representation dimension and finitely generated cohomology abstract we consider selfinjective artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators for such an algebra we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules this provides a unified approach to computing lower bounds for the representation dimension of group algebras exterior algebras and artin complete intersections we also obtain new examples of classes of algebras with arbitrarily large representation dimension

complexity and tor on a complete intersection let r m be a complete intersection that is a local ring whose m adic completion is the quotient of a regular local ring by a regular sequence supposemis a finitely generatedr module it is known that the even and odd betti sequences ofmare eventually given by polynomials of the same degreen the complexity ofmis the nonnegative integern 1 we use this notion of complexity to study the vanishing of torri m n for finitely generated modulesmandnover a complete intersectionr we prove several theorems dealing with rigidity of tor which are generalizations and in certain situations improvements of known results the main idea of these rigidity theorems is that the number of consecutive vanishing tors required in the hypothesis of a rigidity theorem depends more on the minimum of the complexities ofmandnrather than on the codimension ofr we give examples showing that this dependence is sharp we also show that ifm rnhas finite length then for sufficiently high indices two consecutive vanishing tors force the vanishing of all higher tors

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

noise reduction in surface reconstruction from a given gradient field we present a gradient space technique for noise reduction in surfaces reconstructed from a noisy gradient field we first analyze the error sources in the recovered gradient field of a surface using a three image photometric stereo method based on this analysis we propose an additive noise model to describe the errors in the surface gradient estimates we then use a vector space formulation and construct a multiscale orthonormal expansion for gradient fields using the sparse representation properties of this expansion we develop techniques for reducing the gradient field noise by coefficient selection with thresholding the simulation results indicate that the proposed technique provides significant improvement on the noise levels of both the estimated gradient fields and the reconstructed surfaces under heavy noise levels furthermore the experiments using noisy photometric stereo image triplets of real range data suggest that the additive model remains viable after the nonlinear photometric stereo operation to provide accurate noise removal

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

syzygies and tensor product of modules we give an application of the new intersection theorem and prove the following let r be a local complete intersection ring of codimension c and let m and n be nonzero finitely generated r modules assume n is a nonnegative integer and that the tensor product m otimes r n is an n c th syzygy of some finitely generated r module if mathrm tor r 0 m n 0 then both m and n are n th syzygies of some finitely generated r modules

a simplified derivation of the geroch group in two dimensional reduced gravity by generalizing our previous treatment of hidden symmetry in two dimensional chiral models a simple and explicit approach is proposed to the geroch group for the vacuum einstein equations with the metric tensor depending only on two variables two dimensional reduced gravity an infinite number of infinitesimal transformations for the metric tensor preserving the equations of motion are summarized by an explicit parametric transformation and the commutators among them are calculated in a simple and straightforward way these transformations and their group structure are further identified to be those of the well known geroch group

existence and uniqueness in photometric stereo we analyze the problem of recovering the shape of a smooth lambertian surface from two images obtained by consecutive illumination of the surface by distant point light sources from two different directions we discuss the issues of existence and uniqueness for the corresponding system of two first order nonlinear partial differential equations we show that generically this system has up to a constant a unique solution we also examine exceptional cases for which there is no such uniqueness

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

equivariant isomorphisms of ext and tor modules abstract in this article we establish equivariant isomorphisms of ext and tor modules over different relative complete intersections more precisely for a commutative ring q this paper investigates how ext q f m n and tor q f m n change when one varies f among all koszul regular sequences of a fixed length such that f m 0 and f n 0 of notable interest is how the theory of perturbations is used to establish isomorphisms of certain dg modules

g dimension complete intersection dimension complexity and relative homological algebra abstract it is shown that the g dimension and the complete intersection dimension are relative projective dimensions relative auslander buchsbaum formulas are discussed new cohomology theories called complexity cohomology are constructed the new theories play the same role in identifying rings and modules with prescribed complexity as tate vogel cohomology does in identifying modules of finite projective dimension

support varieties an axiomatic approach we provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category where the tensor product is not necessarily symmetric this is illustrated by examples taken in particular from the representation theory of finite dimensional algebras

impossible and ambiguous shading patterns a smooth object depicted in a monochrome image will often exhibit brightness variation or shading a problem much studied in computer vision has been that of how object shape may be recovered from image shading when the imaging conditions are such that an overhead point source illuminates a smooth lambertian surface the problem may be formulated as that of finding a solution to an eikonal equation this article will focus on the existence and uniqueness of such solutions reporting recent results obtained with regard to existence shading patterns are exhibited for which there is no corresponding object shape specifically a necessary and sufficient condition is presented for a circularly symmetric eikonal equation to admit exclusively unbounded solutions additionally a sufficient condition is given for an eikonal equation to have no solution whatsoever in connection with uniqueness we consider eikonal equations defined over a disc such that the euclidean norm of the gradient of any solution is circularly symmetric vanishes exactly at the disc center and diverges to infinity as the circumference of the disc is approached contrary to earlier influential work a class of such eikonal equations is shown to possess simultaneously circularly symmetric and noncircularly symmetric bounded smooth solutions

