precovers and orthogonality in the stable module category abstract we show that any module admits a presentation as the quotient of a gorenstein projective module by a submodule which is itself right orthogonal with respect to the standard ext 1 pairing to the class of gorenstein projective modules of type fp for that purpose we use the concept of orthogonality in the stable module category and examine the orthogonal pair which is induced therein by the class of completely finitary gorenstein projective modules

cohen macaulay modules co torsion pairs and virtually gorenstein algebras abstract we use torsion pairs in stable categories and cotorsion pairs in modules categories to study in general infinitely generated cohen macaulay modules and a generalization of modules of finite projective or injective dimension over an artin algebra we concentrate our investigation to the study of virtually gorenstein algebras which provide a common generalization of gorenstein algebras and algebras of finite representation or cohen macaulay type this class of algebras on the one hand has rich homological structure and satisfies several representation torsion theoretic finiteness conditions and on the other hand it is closed under various operations for instance derived equivalences and stable equivalences of morita type in addition virtual gorensteinness provides a useful tool for the study of the gorenstein symmetry conjecture and modified versions of the telescope conjecture for module or stable categories

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

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

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

monomorphism categories cotilting theory and gorenstein projective modules abstract the monomorphism category s n x is introduced where x is a full subcategory of the module category a mod of an artin algebra a the key result is a reciprocity of the monomorphism operator s n and the left perpendicular operator for a cotilting a module t there is a canonical construction of a cotilting module m t over the upper triangular matrix algebra t n a such that s n t m t as applications s n x is a resolving contravariantly finite subcategory in t n a mod with s n x t n a mod if and only if x is a resolving contravariantly finite subcategory in a mod with x a mod for a gorenstein algebra a the category t n a g proj of gorenstein projective t n a modules can be explicitly determined as s n a also self injective algebras a can be characterized by the property t n a g proj s n a finally we obtain a characterization of those categories s n a which have finite representation type in terms of auslander s representation dimension

comparison of relative cohomology theories with respect to semidualizing modules we compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules we prove a general balance result for relative cohomology over a cohen macaulay ring with a dualizing module and we demonstrate the failure of the naive version of balance one might expect for these functors we prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases and we provide a yoneda type description of the first relative ext functor finally we show by example that each distinct relative cohomology construction does in fact result in a different functor

cohomology over fiber products of local rings abstract let s and t be local rings with common residue field k let r be the fiber product s k t and let m be an s module the poincare series p m r of m has been expressed in terms of p m s p k s and p k t by kostrikin and shafarevich and by dress and kramer here an explicit minimal resolution as well as theorems on the structure of ext r k k and ext r m k are given that illuminate these equalities structure theorems for the cohomology modules of fiber products of modules are also given as an application of these results we compute the depth of cohomology modules over a fiber product

the acyclic closure of an exact category and its triangulation abstract for any exact category a with splitting idempotents a maximal exact category t a containing a as a biresolving subcategory is constructed important types of exact categories including n tilting torsion classes categories of cohen macaulay modules over a cohen macaulay order or categories of gorenstein projectives are shown to be of the form t a the quotient category t a a in the sense of grothendieck always exists and carries a triangulated structure more generally it is proved that any biresolving subcategory a of an exact category m gives a triangulated localization m a for example the unbounded derived category d a of an exact category a is obtained directly with no passage through a homotopy category of complexes as applications some recent developments related to gorenstein projectivity non commutative crepant resolutions singularity categories and cohen macaulay representations are extended and improved in the new framework for example the concept of non commutative resolution of a noetherian frobenius category is extended to arbitrary exact categories which leads to an overarching connection with representation dimension of exact categories and n tilting

vanishing of relative homology and depth of tensor products abstract for finitely generated modules m and n over a gorenstein local ring r one has depth m depth n depth m r n depth r i e the depth formula holds if m and n are tor independent and tate homology tor i m n vanishes for all i z we establish the same conclusion under weaker hypotheses if m and n are g relative tor independent then the vanishing of tor i m n for all i 0 is enough for the depth formula to hold we also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of g relative homology

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

new classes of spectrally arbitrary ray patterns an n n ray pattern matrix s is said to be spectrally arbitrary if for every monic nth degree polynomial f with coefficients from c there is a complex matrix in the ray pattern class of s such that its characteristic polynomial is f in this article we give new classes of spectrally arbitrary ray pattern matrices

spectrally and inertially arbitrary sign patterns we introduce some n by n sign patterns which allow for arbitrary spectrum and hence also arbitrary inertia consequently we demonstrate that some known inertially arbitrary patterns are in fact spectrally arbitrary we demonstrate that all inertially arbitrary patterns of order 3 are spectrally arbitrary and classify all spectrally arbitrary patterns of order 3 we illustrate that in general the class of spectrally arbitrary patterns is distinct from the inertially arbitrary patterns and present some observations about inertially arbitrary patterns

