proper resolutions and gorenstein categories abstract let a be an abelian category and c an additive full subcategory of a we provide a method to construct a proper c resolution resp coproper c coresolution of one term in a short exact sequence in a from that of the other two terms by using these constructions we answer affirmatively an open question on the stability of the gorenstein category g c posed by sather wagstaff sharif and white and also prove that g c is closed under direct summands in addition we obtain some criteria for computing the c dimension and the g c dimension of an object in a

complete cohomology and gorensteinness of schemes abstract we develop and study tate and complete cohomology theory in the category of sheaves of o x modules different approaches are included we study the properties of these theories and show their power in reflecting the gorensteinness of the underlying scheme the connection of these two theories will be discussed

weak liftings of dg modules abstract this paper contains a weak lifting result for differential graded dg modules that is a generalization of auslander ding and solberg s weak lifting for modules and yoshino s weak lifting for complexes

on the auslander reiten conjecture for algebras abstract a recent result of araya asserts that if the auslander reiten conjecture holds in codimension one for a commutative gorenstein ring r then it holds for r this note extends this result to left gorenstein r algebras whenever r is a commutative gorenstein ring this in particular implies that any finitely generated self orthogonal gorenstein projective module is projective provided is an isolated singularity and dim r 2 also some examples of bound quiver algebras satisfying the auslander reiten conjecture are presented

remarks on modules approximated by g projective modules abstract let r be a commutative noetherian henselian local ring denote by modr the category of finitely generated r modules and by g the full subcategory of modr consisting of all g projective r modules in this paper we consider when a given r module has a right g approximation for this we study the full subcategory rap g of modr consisting of all r modules that admit right g approximations we investigate the structure of rap g by observing g g and lap g where lap g denotes the full subcategory of modr consisting of all r modules that admit left g approximations on the other hand we also characterize rap g in terms of tate cohomologies we give several sufficient conditions for g to be contravariantly finite in modr

estimating shear wave velocities from p wave and converted wave data tessmer and behle 1988 show that s wave velocity can be estimated from surface seismic data if both normal p wave data and converted wave data p sv are available the relation of tessmer and behle is vsv vp sv2 vp 1 where vsv is the s wave velocity vp is the p wave velocity and vp sv is the converted wave velocity the growing body of converted wave data suggest a brief examination of the validity of equation 1 for velocities that vary with depth

a variation on a theme of vasconcelos a classical result of w vasconcelos 1967 j algebra 6 309 316 and d ferrand 1967 c r acad sci paris 264 427 428 provides necessary and sufficient conditions for an ideal to be generated by a regular sequence in this paper we provide generally more accessible conditions guaranteeing this property as well as some examples indicating that our hypotheses cannot be significantly weakened the key step in our development is the use of lifting as presented by m auslander et al 1993 j algebra 156 273 317

on liftable and weakly liftable modules abstract let t be a noetherian ring and f a nonzerodivisor on t we study concrete necessary and sufficient conditions for a module over r t f to be weakly liftable to t in the sense of auslander ding and solberg we focus on cyclic modules and obtain various positive and negative results on the lifting and weak lifting problems for a module over t we define the loci for certain properties liftable weakly liftable having finite projective dimension and study their relationships

hyperbolic and elliptic transport barriers in three dimensional unsteady flows abstract we develop a general theory of transport barriers for three dimensional unsteady flows with arbitrary time dependence the barriers are obtained as two dimensional lagrangian coherent structures lcss that create locally maximal deformation along hyperbolic lcss this deformation is induced by locally maximal normal repulsion or attraction along shear lcss the deformation is created by locally maximal tangential shear hyperbolic lcss therefore play the role of generalized stable and unstable manifolds while closed shear lcss elliptic lcss act as generalized kam tori or kam type cylinders all these barriers can be computed from our theory as explicitly parametrized surfaces we illustrate our results by visualizing two dimensional hyperbolic and elliptic barriers in steady and unsteady versions of the abc flow