cohomology of twisted tensor products abstract it is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies we look at twisted tensor products and investigate to which extent this is still true we give an explicit description of the ext algebra of the tensor product of two modules and under certain additional conditions describe an essential part of the hochschild cohomology ring of a twisted tensor product as an application we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the hochschild cohomology ring moreover both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra

the set of semidualizing complexes is a nontrivial metric space abstract we show that the set s r of shift isomorphism classes of semidualizing complexes over a local ring r admits a nontrivial metric we investigate the interplay between the metric and several algebraic operations motivated by the dagger duality isometry we prove the following if k l are homologically bounded below and degreewise finite r complexes such that k r l k r l l is semidualizing then k is shift isomorphic to r in investigating the existence of nontrivial open balls in s r we prove that s r contains elements that are not comparable in the reflexivity ordering if and only if it contains at least three distinct elements

modules with reducible complexity abstract for a commutative noetherian local ring we define and study the class of modules having reducible complexity a class containing all modules of finite complete intersection dimension various properties of this class of modules are given together with results on the vanishing of homology and cohomology

on the growth of the betti sequence of the canonical module we study the growth of the betti sequence of the canonical module of a cohen macaulay local ring it is an open question whether this sequence grows exponentially whenever the ring is not gorenstein we answer the question of exponential growth affirmatively for a large class of rings and prove that the growth is in general not extremal as an application of growth we give criteria for a cohen macaulay ring possessing a canonical module to be gorenstein

on complete integrals and uniqueness in shape from shading in this paper we analyze the problem of representing solutions of a first order partial differential equation in terms of complete integrals we show that some of the existing results referring to the representability problem are incomplete additionally we analyze in the context of complete integrals the uniqueness problem for the shape recovery of a smooth lambertian surface from an image obtained by illuminating this surface by an overhead distant point light source specifically we revisit the uniqueness results already existing in the shape from shading literature that concern eikonal equations corresponding to the images of a lambertian hemisphere and a lambertian plane we show that the latter results are incomplete and indicate how to fill the gaps in the corresponding proofs

1 torsion of finite modules over semiperfect rings abstract we initiate the study of 1 torsion of finite modules over two sided noetherian semiperfect rings in particular we give a criterion for determining when the 1 torsion submodule contains minimal generators of the module we also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module some of the obtained results hold without the noetherian assumption we also give several applications to local algebra

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

homology of perfect complexes abstract it is proved that the sum of the loewy lengths of the homology modules of a finite free complex f over a local ring r is bounded below by a number depending only on r this result uncovers in the structure of modules of finite projective dimension obstructions to realizing r as a closed fiber of some flat local homomorphism other applications include as special cases uniform proofs of known results on free actions of elementary abelian groups and of tori on finite cw complexes the arguments use numerical invariants of objects in general triangulated categories introduced here and called levels they allow one to track through changes of triangulated categories homological invariants like projective dimension as well as structural invariants like loewy length an intermediate result sharpens with a new proof the new intersection theorem for commutative algebras over fields under additional hypotheses on the ring r stronger estimates are proved for loewy lengths of modules of finite projective dimension

some remarks on the shape from shading problem in computer vision the shape from shading problem in computer vision see horn 11 deals with the reconstruction of the shape of an object from a two dimensional image we consider an object illuminated by a point source the reflected light being recorded on a photographic plate at a fixed position the intensity of the reflected light is a function of the position of the source the position of the plate and the normal to the surface of the object if the surface is given as u u x jj this leads see horn i to a first order partial differential equation of the form

homological algebra modulo a regular sequence with special attention to codimension two abstract let m be a finite module over a ring r obtained from a commutative ring q by factoring out an ideal generated by a regular sequence the homological properties m over r and over q are intimately related their links are analyzed here from the point of view of differential graded homological algebra over a koszul complex that resolves r over q one outcome of this approach is a transparent derivation of some central results of the theory another is a new insight into codimension two phenomena yielding an explicit finitistic construction of the generally infinite minimal r free resolution of m it leads to theorems on the structure and classification of finite modules over codimension two local complete intersections that are exact counterparts of eisenbud s results for modules over hypersurfaces

asymptotic behavior of ext functors for modules of finite complete intersection dimension let r be a local ring and let m and n be finitely generated r modules such that m has finite complete intersection dimension in this paper we define and study under certain conditions a pairing using the modules rm ext r i m n which generalizes buchweitz s notion of the herbrand difference we exploit this pairing to examine the number of consecutive vanishing of rm ext r i m n needed to ensure that rm ext r i m n 0 for all i gg 0 our results recover and improve on most of the known bounds in the literature especially when r has dimension at most two