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

totally reflexive modules and poincare series abstract we study cohen macaulay non gorenstein local rings r m k admitting certain totally reflexive modules more precisely we give a description of the poincare series of k by using the poincare series of a non zero totally reflexive module with minimal multiplicity our results generalize a result of yoshino to higher dimensional cohen macaulay local rings moreover from a quasi gorenstein ideal satisfying some conditions we construct a family of non isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators

when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module abstract in this paper we study when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module applications go in three directions the first is to characterize when the little finitistic dimension is finite the second is to obtain equivalent formulations for a wakamatsu tilting module to be a tilting module the third is to give some new characterizations of gorenstein rings

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

on the finiteness of gorenstein homological dimensions abstract in this paper we study certain properties of modules of finite gorenstein projective injective and flat dimensions we examine conditions which imply that all gorenstein projective modules are gorenstein flat and establish the balance of the gorenstein tor functor for modules of finite gorenstein projective dimension we also examine the class of rings that have finite gorenstein global and weak dimensions and compute these dimensions in terms of certain cohomological invariants of the ring finally we provide some examples of rings of finite gorenstein global and weak dimensions

on evaluating the donsker varadhan i function let x t be a feller process on a complete separable metric space a and consider the occupation measure l t omega cdot int t 0 chi cdot x s ds the i function is defined for mu in mathscr p a the set of probability measures on a by i mu inf u in mathscr d int a lu u d mu where l mathscr d is the generator of the process and mathscr d subset mathscr d consists of the strictly positive functions in mathscr d the i function determines the asymptotic rate of decay of p 1 t l t omega cdot in g for g subset mathscr p a the first difficulty encountered in evaluating i mu is that the domain mathscr d is generally not known explicitly in this paper we prove a theorem which allows us to restrict the calculation of the infimum to a nice subdomain we then apply this general result to diffusion processes with boundaries

the ray nonsingularity of certain uniformly random ray patterns abstract a uniformly random ray pattern matrix a with a given zero nonzero pattern described by a digraph d with no multi arcs or loops is the matrix whose nonzero entries are mutually independent random variables uniformly distributed over the unit circle s 1 in the complex plane it is shown in this paper that the probability of i a to be ray nonsingular is completely determined by the cycle graph cg d of d i e the adjacency structure of the directed cycles in d if cg d is a tree a formula is given to compute the probability when cg d is a tree and it is also shown that as the order of cg d tends to infinity the limit of the probability is 0

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

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

gorenstein singularity categories abstract the aim of this paper is to introduce gorenstein singularity category d g p s g b a as the verdier quotient of the gorenstein derived category d g p b a by the triangulated subcategory k b gp a the main result shows that if a is cm contravariantly finite abelian category then the gorenstein singularity category d g p s g b a is triangle equivalent to the gorenstein defect category d defect b a in the sense of 7 we point out several classes of algebras with non zero singularity categories but their gorenstein singularity categories become zero we also prove that a gorenstein derived equivalence between cm contravariantly finite abelian categories with infinite direct sums induces a gorenstein singular equivalence

a construction of gorenstein projective modules abstract we determine all the gorenstein projective modules over the t 2 extension of a gorenstein algebra and over a m 0 b where a and b are self injective algebras and m is an a b bimodule with m a and m b projective using this we obtain a criterion for the cm finiteness of the t 2 extensions of cm finite gorenstein algebras as an application we get non trivial examples of cm finite gorenstein algebras

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

three results on frobenius categories this paper consists of three results on frobenius categories 1 we give sufficient conditions on when a factor category of a frobenius category is still a frobenius category 2 we show that any frobenius category is equivalent to an extension closed exact subcategory of the frobenius category formed by cohen macaulay modules over some additive category this is an analogue of gabriel quillen s embedding theorem for frobenius categories 3 we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure with which the given category becomes a frobenius category several applications of the results are discussed

contravariantly finite resolving subcategories over commutative rings contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras in this paper we study contravariantly finite resolving subcategories over commutative rings the main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian gorenstein local ring in fact there exist only three such subcategories our method to obtain this classification also recovers as a by product the theorem of christensen piepmeyer striuli and takahashi concerning the relationship between the contravariant finiteness of the full subcategory of totally reflexive modules and the gorenstein property of the base ring

spectrally arbitrary patterns reducibility and the 2n conjecture for n 5 a sign pattern z a matrix whose entries are elements of 0 is spectrally arbitrary if for any self conjugate spectrum there is a real matrix with sign pattern z having the given spectrum spectrally arbitrary sign patterns were introduced in 5 where it was incorrectly stated that if a sign pattern z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern then z is a spectrally arbitrary sign pattern and it was conjectured that the converse is true as well we present counterexamples to both of these statements in 2 it was conjectured that any n n spectrally arbitrary sign pattern must have at least 2n nonzero entries we establish that this conjecture is true for 5 5 sign patterns we also establish analogous results for nonzero patterns

