stratification for module categories of finite group schemes the second author was partly supported by nsf grant dms 1503044 the fourth author was partly supported by nsf grant dms 0953011

factoring the adjoint and maximal cohen macaulay modules over the generic determinant a question of bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices we show that such factorizations indeed exist over any coefficient ring when the matrix has even size establishing a correspondence between such factorizations and extensions of maximal cohen macaulay modules over the generic determinant we exhibit all factorizations where one of the factors has determinant equal to the generic determinant the classification shows not only that the cohen macaulay representation theory of the generic determinant is wild in the tame wild dichotomy but that it is quite wild even in rank two the isomorphism classes cannot be parametrized by a finite dimensional variety over the coefficients we further relate the factorization problem to the multiplicative structure of the mathop rm ext nolimits algebra of the two nontrivial rank one maximal cohen macaulay modules and determine it completely

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

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

classifying thick subcategories of the stable category of cohen macaulay modules abstract various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics in this paper as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p group due to benson carlson and rickard we consider classifying thick subcategories of the stable category of cohen macaulay modules over a gorenstein local ring the main result of this paper yields a complete classification of the thick subcategories of the stable category of cohen macaulay modules over a local hypersurface in terms of specialization closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus

periodic modules over gorenstein local rings abstract it is proved that the minimal free resolution of a module m over a gorenstein local ring r is eventually periodic if and only if the class of m is torsion in a certain z t 1 module associated to r this module denoted j r is the free z t 1 module on the isomorphism classes of finitely generated r modules modulo relations reminiscent of those defining the grothendieck group of r the main result is a structure theorem for j r when r is a complete gorenstein local ring the link between periodicity and torsion stated above is a corollary

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

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

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

multiplicatively regularized source reconstruction method for phaseless planar near field antenna measurements this paper approaches the source reconstruction method srm for phaseless planar near field nf antenna measurements by optimizing a multiplicatively regularized cost functional over the equivalent current distribution of the antenna under test aut the utilized multiplicative regularization scheme originally developed for image deblurring and inverse scattering problems is adapted to phaseless nf antenna measurements performed over two parallel planes in front of the aut current phaseless nf to far field ff transformation techniques are highly dependent on the accuracy of the initial phase guess and the pattern features of the aut the proposed multiplicatively regularized srm mr srm provides a robust and automated framework that can reconstruct the ff pattern of the aut based on a nonsophisticated initial phase guess in addition advantages of the srm are inherently incorporated into the mr srm such as the capability for antenna diagnostics and allowing for extension to arbitrary measurement domains without the need for data interpolation the cost functional associated with the mr srm is minimized with the conjugate gradient algorithm using closed form expressions for gradient operators the developed algorithm is presented in detail along with the synthetic and experimental examples demonstrating the method s performance in different measurement scenarios along with a comparison to alternative methods

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

planar near field to far field transformation using an equivalent magnetic current approach an alternative method is presented for computing far field antenna patterns from near field measurements the method utilizes the near field data to determine equivalent magnetic current sources over a fictitious planar surface that encompasses the antenna and these currents are used to ascertain the far fields under certain approximations the currents should produce the correct far fields in all regions in front of the antenna regardless of the geometry over which the near field measurements are made an electric field integral equation efie is developed to relate the near fields to the equivalent magnetic currents the method of moments is used to transform the integral equation into a matrix one the matrix equation is solved with the conjugate gradient method and in the case of a rectangular matrix a least squares solution for the currents is found without explicitly computing the normal form of the equation near field to far field transformation for planar scanning may be efficiently performed under certain conditions numerical results are presented for several antenna configurations

