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

purity in compactly generated derivators and t structures with grothendieck hearts we study t structures with grothendieck hearts on compactly generated triangulated categories mathcal t that are underlying categories of strong and stable derivators this setting includes all algebraic compactly generated triangulated categories we give an intrinsic characterisation of pure triangles and the definable subcategories of mathcal t in terms of directed homotopy colimits for a left nondegenerate t structure mathbf t mathcal u mathcal v on mathcal t we show that mathcal v is definable if and only if mathbf t is smashing and has a grothendieck heart moreover these conditions are equivalent to mathbf t being homotopically smashing and to mathbf t being cogenerated by a pure injective partial cosilting object finally we show that finiteness conditions on the heart of mathbf t are determined by purity conditions on the associated partial cosilting object

construction of t structures and equivalences of derived categories we associate a t structure to a family of objects in d a the derived category of a grothendieck category a using general results on t structures we give a new proof of rickard s theorem on equivalence of bounded derived categories of modules also we extend this result to bounded derived categories of quasi coherent sheaves on separated divisorial schemes obtaining in particular bei linson s equivalences

towards the homotopy of the k 2 local moore spectrum at p 2 abstract let v 0 be the mod 2 moore spectrum and let c be the supersingular elliptic curve over f 4 defined by the weierstrass equation y 2 y x 3 let f c be its formal group law and e c be the spectrum classifying the deformations of f c the group of automorphisms of f c which we denote by s c acts on e c further s c admits a surjective homomorphism to z 2 whose kernel we denote by s c 1 the cohomology of s c 1 with coefficients in e c v 0 is the e 2 term of a spectral sequence converging to the homotopy groups of e c h s c 1 v 0 a spectrum closely related to l k 2 v 0 in this paper we use the algebraic duality resolution spectral sequence to compute an associated graded for h s c 1 e c v 0 these computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2

the flat model structure on given a cotorsion pair a b in an abelian category c with enough a objects and enough b objects we define two cotorsion pairs in the category ch c of unbounded chain complexes we see that these two cotorsion pairs are related in a nice way when a b is hereditary we then show that both of these induced cotorsion pairs are complete when a b is the flat cotorsion pair of r modules this proves the flat cover conjecture for possibly unbounded chain complexes and also gives us a new flat model category structure on ch r in the last section we use the theory of model categories to show that we can define ext n r m n using a flat resolution of m and a cotorsion coresolution of n

t structures and cotilting modules over commutative noetherian rings for a commutative noetherian ring r we establish a bijection between the resolving subcategories consisting of finitely generated r modules of finite projective dimension and the compactly generated t structures in the unbounded derived category mathcal d r that contain r 1 in their heart under this bijection the t structures mathcal u mathcal v such that the aisle mathcal u consists of objects with homology concentrated in degrees n correspond to the n cotilting classes in mathrm mod text r as a consequence of these results we prove that the little finitistic dimension findim r of r equals an integer n if and only if the direct sum bigoplus k 0 n e k r of the first n 1 terms in a minimal injective coresolution 0 rightarrow r rightarrow e 0 r rightarrow e 1 r rightarrow cdots of r is an injective cogenerator of mathrm mod text r

t structures for hereditary categories we study aisles equivalently t structures in the derived category of a hereditary abelian category given an aisle we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle we then obtain a sequence called a narrow sequence we then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories together with a tilting torsion class in each of these wide subcategories studying the extra conditions that the narrow sequences coming from aisles must satisfy we get a bijection between coreflective narrow seqeunces and t structures in the derived category in some cases including the case of finite dimensional modules over a finite dimensional hereditary algebra we refine our results and show that the t structures are determined by an increasing sequence of coreflective wide subcategories together with a tilting torsion class in the orthogonal of one wide subcategory in the next effectively decoupling the different tilting torsion theories one chooses in the wide subcategories these results are sufficient to recover known classifications of t structures for smooth projective curves and for finitely generated modules over a dedekind ring