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

the homological theory of quasi resolving subcategories abstract as a generalization of a resolving subcategory which was introduced by auslander and bridger in 2 the notion of a quasi resolving subcategory of an abelian category is introduced the homological theory on quasi resolving subcategories is studied it is shown that many results in 1 2 4 5 6 8 13 14 17 18 22 25 26 27 are unified and naturally generated

construction of modules with finite homological dimensions a new homological dimension called g dimension is defined for every finitely generated module m over a local noetherian ring r it is modeled on the ci dimension of avramov gasharov and peeva and has parallel properties in particular a ring r is gorenstein if and only if every finitely generated r module has finite g dimension the g dimension lies between the ci dimension and the g dimension of auslander and bridger this relation belongs to a longer sequence of inequalities where a strict inequality in any place implies equalities to its right and left over general local rings we construct classes of modules that show that a strict inequality can occur at almost every place in the sequence

exact pairs of homogeneous zero divisors abstract let s be a standard graded artinian algebra over a field k we identify constraints on the hilbert function of s which are imposed by the hypothesis that s contains an exact pair of homogeneous zero divisors as a consequence we prove that if s is a compressed level algebra then s does not contain any homogeneous zero divisors

stable t structures and homotopy category of gorenstein projective modules abstract we study the homotopy category of unbounded complexes of gorenstein projective modules with bounded relative homologies we show the existence of a right recollement of the above homotopy category and it has the homotopy category of gorenstein acyclic complexes as a triangulated subcategory in some case we also show that the bounded gorenstein derived category of a cm finite gorenstein artin algebra is triangle equivalent to the bounded derived category of the endomorphism ring of some tilting object

cosupport computations for finitely generated modules over commutative noetherian rings abstract we show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure injective resolution of the ring as an application of this we prove that every countable commutative noetherian ring has full cosupport we also settle the comparison of cosupport and support of finitely generated modules over any commutative noetherian ring of finite krull dimension finally we give an example showing that the cosupport of a finitely generated module need not be a closed subset of spec r providing a negative answer to a question of sather wagstaff and wicklein 29

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

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

the minimum number of nonzeros in a spectrally arbitrary ray pattern abstract a ray pattern a is spectrally arbitrary if given any monic polynomial r x of order n with complex coefficients there exists b q r a such that the characteristic polynomial of b is r x in 1 the authors first presented that a family of n n irreducible ray patterns with exactly 3n nonzeros is spectrally arbitrary then they proved that every n n irreducible spectrally arbitrary ray pattern has at least 3 n 1 nonzeros so as pointed in 1 the minimum number of nonzeros in an n n irreducible spectrally arbitrary ray pattern is either 3n or 3 n 1 in this paper we provide an irreducible spectrally arbitrary ray pattern of order n with 3 n 1 nonzeros so the minimum number of nonzeros in an n n irreducible spectrally arbitrary ray pattern is 3 n 1

spectrally arbitrary star sign patterns abstract an n n sign pattern s n is spectrally arbitrary if for any given real monic polynomial g x of degree n there is a real matrix having sign pattern s n and characteristic polynomial g x all n n star sign patterns that are spectrally arbitrary and all minimal such patterns are characterized this subsequently leads to an explicit characterization of all n n star sign patterns that are potentially nilpotent it is shown that any super pattern of a spectrally arbitrary star sign pattern is also spectrally arbitrary

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

gorenstein algebras of finite cohen macaulay type abstract an artin algebra a is said to be cm finite if there are only finitely many isomorphism classes of indecomposable finitely generated gorenstein projective a modules inspired by auslander s idea on representation dimension we prove that for 2 n a is a cm finite n gorenstein algebra if and only if there is a resolving gorenstein projective a module e such that gl dim end a e op n

spectrally arbitrary patterns abstract an n n sign pattern matrix s is an inertially arbitrary pattern iap if each nonnegative triple n 1 n 2 n 3 with n 1 n 2 n 3 n is the inertia of a matrix with sign pattern s analogously s is a spectrally arbitrary pattern sap if for any given real monic polynomial r x of order n there is a matrix with sign pattern s and characteristic polynomial r x focusing on tree sign patterns consider the n n tridiagonal sign pattern t n with each superdiagonal entry positive each subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other entry zero it is conjectured that t n is an iap by constructing matrices a n with pattern t n it is proved that t n allows any inertia with n 3 0 1 2 n 1 n for all n 2 this leads to a proof of the conjecture for n 5 the truth of the conjecture is extended to n 7 by showing the stronger result that t n is a sap the proof of this latter statement involves finding a matrix a n with pattern t n that is nilpotent further questions about patterns that are saps and iaps are considered

