of pure projective modules

some adjoints in homotopy categories let r be a ring in a previous paper 11 we found a new description for the category k r proj it is equivalent to the verdier quotient k r flat s for some suitable s c k r flat in this article we show that the quotient map from k r flat to k r flat s always has a right adjoint this gives a new fully faithful embedding of k r proj into k r flat its virtue is that it generalizes to nonaffine schemes

approximations and adjoints in homotopy categories we provide a criterion for the existence of right approximations in cocomplete additive categories it is a straightforward generalisation of a result due to el bashir this criterion is used to construct adjoint functors in homotopy categories applications include the study of pure derived categories for instance it is shown that the pure derived category of any module category is compactly generated

flat covers of complexes abstract in this article we define and study flat complexes over any ring also we prove that any complex over a commutative noetherian ring with finite krull dimension has a flat cover and a dg flat cover

minimal semi flat cotorsion replacements and cosupport abstract over a commutative noetherian ring r of finite krull dimension we show that every complex of flat cotorsion r modules decomposes as a direct sum of a minimal complex and a contractible complex moreover we define the notion of a semi flat cotorsion complex as a special type of semi flat complex and provide functorial ways to construct a quasi isomorphism from a semi flat complex to a semi flat cotorsion complex consequently every r complex can be replaced by a minimal semi flat cotorsion complex in the derived category over r furthermore we describe structure of semi flat cotorsion replacements by which we recover classic theorems for finitistic dimensions in addition we improve some results on cosupport and give a cautionary example we also explain that semi flat cotorsion replacements always exist and can be used to describe the derived category over any associative ring

on the homotopy categories of projective and injective representations of quivers abstract let r be a ring and q be a quiver we study the homotopy categories k prj q and k inj q consisting respectively of projective and injective representations of q by r modules we show that for certain quivers these triangulated categories are compactly generated and provide explicit descriptions of compact generating sets moreover in case r is commutative and noetherian with a dualizing complex d the dualizing functor d r k prj r k inj r is extended to a triangulated functor k prj q k inj op q which is an equivalence of triangulated categories this functor establishes an equivalence on k prj q and k inj q whenever q is finite

totally acyclic complexes abstract it is known that over an iwanaga gorenstein ring the gorenstein injective gorenstein projective gorenstein flat modules are simply the cycles of acyclic complexes of injective projective flat modules we consider the question are these characterizations only working over iwanaga gorenstein rings we prove that if r is a commutative noetherian ring of finite krull dimension then the following are equivalent 1 r is an iwanaga gorenstein ring 2 every acyclic complex of injective modules is totally acyclic 3 the cycles of every acyclic complex of gorenstein injective modules are gorenstein injective 4 every acyclic complex of projective modules is totally acyclic 5 the cycles of every acyclic complex of gorenstein projective modules are gorenstein projective 6 every acyclic complex of flat modules is f totally acyclic 7 the cycles of every acyclic complex of gorenstein flat modules are gorenstein flat thus we improve slightly on a result of iyengar and krause in 22 they proved that for a commutative noetherian ring r with a dualizing complex the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if r is gorenstein we replace the dualizing complex hypothesis by the finiteness of the krull dimension and add more equivalent conditions in the second part of the paper we focus on the noncommutative case we prove that for a two sided noetherian ring r of finite finitistic flat dimension that satisfies the auslander condition the following are equivalent 1 every complex of injective left and respectively right r modules is totally acyclic 2 r is iwanaga gorenstein

existence and uniqueness of pseudo almost periodic solutions to some classes of partial evolution equations abstract some sufficient conditions for the existence and uniqueness of pseudo almost periodic mild solutions to some classes of partial evolution equations are given we then make use of our abstract results to discuss the existence of pseudo almost periodic solutions to some partial differential equations

the homotopy category of flat modules and grothendieck duality let r be a ring we prove that the homotopy category k r proj is always aleph 1 compactly generated and depending on the ring r it may or may not be compactly generated we use this to give a description of k r proj as a quotient of k r flat the remarkable fact is that this new description of k r proj generalizes to non affine schemes this will appear in murfet s thesis

existence and uniqueness of pseudo almost periodic solutions to some classes of semilinear differential equations and applications this paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to the class of semilinear differential equations of the form u t au t f t bu t where a is the infinitesimal generator of an analytic semigroup acting on a complex banach space x b d b x x is a densely defined closed linear operator and f r x x is a jointly continuous function under some suitable assumptions on a b and f the existence and uniqueness of a pseudoalmost periodic solution to is obtained

presheaves of triangulated categories and reconstruction of schemes to any triangulated category with tensor product k otimes we associate a topological space rm spc k otimes by means of thick subcategories of k a la hopkins neeman thomason moreover to each open subset u of this space spc k otimes we associate a triangulated category cal k u producing what could be thought of as a presheaf of triangulated categories applying this to the derived category k otimes rm d rm perf x otimes l of perfect complexes on a noetherian scheme x the topological space rm spc k otimes turns out to be the underlying topological space of x moreover for each open u subset x the category cal k u is naturally equivalent to rm d rm perf u as an application we give a method to reconstruct any reduced noetherian scheme x from its derived category of perfect complexes rm d rm perf x considering the latter as a tensor triangulated category with otimes l

on exact categories and applications to triangulated adjoints and model structures abstract we show that quillen s small object argument works for exact categories under very mild conditions this has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures in particular the interplay of different exact structures on the category of complexes of quasi coherent sheaves leads to a streamlined and generalized version of recent results obtained by estrada gillespie guil asensio hovey jorgensen neeman murfet prest trlifaj and possibly others

koszul duality for lie algebroids abstract this paper studies the role of dg lie algebroids in derived deformation theory more precisely we provide an equivalence between the homotopy theories of formal moduli problems and dg lie algebroids over a commutative dg algebra of characteristic zero at the level of linear objects we show that the category of representations of a dg lie algebroid is an extension of the category of quasi coherent sheaves on the corresponding formal moduli problem we describe this extension geometrically in terms of pro coherent sheaves