models for singularity categories abstract in this article we construct various models for singularity categories of modules over differential graded rings the main technique is the connection between abelian model structures cotorsion pairs and deconstructible classes and our constructions are based on more general results about localization and transfer of abelian model structures we indicate how recollements of triangulated categories can be obtained model categorically discussing in detail krause s recollement k ac inj r k inj r d r in the special case of curved mixed z graded complexes we show that one of our singular models is quillen equivalent to positselski s contraderived model for the homotopy category of matrix factorizations

enriched two dimensional mixed finite element models for linear elasticity with weak stress symmetry abstract the purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry these approximations are based on the application of enriched versions of classic poisson compatible spaces for stress and displacement variables and or on enriched stokes compatible space configurations for the choice of rotation spaces used to weakly enforce stress symmetry accordingly the stress space has to be adapted to ensure stability such enrichment procedures are done via space increments with extra bubble functions which have their support on a single element in the case of h 1 conforming approximations or with vanishing normal components over element edges in the case of h div conforming spaces the advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed in a way that the number of equations to be solved and the matrix structure are not affected enhanced approximations are observed when using the resulting enriched space configurations which may have different orders of accuracy for the different variables a general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables individually the use of enriched poisson spaces improves the rates of convergence of stress divergence and displacement variables stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order reaching the maximum rate given by the normal traces which are not affected

model category structures arising from drinfeld vector bundles we present a general construction of model category structures on the category ch qco x of unbounded chain complexes of quasi coherent sheaves on a semi separated scheme x the construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of x it does not require closure under direct limits as previous methods we apply it to describe the derived category mathbb d qco x via various model structures on ch qco x as particular instances we recover recent results on the flat model structure for quasi coherent sheaves our approach also includes the case of infinite dimensional vector bundles and of restricted flat mittag leffler quasi coherent sheaves as introduced by drinfeld finally we prove that the unrestricted case does not induce a model category structure as above in general

h div finite elements based on nonaffine meshes for 3d mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

the auslander reiten formula for complexes of modules an auslander reiten formula for complexes of modules is presented this formula contains as a special case the classical auslander reiten formula the auslander reiten translate of a complex is described explicitly and various applications are discussed

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

cotilting modules and homological ring epimorphisms abstract we show that every injective homological ring epimorphism f r s where s r has flat dimension at most one gives rise to a 1 cotilting r module and we give sufficient conditions under which the converse holds true specializing to the case of a valuation domain r we illustrate a bijective correspondence between equivalence classes of injective homological ring epimorphisms originating in r and cotilting classes of certain type and in turn a bijection with a class of smashing localizing subcategories of the derived category of r moreover we obtain that every cotilting class over a valuation domain is a tor orthogonal class hence it is of cocountable type even though in general cotilting classes are not of cofinite type

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence abstract the purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal potential variables one for triangles and the other one for quadrilateral elements to be used in discrete versions of the mixed finite element method for elliptic problems they can be interpreted as enriched versions of b d f m k 1 spaces based on triangles and of r t k spaces for quadrilateral elements the new flux approximations are incremented with properly chosen internal shape functions with vanishing normal components on the edges of degree k 2 and matching primal functions of degree k 1 higher than the border fluxes which are kept of degree k in all these cases the divergence of the flux space coincide with the primal approximation space on the master element producing stable simulations using static condensation the global condensed system to be solved in the enriched cases has same dimension and structure of the original ones which is proportional to the space dimension of the border fluxes for each element geometry measuring the errors with l 2 norms the enriched space configurations give higher convergence rate of order k 2 for the primal variable while keeping the order k 1 for the flux for affine meshes the divergence errors have the same improved accuracy rate as for the error in the primal variable for quadrilateral non affine meshes for instance trapezoidal elements the divergence error has order k 1 one unit more than the order k occurring for r t k spaces on this kind of deformed meshes this fact also holds for a b f k elements but for them the potential order of accuracy does not improve keeping order k 1

covers and envelopes in grothendieck categories flat covers of complexes with applications abstract in the general setting of grothendieck categories with enough projectives we prove theorems that make possible to restrict the study of the problem of the existence of f covers and envelopes to the study of some properties of the class f we then prove the existence of flat covers and cotorsion envelopes of complexes giving some examples this generalizes the earlier work j algebra201 1998 86 102 and finishes the problem of the existence of flat covers of complexes