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

gorenstein projective dimension for complexes we define and study a notion of gorenstein projective dimension for complexes of left modules over associative rings for complexes of finite gorenstein projective dimension we define and study a tate cohomology theory tate cohomology groups have a natural transformation to classical ext groups in the case of module arguments we show that these maps fit into a long exact sequence where every third term is a relative cohomology group defined for left modules of finite gorenstein projective dimension

spectrally arbitrary ray patterns abstract an n n ray pattern a is said to be spectrally arbitrary if for every monic nth degree polynomial f x with coefficients from c there is a matrix in the pattern class of a such that its characteristic polynomial is f x in this article the authors extend the nilpotent jacobi method for sign patterns to ray patterns establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary they use this method to establish that a particular family of n n irreducible ray patterns with exactly 3 n nonzeros is spectrally arbitrary they then show that every n n irreducible spectrally arbitrary ray pattern has at least 3 n 1 nonzeros

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

the radical of the homotopy lie algebra introduction let t be a simply connected cw complex of finite type its homotopy lie algebra is the graded rational lie algebra 7r 9 t 0 q equipped with the samelson product let a be a local noetherian ring with residue field k of arbitrary characteristic its homotopy lie algebra 7r a is the naturally defined graded lie algebra whose enveloping algebra is exta k k cf av these graded lie algebras have finite type i e are finite dimensional in each degree and are concentrated in strictly positive degrees thus for the sake of simplicity we shall use graded lie algebra to mean graded lie algebra of finite type concentrated in strictly positive degrees throughout this paper on the other hand these graded lie algebras are usually nonzero in infinitely many degrees thus while we may and do form the sum r of all the solvable ideals in such a graded lie algebra and call it the radical it may well happen that r itself is not solvable indeed this does happen in the topological context since by quillen s result q every graded rational lie algebra of finite type occurs as the homotopy lie algebra of a space by contrast in the case of local rings or of cw complexes with finite lusternik schnirelmann l s category serious restrictions apply to the homotopy lie algebra indeed our first main result is the remarkable

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

separated monic representations i gorenstein projective modules abstract for a finite acyclic quiver q an ideal i of a path algebra kq generated by monomial relations and a finite dimensional k algebra a we introduce the separated monic representations of a bound quiver q i over a they differ from the usual monic representations the category smon q i a of the separated monic representations of q i over a coincides with the category mon q i a of the usual monic representations if and only if i 0 and each vertex of q is the ending vertex of at most one arrow we give properties of the structural maps of separated monic representations and prove that smon q i a is a resolving subcategory of rep q i a we introduce the condition g let a k q i by the equivalence rep q i a mod of categories the main result claims that a module is gorenstein projective if and only if it is in smon q i a and has a local a gorenstein projective property g as consequences the separated monic modules are exactly the projective modules if and only if a is semi simple and they are exactly the gorenstein projective modules if and only if a is self injective and if and only if smon q i a is a frobenius category

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

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

gorenstein defect categories of triangular matrix algebras abstract we apply the technique of recollement to study the gorenstein defect categories of triangular matrix algebras first we construct a left recollement of gorenstein defect categories for a triangular matrix algebra under some conditions using it we give a categorical interpretation of the gorenstein properties of the triangular matrix algebras obtained by x w chen b l xiong and p zhang respectively second under some additional conditions a recollement of gorenstein defect categories for a triangular matrix algebra is constructed as an application for a special kind of triangular matrix algebras which are called simple gluing algebras we describe their singularity categories and gorenstein defect categories

triangulated equivalence between a homotopy category and a triangulated quotient category abstract given two complete hereditary cotorsion pairs q r and q r in a bicomplete abelian category g such that q q and q r q r becker showed that there exists a hereditary abelian model structure m q w r on g where w is a thick subcategory of g we prove that the homotopy category ho m of m is triangulated equivalent to the triangulated quotient category d b g q r k b q r where d b g q r is the subcategory of d b g consisting of all homology bounded complexes with both finite q dimension and r dimension and k b q r is the bounded homotopy category of q r core objects applications are given in the category of modules it is shown that the homotopy category of the gorenstein flat resp ding projective and gorenstein ac projective model structure on the category of modules established by gillespie and his coauthors can be realized as a certain triangulated quotient category