strongly gorenstein projective injective and flat modules abstract in this paper we study some properties of strongly gorenstein projective injective and flat modules and we discuss some connections between strongly gorenstein projective injective and flat modules and we consider these properties under change of rings

tilting complexes and auslander reiten conjecture we studied the properties of tilting complexes and proved that derived equivalences preserve the validity of the auslander reiten conjecture

tate cohomology with respect to semidualizing modules abstract we investigate tate cohomology of modules over a commutative noetherian ring with respect to semidualizing modules we identify classes of modules admitting tate resolutions and analyze the interaction between the corresponding relative and tate cohomology modules as an application of our approach we prove a general balance result for tate cohomology our results are based on an analysis of tate cohomology in abelian categories

a geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of 2 we give a proof based in geometric perturbation theory of a result proved by j n mather using variational methods namely the existence of orbits with unbounded energy in perturbations of a generic geodesic flow in 2 by a generic periodic potential

on a conjecture of auslander and reiten abstract in studying nakayama s 1958 conjecture on rings of infinite dominant dimension auslander and reiten proposed the following generalization let be an artin algebra and m a generator such that ext i m m 0 for all i 1 then m is projective this conjecture makes sense for any ring we establish auslander and reiten s conjecture for excellent cohen macaulay normal domains containing the rational numbers and slightly more generally

vanishing of ext cluster tilting modules and finite global dimension of endomorphism rings let r be a cohen macaulay ring and m a maximal cohen macaulay r module inspired by recent striking work by iyama burban iyama keller reiten and van den bergh we study the question of when the endomorphism ring of m has finite global dimension via certain conditions about vanishing of ext modules we are able to strengthen certain results by iyama on connections between a higher dimension version of auslander correspondence and existence of non commutative crepant resolutions we also recover and extend to positive characteristics a recent theorem by burban iyama keller reiten on cluster tilting objects in the category of maximal cohen macaulay modules over reduced 1 dimensional hypersurfaces

relative cohomology of complexes abstract let r be an arbitrary ring and c a complex with finite gorenstein projective dimension that is the supremum of gorenstein projective dimension of all r modules in c is finite for each complex d we define the nth relative cohomology group ext gp n c d by the equality ext gp n c d h n h om g d where g c is a strict gorenstein projective precover of c if d is a complex with finite gorenstein injective dimension that is the supremum of gorenstein injective dimension of all r modules in d is finite then one can use a dual argument to define a notion of relative cohomology group ext gi n c d we show that if c is a complex with finite gorenstein projective dimension and d a complex with finite gorenstein injective dimension then for each n z there exists an isomorphism ext gp n c d ext gi n c d this shows that the relative cohomology functor of complexes is balanced some induced exact sequences concerning relative cohomology groups are considered

deformations of complexes for finite dimensional algebras abstract let k be a field and let be a finite dimensional k algebra we prove that every bounded complex v of finitely generated modules has a well defined versal deformation ring r v which is a complete local commutative noetherian k algebra with residue field k we also prove that nice two sided tilting complexes between and another finite dimensional k algebra preserve these versal deformation rings additionally we investigate stable equivalences of morita type between self injective algebras in this context we apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by erdmann and shown by holm to be not derived equivalent to any block of a group algebra

the new intersection theorem and descent of flatness for integral extensions abstract the new intersection theorem is used to derive a criteria for flat descent in the setting of integral ring extensions applications such as purity of branch locus for extensions of normal domains are noted