growth of hilbert coefficients of syzygy modules abstract let a m be a local complete intersection ring of dimension d and let i be an m primary ideal let m be a maximal cohen macaulay a module for i 0 1 d let e i i m denote the i th hilbert coefficient of m with respect to i we prove that for i 0 1 2 the function j e i i syz j a m is of quasi polynomial type with period 2 let g i m be the associated graded module of m with respect to i if g i a is cohen macaulay and dim a 2 we also prove that the functions j depth g i syz 2 j i a m are eventually constant for i 0 1 let i m lim l depth g i l m finally we prove that if dim a 2 and g i a is cohen macaulay then the functions j i syz 2 j i a m are eventually constant for i 0 1

huneke ulrich almost complete intersections of cohen macaulay type two abstract let r m k be a commutative noetherian local ring n 2 an integer x a 2n 1 2n 1 alternating matrix with entries from m y a 1 2n 1 matrix with entries from m i the ideal i i1 yx and a the quotient ring r i assume that the grade of i is at least 2n in this case i is a perfect ideal of grade equal to 2n and i is minimally generated by 2n 1 elements we prove that the minimal resolution of a by free r modules is a dg algebra furthermore we identify the algebra torr a k and prove that if r is regular and char k 0 or n 2 2 char k then the poincare series pma z i 0 dimk torai m k zi is a rational function for every finitely generated a module m as a consequence we deduce that if the projective dimension of m is infinite then eventually the betti numbers of m form an increasing sequence with strong exponential growth

free resolutions and change of rings abstract projective resolutions of modules over a ring r are constructed starting from appropriate projective resolutions over a ring q mapping to r it is shown that such resolutions may be chosen to be minimal in codimension 2 but not in codimension 3 this is used to obtain minimal resolutions for essentially all modules over local or graded rings r with codimension 2 explicit resolutions are given for cyclic modules over multigraded rings and necessary and sufficient conditions are obtained for their minimality

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

vanishing theorems for complete intersections the starting point for this work is auslander s seminal paper a the main focus of his paper was to understand when the tensor product m rn of finitely generated modules m and n over a regular local ring r is torsion free this condition forces the vanishing of a certain tor module associated to m and n which in turn by auslander s famous rigidity theorem implies that tori m n 0 for all i 1 the vanishing of tor carries a great deal of information for example it implies the depth formula depth m depth n dim r depth m r n from this formula one can deduce for example the highly non trivial fact that if m r n is torsion free and non zero then m and n must both be torsion free in hw huneke and wiegand generalized some of auslander s results to hypersurfaces though with some unavoidable extra hypotheses in particular they proved a rigidity theorem hw 2 4 and the following vanishing theorem

on the moyal deformation of nahm equations in seven dimensions we show how the reduced anti self dual yang mills equations to seven dimensions described by the nahm equations can be carried over to the weyl wigner moyal formalism in the process some new solutions for the cases of gauge groups su 2 and sl 2 r are explicitly obtained

the derived category of a locally complete intersection ring abstract in this paper we answer a question of dwyer greenlees and iyengar by proving a local ring r is a complete intersection if and only if every complex of r modules with finitely generated homology is proxy small moreover we establish that a commutative noetherian ring r is locally a complete intersection if and only if every complex of r modules with finitely generated homology is virtually small

height and gradient from shading the method described here for recovering the shape of a surface from a shaded image can deal with complex wrinkled surfaces integrability can be enforced easily because both surface height and gradient are represented the robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell the new scheme can find an exact solution of a given shape from shading problem even though a regularizing term is included this is a reflection of the fact that shape from shading problems are it not ill posed when boundary conditions are available or when the image contains singular points

new invariants of noetherian local rings abstract we investigate properties of certain invariants of noetherian local rings including their behavior under flat local homomorphisms we show that these invariants are bounded by the multiplicity for cohen macaulay local rings with infinite residue fields and they all agree with the multiplicity when such rings are hypersurfaces we also show that these invariants are all equal to 2 for a non regular cohen macaulay local ring a if and only if a has a minimal multiplicity provided its residue field is infinite

reduced bass numbers auslander s δ invariant and certain homological conjectures in the category of finitely generated modules over a local cohen macaulay ring with dualizing module one can following auslander buchweitz consider maximal cohen macaulay approximations and hulls of finite injective dimension the former give rise to y and for a gorenstein ring invariants the latter to vd invariants which are in some sense dual this last type of invariant has in the guise of what we call a reduced bass number been encountered in the study of the canonical element conjecture cec by hochster in this paper we establish the basic properties of these invariants some known some not we go on to explore the connection with cec the latter is through linkage seen to imply a fact about delta invariants this in turn shows that cec guarantees the existence of certain maximal cohen macaulay modules over a gorenstein ring