on gorenstein projective injective and flat dimensions a functorial description with applications gorenstein homological dimensions are refinements of the classical homological dimensions and finiteness singles out modules with amenable properties reflecting those of modules over gorenstein rings as opposed to their classical counterparts these dimensions do not immediately come with practical and robust criteria for finiteness not even over commutative noetherian local rings in this paper we enlarge the class of rings known to admit good criteria for finiteness of gorenstein dimensions

on certain cohomological invariants of groups abstract let r be a left and right 0 noetherian ring we show that if all projective left and all projective right r modules have finite injective dimension then all injective left and all injective right r modules have finite projective dimension using this result we prove that the invariants silp z g and spli z g which were introduced by gedrich and gruenberg 1987 15 are equal for any group g as an application of the latter equality we show that a group g is finite if and only if cd g 0 where cd is the generalized cohomological dimension of groups introduced by ikenaga 1984 21

realizability of modules over tate cohomology let k be a field and let g be a finite group there is a canonical element in the hochschild cohomology of the tate cohomology g hh 3 h1 h g k with the following property given a graded h g k module x the image of g in ext 3 1 h g k x x vanishes if and only if x is isomorphic to a direct summand of h g m for some kg module m the description of the realizability obstruction works in any triangulated category with direct sums we show that in the case of the derived category of a differential graded algebra a there is also a canonical element of hochschild cohomology hh 3 1 h a which is a predecessor for these obstructions

stable cohomology over local rings abstract for a commutative noetherian ring r with residue field k stable cohomology modules ext r n k k have been defined for each n z but their meaning has remained elusive it is proved that the k rank of any ext r n k k characterizes important properties of r such as being regular complete intersection or gorenstein these numerical characterizations are based on results concerning the structure of z graded k algebra carried by stable cohomology it is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras ext r k k ext r k k some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings

tate cohomology and gorensteinness for triangulated categories abstract motivated by the classical structure of tate cohomology we develop and study a tate cohomology theory in a triangulated category c let e be a proper class of triangles by using e projective as well as e injective objects we give two alternative approaches to this theory that in general are not equivalent so in the second part of the paper we study triangulated categories in which these two theories are equivalent this leads us to study the categories in which all objects have finite e g projective as well as finite e g injective dimension these categories will be called e gorenstein triangulated categories we give a characterization of these categories in terms of the finiteness of two invariants e silp c the supremum of the e injective dimension of e projective objects of c and e spli c the supremum of the e projective dimension of e injective objects of c where finiteness of each of these invariants for a category implies the finiteness of the other finally we show that over e gorenstein triangulated categories the class of objects of finite e projective dimension and the class of e g injective objects form an e complete cotorsion theory

on the large deviation rate function for the empirical measures of reversible jump markov processes abstract the large deviations principle for the empirical measure for both continuous and discrete time markov processes is well known various expressions are avail able for the rate function but these expressions are usually as the solution to a variational problem and in this sense not explicit an interesting class of con tinuous time reversible processes was identi ed in the original work of donsker and varadhan for which an explicit expression is possible while this class in cludes many reversible processes of interest it excludes the case of continuous time pure jump processes such as a reversible nite state markov chain in this paper we study the large deviations principle for the empirical measure of pure jump markov processes and provide an explicit formula of the rate function under reversibility

allow problems concerning spectral properties of sign pattern matrices a survey abstract an n n sign pattern matrix has entries in 0 this paper surveys the following problems concerning spectral properties of sign pattern matrices sign patterns that allow all possible spectra spectrally arbitrary sign patterns sign patterns that allow all inertias inertially arbitrary sign patterns sign patterns that allow nilpotency potentially nilpotent sign patterns and sign patterns that allow stability potentially stable sign patterns relationships between these four classes of sign patterns are given and several open problems are identified

asymmetry of ext groups in this paper we will use techniques of noncommutative projective geometry to construct examples of algebras r over a field k not satisfying the following two types of symmetric behaviors of ext groups ee for any pair of finitely generated r modules m n dimkextri m n for all i n if and only if dimkextri n m for all i n ee for any pair of finitely generated r modules m n extri m n 0 for all i 0 if and only if extri n m 0 for all i 0 in particular and contrary to the commutative case we give a simple example of a noncommutative noetherian gorenstein frobenius local algebra satisfying uac uniform auslander condition but not satisfying ee