flat covers and factorizations abstract the flat cover conjecture saying that every module has a flat pre cover has been recently proved by bican el bashir and enochs we relate flat precovers and cotorsion preenvelopes to weak factorizations and prove that flat monomorphisms form a left part of a weak factorization system

error analysis of variable degree mixed methods for elliptic problems via hybridization a new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree raviart thomas method for second order elliptic problems the approach gives error estimates for the lagrange multipliers without using error estimates for the other variables error estimates for the primal and flux variables then follow from those for the lagrange multipliers in contrast traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the lagrange multipliers the new approach not only gives new error estimates for the new variable degree raviart thomas method but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates the error analysis is achieved by using a variational characterization of the lagrange multipliers wherein the other unknowns do not appear this approach can be applied to other hybridized mixed methods as well

invariants of t structures and classification of nullity classes abstract we construct an invariant of t structures on the derived category of a commutative noetherian ring this invariant is complete when restricting to the category of complexes with finitely generated bounded homology and also gives a classification of nullity classes with the same restriction on the full derived category of z we show that the class of distinct t structures do not form a set

hierarchical high order finite element bases for h div spaces based on curved meshes for two dimensional regions or manifolds the mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential primal variable and of the flux field dual variable this work focuses on new h div conforming finite element spaces which are suitable for flux approximations based on curved meshes of a planar region or a manifold domain embedded in r 3 the adopted methodology for the construction of h div bases consists in using hierarchical h 1 conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure they are classified as being of edge or internal type the normal component of an edge function coincides on the corresponding edge with the associated scalar shape function and vanishes over the other edges and the normal components of an internal shape function vanishes on all element edges these properties are fundamental for the global assembly of h div conforming functions locally defined by these vectorial shape functions for applications to the mixed formulation the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular cylindrical and spherical regions demonstrating stable convergence with optimal convergence rates coinciding for primal and dual variables innovative two dimensional hdiv approximation spaces for curved quadrilateral and triangular elements hdiv bases properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure stable convergence with optimal convergence rates coinciding for primal and dual variables study of number of condensed equations as a function of the polynomial order of the shape functions

a multiscale hybrid method for darcy s problems using mixed finite element local solvers abstract multiscale hybrid mixed mhm method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions for fluid flow normal fluxes multiplier over macro element boundaries and coarse piecewise constant potential approximations in each macro element are computed upscaling then small details are resolved by local problems using fine representations inside the macro elements setting the multiplier as neumann boundary conditions downscaling in this work a variant of the method is developed denoted by mhm h div adopting mixed finite elements at the downscaling stage instead of continuous finite elements used in all previous publications of the method thus this alternative mhm method inherits improvements typical of mixed methods as better flux accuracy and local mass conservation at the micro scale level inside the macro elements which are important properties for multi phase flows in rough heterogeneous media different two scale stable space settings are considered vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries in each macro element the internal flux components with vanishing normal traces and the potential approximations may be enriched in different extents with respect to internal mesh size internal polynomial degree or both the choice being determined by the problem at hands a unified general error analysis of the mhm h div method is presented for all these two scale space scenarios both mhm versions are compared for 2d test problems with smooth solutions for convergence rates verification and for darcy s flow in heterogeneous media mhm h div 3d simulations are presented for a known singular darcy s solution using adaptive macro partitions and for an oscillatory permeability scenario

a resolution of the k 2 local sphere at the prime 3 we develop a framework for displaying the stable homotopy theory of the sphere at least after localization at the second morava k theory k 2 at the prime 3 we write the spectrum lk 2 s 0 as the inverse limit of a tower of fibrations with four layers the successive fibers are of the form e hf where f is a finite subgroup of the morava stabilizer group and e2 is the second morava or lubin tate homology theory we give explicit calculation of the homotopy groups of these fibers the case n 2a tp 3 represents the edge of our current knowledge n 1 is classical and at n 2 the prime 3 is the largest prime where the morava stabilizer group has a p torsion subgroup so that the homotopy theory is not entirely algebraic the problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics a period of calculation beginning with serre s computation of the cohomology of eilenberg maclane spaces and the advent of the adams spectral sequence culminated in the late 1970s with the work of miller ravenel and wilson on periodic phenomena in the homotopy groups of spheres and ravenel s nilpotence conjectures the solutions to most of these conjectures by devinatz hopkins and smith in the middle 1980s established the primacy of the chromatic point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory computations passed from center stage to some extent although there has been steady work in the wings most notably by shimomura and his coworkers and ravenel and more lately by hopkins and