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

phaseless bi polar planar near field measurements and diagnostics of array antennas antenna near field measurements typically require very accurate measurement of the near field phase there are applications where an accurate phase measurement may not be practically achievable phaseless measurements are beginning to emerge as an alternative microwave antenna measurements technique when phase cannot be directly measured there are many important aspects for successful implementation of a phaseless measurement algorithm this paper presents appropriate phaseless measurement requirements and a phase retrieval algorithm tailored for the bi polar planar near field antenna measurement technique two amplitude measurements and a squared amplitude optimal sampling interpolation method are integrated with an iterative fourier procedure to first retrieve the phase information and then construct both the far field pattern and diagnostic characteristics of the antenna under test in order to critically examine the methodologies developed in this paper phaseless measurement results for two different array antennas are presented and compared to results obtained when the near field amplitude and phase are directly measured

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

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

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

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

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

reconstruction of equivalent currents distribution over arbitrary three dimensional surfaces based on integral equation algorithms a technique for the determination of the equivalent currents distribution from a known radiated field is described this inverse radiation problem is solved through an integral equation algorithm that allows the characterization of antennas of complex geometry both for near field to far field nf ff transformation purposes as well as for diagnostic tasks the algorithm is based on the representation of the radiating structure by means of a set of equivalent currents over a three dimensional 3d surface that can be fitted to the arbitrary geometry of the antenna the innovative formulation uses an integral equation involving the electric field due to the currents tangential components to the represented antenna 3d surface for that purpose both the magnetic and electric equivalent currents are considered in the integral equations regularization techniques are also introduced to improve the convergence of the proposed iterative solution the paper concludes with several results related to the practical verification of the equivalence principle and the characterization of a horn antenna

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

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

decent intersection and tor rigidity for modules over local hypersurfaces first published in transactions of the american mathematical society in 2013 published by the american mathematical society also available electronically from http www ams org journals tran 2013 365 06 s0002 9947 2012 05574 7 home html

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

tensor products of modules and the rigidity of tor let r be a hypersurface domain that is a ring of the form s f where s is a regular local ring and f is a prime element suppose m and n are finitely generated r modules we prove two rigidity theorems on the vanishing of tor in the first theorem we assume that the regular local ring s is unramified that m r n has finite length and that dim m dim n dim r with these assumptions if tor r m n 0 for some j 0 then tor r m n 0 for all i j the second rigidity theorem states that if m r n is reflexive then tor r m n 0 for all i 1 we use these theorems to prove the following theorem valid even if s is ramified if m r n is a maximal cohen macaulay r module then both m and n are maximal cohen macaulay modules and at least one of them is free

on data increasing in phase retrieval via quadratic inversion flattening manifold and local minima in this article we address the question of traps in phase retrieval via quadratic inversion first we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional later we recall a mathematical condition for the lack of traps afterward we illustrate how the ratio between the dimension of data space m and the number of real unknowns n r impacts on the manifold of data and consequently on the presence of local minima in particular we show that if the dimension of data space is high enough the manifold of data is sufficiently large and no traps appear in the functional in the last part of this article with reference to a particular unknown sequence we establish the minimum value of the ratio m n r such that no traps should appear in the functional in this condition phase retrieval via quadratic inversion exhibits a global convergence behavior consequently the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random

phantom depth and stable phantom exactness phantom depth phantom nonzerodivisors and phantom exact sequences are analogues of the non phantom notions which have been useful in tackling the very difficult localization problem in tight closure theory in the present paper these notions are developed further and partially reworked for instance although no analogue of a long exact sequence arises from a short stably phantom exact sequence of complexes we provide a method for recovering the kind of information obtainable from such a long sequence also we give alternate characterizations of the notion of phantom depth including one based on koszul homology which we use to show that with very mild conditions on a finitely generated module m any two maximal phantom m regular sequences in an ideal ihave the same length in order to do so we prove a nakayama lemma for tight closure which is of independent interest we strengthen the connection of phantom depth with minheight we explore several analogues of associated prime in tight closure theory and we discuss a connection with the problem of when tight closure commutes with localization