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 transpose of modules relative to subcategories abstract let r be a left noetherian ring and s a right noetherian ring and let x be a subcategory of finitely generated left r modules and b s r a finite r s bimodule as a generalization of the auslander transpose replacing a projective presentation by an x presentation and the functor hom r r by the functor hom r b we introduce the x b transpose of an r module m admitting an x presentation for a suitable subcategory x some useful properties of x b transposes are obtained it is shown that the x b transpose shares many nice properties with the auslander transpose some known results are obtained as corollaries

liftings and quasi liftings of dg modules abstract we prove lifting results for dg modules that are akin to auslander ding and solberg s famous lifting results for modules

gorenstein projective modules and symmetric recollements we introduce compatible bimodules if m is a compatible a b bimodule then the gorenstein projective modules over algebra am0b are explicitly described and if is gorenstein then this description implies that m is compatible as an application if m is compatible then there is a left recollement of the stable category gp and if is gorenstein and ma is projective then there is a symmetric recollement of the singularity category dsgb

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

efficient computation of null geodesics with applications to coherent vortex detection recent results suggest that boundaries of coherent fluid vortices elliptic coherent structures can be identified as closed null geodesics of appropriate lorentzian metrics defined on the flow domain here we derive an automated method for computing such null geodesics based on the geometry of the underlying geodesic flow our approach simplifies and improves existing procedures for computing variationally defined eulerian and lagrangian vortex boundaries as an illustration we compute objective vortex boundaries from satellite inferred ocean velocity data a matlab implementation of our method is available at https github com mattiaserra closed null geodesics 2d

auslander reiten conjecture and auslander reiten duality abstract motivated by a result of araya we extend the auslander reiten duality theorem to cohen macaulay local rings we also study the auslander reiten conjecture which is rooted in nakayama s work on finite dimensional algebras one of our results detects a certain condition that forces the conjecture to hold over local rings of positive depth

finite gorenstein representation type implies simple singularity let r be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive r modules we prove that if this set is finite then either it has exactly one element represented by the rank 1 free module or r is gorenstein and an isolated singularity if r is complete then it is even a simple hypersurface singularity the crux of our proof is to argue that if the residue field has a totally reflexive cover then r is gorenstein or every totally reflexive r module is free

gorenstein complexes and recollements from cotorsion pairs we describe a general correspondence between injective resp projective recollements of triangulated categories and injective resp projective cotorsion pairs this provides a model category description of these recollement situations our applications focus on displaying several recollements that glue together various full subcategories of k r the homotopy category of chain complexes of modules over a general ring r when r is left noetherian these recollements involve complexes built from the gorenstein injective modules when r is a left coherent ring for which all flat modules have finite projective dimension we obtain the duals these results extend to a general ring r by replacing the gorenstein modules with the gorenstein ac modules introduced in 9 we also see that in any abelian category with enough injectives the gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair

short note fast geodesics computation with the phase flow method this paper introduces a novel approach for rapidly computing a very large number of geodesics on a smooth surface the idea is to apply the recently developed phase flow method l ying e j candes the phase flow method j comput phys to appear an efficient and accurate technique for constructing phase maps for nonlinear ordinary differential equations on invariant manifolds which are here the unit tangent bundles of the surfaces under study we show how to rapidly construct the whole geodesic flow map which then allows computing any geodesic by straightforward local interpolation an operation with constant complexity a few numerical experiments complement our study and demonstrate the effectiveness of our approach

remarks on non commutative crepant resolutions of complete intersections abstract we study obstructions to existence of non commutative crepant resolutions in the sense of van den bergh over local complete intersections

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

symmetry in the vanishing of ext over stably symmetric algebras abstract a frobenius algebra over a field k is called symmetric if the nakayama automorphism is an inner automorphism a stably symmetric algebra is defined to be a generalization of a symmetric k algebra in this paper we will study symmetry in the vanishing of ext for such algebras r namely for all finitely generated r modules m and n ext r i m n 0 for all i 0 if and only if ext r i n m 0 for all i 0 we show that a certain class of noetherian stably symmetric gorenstein algebras such as the group algebra of a finite group and the exterior algebra k n when n is odd have this symmetry using serre duality we also show that every exterior algebra k n whether n is even or odd has this symmetry for graded modules using koszul duality