three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy abstract there are different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal pressure variables to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations three cases shall be studied and compared for discretized three dimensional formulations based on tetrahedral hexahedral and prismatic meshes the principle guiding the constructions is the property that the divergence of the dual space and the primal approximation space should coincide while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable there is the classic case of b d m k spaces based on tetrahedral meshes and polynomials of total degree k for the dual variable and k 1 for the primal variable showing stable simulations with optimal convergence rates of orders k 1 and k respectively another case is related to r t k and b d f m k 1 spaces for hexahedral and tetrahedral meshes respectively but holding for prismatic elements as well it gives identical approximation order k 1 for both primal and dual variables an improvement in accuracy obtained by increasing the degree of primal functions to k and by enriching the dual space with some properly chosen internal shape functions of degree k 1 while keeping degree k for the border fluxes a new type of approximation is proposed by further incrementing the order of some internal flux functions to k 2 and matching primal functions to k 1 higher than the border fluxes of degree k thus higher convergence rate of order k 2 is obtained for the primal variable using static condensation the global condensed system to be solved in all the cases has same dimension and structure which is proportional to the space dimension of the border fluxes for each element geometry illustrating results comparing the three different space configurations are presented for simulations based on hierarchical high order shape functions for h div conforming spaces which are specially constructed for affine tetrahedral hexahedral and prismatic meshes expected convergence rates are obtained for the flux pressure and divergence variables

additivity for derivator k theory abstract we prove the additivity theorem for the k theory of triangulated derivators this solves one of the conjectures made by maltsiniotis in g maltsiniotis la k theorie d un derivateur triangule in alexei davydov michael batanin michael johnson stephen lack amnon neeman eds categories in algebra geometry and physics conference and workshop in honor of ross street s 60th birthday in contemp math vol 431 amer math soc 2007 pp 341 368 we also review some basic definitions and results in the theory of derivators in the sense of grothendieck

the homotopy of the k 2 local moore spectrum at the prime 3 revisited in this paper we use the approach introduced in goerss et al ann math 162 2 777 822 2005 in order to analyze the homotopy groups of l k 2 v 0 the mod 3 moore spectrum v 0 localized with respect to morava k theory k 2 these homotopy groups have already been calculated by shimomura j math soc japan 52 1 65 90 2000 the results are very complicated so that an independent verification via an alternative approach is of interest in fact we end up with a result which is more precise and also differs in some of its details from that of shimomura j math soc japan 52 1 65 90 2000 an additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the k 2 localization of the spectrum tmf of topological modular forms and related spectra even more the adams novikov differentials for l k 2 v 0 can be read off from those for tmf

derived equivalences induced by big cotilting modules abstract we prove that given a grothendieck category g with a tilting object of finite projective dimension the induced triangle equivalence sends an injective cogenerator of g to a big cotilting module moreover every big cotilting module can be constructed like that in an essentially unique way we also prove that the triangle equivalence is at the base of an equivalence of derivators which in turn is induced by a quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

cotilting sheaves on noetherian schemes we develop theory of possibly large cotilting objects of injective dimension at most one in general grothendieck categories we show that such cotilting objects are always pure injective and that they characterize the situation where the grothendieck category is tilted using a torsion pair to another grothendieck category we prove that for noetherian schemes with an ample family of line bundles a cotilting class of quasi coherent sheaves is closed under injective envelopes if and only if it is invariant under twists by line bundles and that such cotilting classes are parametrized by specialization closed subsets disjoint from the associated points of the scheme finally we compute the cotilting sheaves of the latter type explicitly for curves as products of direct images of indecomposable injective modules or completed canonical modules at stalks

compactly generated t structures on the derived category of a noetherian ring abstract we study t structures on d r the derived category of modules over a commutative noetherian ring r generated by complexes in d fg r we prove that they are exactly the compactly generated t structures on d r and describe them in terms of decreasing filtrations by supports of spec r a decreasing filtration by supports z spec r satisfies the weak cousin condition if for any integer i the set i contains all the immediate generalizations of each point in i 1 if a compactly generated t structure on d r restricts to a t structure on d fg r then the corresponding filtration satisfies the weak cousin condition if r has a pointwise dualizing complex the converse is true if the ring r has dualizing complex then these are exactly all the t structures on d fg b r