jacobian criteria for complete intersections the graded case letp be a positively graded polynomial ring over a fieldk of characteristic zero leti be a homogeneous ideal ofp and setr p i the paper investigates the homological properties of somer modules canonically associated withr among them the module r k of kahler differentials and the conormal modulei i 2 it is shown that a subexponential bound on the betti numbers of any of these modules implies thati is generated by ap regular sequence in particular the finiteness of the projective dimension of the conormal module impliesr is a complete intersection similarly the finiteness of the projective dimension of the differential module impliesr is a reduced complete intersection this provides strong converses to some well known properties of complete intersections and establishes special cases of conjectures of vasconcelos the proofs of these results make extensive use of differential graded homological algebra the crucial step is to show that any homomorphism of complexes from the minimal cotangent complexl r k of andre and quillen into the minimal free resolution of the irrelevant maximal ideal m ofr which extends the euler map r k is a split embedding of gradedr modules

complexity and elementary abelian p groups one of the main differences between ordinary and modular representation theory of finite groups is the existence of nonprojective modules in the modular representation algebra however given a finitely generated module m over the group algebra kg where k is a field of characteristic p 0 there exists a uniquely determined up to kg isomorphism projective module p called the projective cover of m and an epimorphism f

bi differential calculi and integrable models the existence of an infinite set of conserved currents in completely integrable classical models including chiral and toda models as well as the kp and self dual yang mills equations is traced back to a simple construction of an infinite chain of closed respectively covariantly constant 1 forms in a gauged bi differential calculus the latter consists of a differential algebra on which two differential maps act in a gauged bi differential calculus these maps are extended to flat covariant derivatives

boundedness versus periodicity over commutative local rings over commutative graded local artinian rings examples are constructed of periodic modules of arbitrary minimal period and modules with bounded betti numbers which are not eventually periodic they provide counterexamples to a conjecture of d eisenbud that every module with bounded betti numbers over a commutative local ring is eventually periodic of period 2 it is proved however that the conjecture holds over rings of small length

a connection between the einstein and yang mills equations it is our purpose here to show an unusual relationship between the einstein equations and the yang mills equations we give a correspondence between solutions of the self dual einstein vacuum equations and the self dual yang mills equations with a special choice of gauge group the extension of the argument to the full yang mills equations yields einstein s unifield equations we try to incorporate the full einstein vacuum equations but the approach is incomplete we first consider yang mills theory for an arbitrary lie algebra with the condition that the connection 1 form and curvature are constant on minkowski space this leads to a set of algebraic equations on the connection components we then specialize the lie algebra to be the infinite dimensional lie algebra of a group of diffeomorphisms of some manifold the algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act in the self dual case if we choose the connection components from the lie algebra of the volume preserving 4 dimensional diffeomorphism group the resulting equations are the same as those obtained by ashtekar jacobsen and smolin in their remarkable simplification of the self dual einstein vacuum equations an alternative derivation of the same equations begins with the self dual yang mills connection now depending only on the time then choosing the lie algebra as that of the volume preserving 3 dimensional diffeomorphisms when the reduced full yang mills equations are used in the same context we get einstein s equations for his unified theory based on absolute parallelism to incorporate the full einsteinvacuum equations we use as the lie group the semi direct product of the diffeomorphism group of a 4 dimensional manifold with the group of frame rotations of anso 1 3 bundle over the 4 manifold this last approach however yields equations more general than the vacuum equations

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

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

three themes of syzygies we present three exciting themes of syzygies where major progress was made recently boij soderberg theory stillman s question and syzygies over complete intersections

tor and torsion on a complete intersection abstract let r m be a complete intersection that is a ring whose m adic completion is the quotient of a regular local ring by a regular sequence supposemandnare finitely generatedr modules we give a necessary condition for the vanishing of torri m n for alli 0 in terms of the intersection of certain affine algebraic sets associated tomandn we apply this condition to the study of torsion in tensor products for example we show that ifris a domain andmis anr module of infinite projective dimension then there exist infinitely manynfor which the tensor product ofmwith one of itsnth syzygy modules has torsion we also give a sufficient condition for the vanishing torri m n for alli 0 in terms of the ability to leftmandnto disjoint complete intersections of smaller codimension we use this condition to construct tensor products of non free modules which are maximal cohen macaulay

the affine symmetry of self dual gravity self dual gravity may be reformulated as the two dimensional principal chiral model with the group of area preserving diffeomorphisms as its gauge group using this formulation it is shown that self dual gravity contains an infinite dimensional hidden symmetry whose generators form the affine kac moody algebra associated with the lie algebra of area preserving diffeomorphisms this result provides an observable algebra and a solution generating technique for self dual gravity