on the category of modules of gorenstein dimension zero let r be a commutative noetherian henselian non gorenstein local ring of depth zero denote by modr the category of all finitely generated r modules and by open image in new window the full subcategory of modr consisting of all r modules of gorenstein dimension zero we prove in this paper that if open image in new window contains a non free module then it is not precovering in modr in particular there exist infinitely many isomorphism classes of indecomposable r modules of gorenstein dimension zero

cohomology theories based on gorenstein injective modules in this paper we study relative and tate cohomology of modules of finite gorenstein injective dimension using these cohomology theories we present variations of grothendieck local cohomology modules namely gorenstein and tate local cohomology modules by applying a sort of avramov martsinkovsky exact sequence we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by j herzog we discuss some properties of these modules and give some results concerning their vanishing and non vanishing

geodesic theory of transport barriers in two dimensional flows abstract we introduce a new approach to locating key material transport barriers in two dimensional non autonomous dynamical systems such as unsteady planar fluid flows seeking transport barriers as minimally stretching material lines we obtain that such barriers must be shadowed by minimal geodesics under the riemannian metric induced by the cauchy green strain tensor as a result snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations using this approach we locate hyperbolic barriers generalized stable and unstable manifolds elliptic barriers generalized kam curves and parabolic barriers generalized shear jets in temporally aperiodic flows defined over a finite time interval our approach also yields a metric geodesic deviation that determines the minimal computational time scale needed for a robust numerical identification of generalized lagrangian coherent structures lcss as we show an extension of our transport barrier theory to non euclidean flow domains such as a sphere follows directly we illustrate our main results by computing key transport barriers in a chaotic advection map and in a geophysical model flow with chaotic time dependence

p sv reflection moveouts for transversely isotropic media with a vertical symmetry axis in a recently published short note f k levin 1989 discusses the relation between the moveout velocities of p p p sv and sv sv reflections from the bottom of a transversely isotropic layer with a vertical symmetry axis we refer to such a medium as one exhibiting polar anisotropy levin s note was prompted by a paper of tessmer and behle 1988 and it is relevant to a paper by iverson and others 1989 both of which discuss the computation of shear velocities from moveout velocities obtained with p p and p s reflections levin s note addresses the practically important question of the use of this method in the presence of polar anisotropy a phenomenon which we believe occurs almost universally in the sedimentary layers of the real earth levin suggests that polar anisotropy of typical magnitude must be considered in this problem he uses as an estimate of typical magnitudes data given by thomsen 1986 and concludes from numerical examples that the method of estimating shear velocities prop

deformations of maximal cohen macaulay modules let r m be a complete cohen macaulay isolated singularity over a field k which is either perfect or k k p 0 according to dieterich di and yo ch 6 see also pol pr or more generally chp there exists a system of parameters x xl xr of r such that the base change functor r x rdefines an injection v preserving the indecomposability from the set of isomorphism classes of maximal cohen macaulay r modules to the set of isomorphism classes of r x modules in this case the ideal generated by x is called after di a reduction ideal trying to describe the image of v we noticed in po2 that a finitely generated r x module is in imv if and only if it has the form xl x p for a finitely generated r2 r x 2 x module satisfying

syzygy modules with semidualizing or g projective summands abstract let r be a commutative noetherian local ring with residue class field k in this paper we mainly investigate direct summands of the syzygy modules of k we prove that r is regular if and only if some syzygy module of k has a semidualizing summand after that we consider whether r is gorenstein if and only if some syzygy module of k has a g projective summand

nonhyperbolic reflection moveout in anisotropic media the standard hyperbolic approximation for reflection moveouts in layered media is accurate only for relatively short spreads even if the layers are isotropic velocity anisotropy may significantly enhance deviations from hyperbolic moveout nonhyperbolic analysis in anisotropic media is also important because conventional hyperbolic moveout processing on short spreads is insufficient to recover the true vertical velocity hence the depth we present analytic and numerical analysis of the combined influence of vertical transverse isotropy and layering on long spread reflection moveouts qualitative description of nonhyperbolic moveout on intermediate spreads offset to depth ratio x z 1 7 2 is given in terms of the exact fourth order taylor series expansion for p sv and p sv traveltime curves valid for multilayered transversely isotropic media with arbitrary strength of anisotropy we use this expansion to provide an analytic explanation for deviations from hyperbolic moveout such as the strong