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

a classification theorem for t structures abstract we give a classification theorem for a relevant class of t structures in triangulated categories which includes in the case of the derived category of a grothendieck category a large class of t structures whose hearts have at most n fixed consecutive non zero cohomologies moreover by this classification theorem we deduce the construction of the t tree a new technique which generalizes the filtration induced by a torsion pair at last we apply our results in the tilting context generalizing the 1 tilting equivalence proved by happel reiten and smalo 16 the last section provides applications to classical n tilting objects examples of t trees for modules over a path algebra and new developments on compatible t structures 22 20

compactly generated t structures in the derived category of a commutative ring we classify all compactly generated t structures in the unbounded derived category of an arbitrary commutative ring generalizing the result of alonso tarrio et al j algebra 324 3 313 346 2010 for noetherian rings more specifically we establish a bijective correspondence between the compactly generated t structures and infinite filtrations of the zariski spectrum by thomason subsets moreover we show that in the case of a commutative noetherian ring any bounded below homotopically smashing t structure is compactly generated as a consequence all cosilting complexes are classified up to equivalence

centralizers of elementary abelian p subgroups and mod p cohomology of profinite groups 1 1 let g be a profinite group and p be a fixed prime in this paper we will be concerned with h c g fp the continuous cohomology of g with coefficients in the trivial module fp we will abbreviate h c g fp by h g fp or simply by h g if p is understood from the context we recall that if g is the inverse limit of finite groups gi then h g colim h gi throughout this paper we will assume that h g is finitely generated as fp algebra by work of lazard la it is known that this holds for many interesting groups for example for profinite p analytic groups like gl n zp the general linear groups over the p adic integers in case h g is finitely generated as fp algebra quillen has shown q1 that there are only finitely many conjugacy classes of elementary abelian p subgroups of g i e groups isomorphic to z p for some natural number n in other words the following category a g is equivalent to a finite category objects of a g are all elementary abelian p subgroups of g and if e1 and e2 are elementary abelian p subgroups of g then the set of morphisms from e1 to e2 in a g consists precisely of those homomorphisms e1 e2 of abelian groups for which there exists an element g g with e geg 1 e e1 the category a g plays an important role both in quillen s results and in the work presented here this category entered into quillen s work as follows the assignment e 7 h e extends to a functor from the opposite category a g op to graded fp algebras and the restriction homomorphisms h g h e for e running through the elementary abelian p subgroups of g induce a canonical map of algebras q h g lima g op h e

a new procedure for the construction of hierarchical high order hdiv and hcurl finite element spaces this paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for hdiv and hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains the principle is to choose an appropriate set of vectors based on the geometry of each element which are multiplied by an available set of h 1 hierarchical scalar basic functions this strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces properties that characterise functions in hdiv or hcurl respectively we also present a numerical study to evaluate the correct balancedness of the resulting hdiv spaces of degree k and l 2 spaces of degree k 1 on the resolution of the mixed formulation for a steklov eigenvalue problem

the homotopy groups of se 2 at p 5 revisited abstract we present a new technique for analyzing the v 0 bockstein spectral sequence studied by shimomura and yabe employing this technique we derive a conceptually simpler presentation of the homotopy groups of the e 2 local sphere at primes p 5 we identify and correct some errors in the original shimomura yabe calculation we deduce the related k 2 local homotopy groups and discuss their manifestation of gross hopkins duality

on the heart of a faithful torsion theory abstract in r colpi k r fuller tilting objects in abelian categories and quasitilted rings trans amer math soc in press tilting objects in an arbitrary abelian category h are introduced and are shown to yield a version of the classical tilting theorem between h and the category of modules over their endomorphism rings moreover it is shown that given any faithful torsion theory x y in mod r for a ring r the corresponding heart h x y is an abelian category admitting a tilting object which yields a tilting theorem between the heart and mod r in this paper we first prove that h x y is a prototype for any abelian category h admitting a tilting object which tilts to x y in mod r then we study ab type properties of the heart and commutations with direct limits this allows us to show for instance that any abelian category h with a tilting object is ab4 and to find necessary and sufficient conditions which guarantee that h is a grothendieck or even a module category as particular situations we examine two main cases when x y is hereditary cotilting proving that h x y is grothendieck and when x y is tilting proving that h x y is a module category

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

the homotopy groups of the localized toda smith spectrum 1 at the prime 3 in this paper we try to compute the homotopy groups of the l2 localized toda smith spectrum v 1 at the prime 3 by using the adamsnovikov spectral sequence and have almost done so this computation involves non trivial differentials d5 and dg of the adams novikov spectral sequence different from the case p 3 we also determine the homotopy groups of some l2 localized finite spectra relating to v 1 we further show some of the non trivial differentials on elements relating so called 3 elements in the adams novikov spectral sequence for 7r s0

bousfield localization on formal schemes let x ox be a noetherian formal scheme and consider dqct x its derived category of sheaves with quasi coherent torsion homology we show that there is a bijection between the set of rigid i e ideals localizing subcategories of dqct x and subsets in x generalizing previous work by neeman if moreover x is separated the associated localization and acyclization functors are described in certain cases when z x is a stable for specialization subset its associated acyclization is r z when x is a scheme the corresponding localizing subcategories are generated by perfect complexes and we recover thomason s classification of thick subcategories on the other hand if y x is generically stable we show that the associated localization functor is hom x r x yo x g

a characterization of n cotilting and n tilting modules abstract we consider generalizations of the definitions of one dimensional tilting and cotilting modules which agree with the classical notions of tilting and cotilting modules of finite homological dimension

a remark concerning divergence accuracy order for h div conforming finite element flux approximations abstract the construction of finite element approximations in h div usually requires the piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region it is known that degradation may occur in convergence order if non affine geometric mappings are used on this point we revisit a general procedure for the improvement of two dimensional flux approximations discussed in a recent paper of this journal comput math appl 74 2017 3283 3295 the starting point is an approximation scheme which is known to provide l 2 errors with accuracy of order k 1 for sufficiently smooth flux functions and of order r 1 for flux divergence an example is r t k spaces on quadrilateral meshes where r k or k 1 if linear or bilinear geometric isomorphisms are applied furthermore the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions the goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy r n 1 as desired for any n 1 the enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level k n while keeping the original border fluxes at level k the case n 1 has been discussed in the mentioned publication for two particular examples general stronger enrichment n 1 shall be analyzed and applied to darcy s flow simulations the global condensed systems to be solved having same dimension and structure of the original scheme

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

gorenstein homological algebra and universal coefficient theorems we study criteria for a ring or more generally for a small category to be gorenstein and for a module over it to be of finite projective dimension the goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones among the universal coefficient theorems covered by our methods we find besides all the classic examples several exotic examples arising from the kk theory of c algebras and also neeman s brown adams representability theorem for compactly generated categories

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

realisation functors in tilting theory derived equivalences and t structures are closely related we use realisation functors associated to t structures in triangulated categories to establish a derived morita theory for abelian categories with a projective generator or an injective cogenerator for this purpose we develop a theory of non compact or large tilting and cotilting objects that generalises the preceding notions in the literature within the scope of derived morita theory for rings we show that under some assumptions the realisation functor is a derived tensor product this fact allows us to approach a problem by rickard on the shape of derived equivalences finally we apply the techniques of this new derived morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t structures glueing to a tilting or a cotilting t structure as a further application we answer a question by xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras

cotorsion pairs model category structures and representation theory we make a general study of quillen model structures on abelian categories we show that they are closely related to cotorsion pairs which were introduced by salce sal79 and have been much studied recently by enochs and coauthors ej00 this gives a method of constructing model structures on abelian categories which we illustrate by building two model structures on the category of modules over a possibly noncommutative gorenstein ring the homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory this stable module category has also been studied by benson ben97