homological properties of powers of the maximal ideal of a local ring abstract it is known that the powers m n of the maximal ideal of a local noetherian ring share certain homological properties for all sufficiently large integers n for example the natural homomorphisms r r m n are golod respectively small for all large n we give effective bounds on the smallest integers n for which such properties begin to hold

totally reflexive extensions and modules abstract we introduce the notion of totally reflexive extension of rings it unifies gorenstein orders and frobenius extensions we prove that for a totally reflexive extension a module over the extension ring is totally reflexive if and only if its underlying module over the base ring is totally reflexive

descent via koszul extensions abstract let r be a commutative noetherian local ring with completion r we apply differential graded dg algebra techniques to study descent of modules and complexes from r to r where r is either the henselization of r or a pointed etale neighborhood of r we extend a given r complex to a dg module over a koszul complex we describe this dg module equationally and apply artin approximation to descend it to r this descent result for koszul extensions has several applications when r is excellent we use it to descend the dualizing complex from r to a pointed etale neighborhood of r this yields a new version of p roberts theorem on uniform annihilation of homology modules of perfect complexes as another application we prove that the auslander condition on uniform vanishing of cohomology ascends to r when r is excellent henselian and cohen macaulay

support varieties and cohomology over complete intersections quillen s geometric approach to the cohomology of finite groups 30 has revolutionized modular representation theory the ideology and techniques involved have been generalized and extended to representations of various hopf algebras culminating in the recent work of friedlander and suslin 17 on finite group schemes in this paper we develop geometric methods for the study of finite modules over a local complete intersection r if r is such a ring and m is its maximal ideal then the m adic completion r has the form q f where f is a regular sequence and q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring the least number of equations needed to cut out r from a regular local ring equals the codimension of r where codim r r m dim r and r m denotes the minimal number of generators of an r module m in 5 a cone that is a homogeneous algebraic set vr m is attached to each finite r module m and used to study its minimal free resolution here we prove

cohen macaulay approximation in fibred categories abstract we extend the auslander buchweitz axioms and prove cohen macaulay approximation results for fibred categories we show that these axioms apply for the fibred category of pairs consisting of a finite type flat family of cohen macaulay rings and modules in particular such a pair admits an approximation with a flat family of maximal cohen macaulay modules and a hull with a flat family of modules with finite injective dimension the existence of minimal approximations and hulls in the local flat case implies extension of upper semi continuous invariants as an example of mcm approximation we define a relative version of auslander s fundamental module

algebras that satisfy auslander s condition on vanishing of cohomology auslander conjectured that every artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules the failure of this conjecture by a 2003 counterexample due to jorgensen and ega motivates the consideration of the class of rings that do satisfy auslander s condition we call them ac rings and show that an ac artin algebra that is left gorenstein is also right gorenstein furthermore the auslander reiten conjecture is proved for ac rings and auslander s g dimension is shown to be functorial for ac rings that are commutative or have a dualizing complex

shearless transport barriers in unsteady two dimensional flows and maps abstract we develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two dimensional flows and maps defined over a finite time interval this principle reveals that hyperbolic lagrangian coherent structures lcss and parabolic lcss or jet cores are the two main types of shearless barriers in unsteady flows based on the boundary conditions they satisfy parabolic barriers are found to be more observable and robust than hyperbolic barriers confirming widespread numerical observations both types of barriers are special null geodesics of an appropriate lorentzian metric derived from the cauchy green strain tensor using this fact we devise an algorithm for the automated computation of parabolic barriers we illustrate our detection method on steady and unsteady non twist maps and on the aperiodically forced bickley jet