multilevel fast multipole accelerated inverse equivalent current method employing rao wilton glisson discretization of electric and magnetic surface currents the radiation of any object in a homogeneous environment can be described by a set of equivalent sources different types of equivalent sources are feasible and all of them have their own benefits equivalent current methods are especially advantageous for irregular measurement grids of arbitrary shape and if a priori information about the object shall be used also they can immediately provide diagnostic information about the object in this paper a very flexible equivalent current method is presented which has been derived from a general purpose boundary integral equation solver as such the method works with arbitrary triangular surface meshes and rao wilton glisson basis functions where electric and or magnetic surface current densities can be assumed high efficiency is achieved since the multilevel fast multipole method has been adapted to speed up the inverse solution process results obtained from simulations and measurements are presented where large scale problems with dimensions up to 75 wavelengths are considered

an effective near field far field transformation technique from truncated and inaccurate amplitude only data a general approach to the near field far field transformation from amplitude only near field data is presented the estimation of the far field is stated as an intersection finding problem and is solved by the minimization of a suitable functional the difficulties related to the possible trapping of the algorithm by a false solution common to any nonlinear inverse problem are mitigated by setting the problem in the space of the squared field amplitudes as already done in a number of existing papers and by incorporating all the a priori knowledge concerning the system under test in the formulation of the problem accordingly the a priori information concerning the far field the near field outside the measurement region and the accuracy of the measurement setup and its dynamic range are properly taken into account in the objective functional the intrinsic ill conditioning of the problem is managed by adopting a general flexible and nonredundant sampling representation of the field which takes into account the geometrical characteristics of the source as a consequence the number of unknowns is minimized and a technique is obtained which easily matches the available knowledge concerning the behavior of the field the effectiveness of the approach is shown by reporting the main results of an extensive numerical analysis as well as an experimental validation performed by using a very low cost near field facility available at the electronic engineering department university of napoli italy

the avrunin and scott theorem and a truncated polynomial algebra we prove an analogue of a theorem of avrunin and scott for truncated polynomial algebras m k x 1 x m x 2 i over an algebraically closed field of arbitrary characteristic the avrunin and scott theorem relates the support variety for a finite dimensional ke module to its rank variety where char k p and e is an elementary abelian p group the analogue of the avrunin and scott theorem relates the support variety for a finite dimensional m module using hochschild cohomology to its rank variety developed in k erdmann m holloway rank varieties and projectivity for a class of local algebras math z 247 2004 441 460 using clifford algebras along the way to proving our main result we provide a new proof of the avrunin and scott theorem for elementary abelian p group algebras which we are then able to generalise to the setting of m algebras

nonvanishing cohomology and classes of gorenstein rings abstract we give counterexamples to the following conjecture of auslander given a finitely generated module m over an artin algebra there exists a positive integer n m such that for all finitely generated modules n if ext i m n 0 for all i 0 then ext i m n 0 for all i n m some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local gorenstein rings

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

quantize and forward schemes for the orthogonal multiple access relay channel the multiple access relay channel with two sources a single relay and one destination is considered under the assumption of noisy source relay links causing the relay to be unable to decode without error we propose a framework for designing one and two dimensional quantizers for quantizing the soft information at the relay these quantizers are mutual information preserving simulation results show a that mutual information preserving quantization schemes outperform techniques in which the soft information is forwarded in an analog fashion to the destination b that two dimensional quantization outperforms one dimensional quantization for source relay links of different quality and c that diversity order of two can be gained in block rayleigh fading channels by having the relay adaptively select a two dimensional quantizer from a fixed set of quantizers shared with the destination depending on the channel state on the source relay links

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

cohomological symmetry in triangulated categories abstract we give a criterion for cohomological symmetry in a triangulated category as an application we show that such cohomological symmetry holds for all pairs of modules over any exterior algebra

singularity categories and singular equivalences for resolving subcategories let mathcal x be a resolving subcategory of an abelian category in this paper we investigate the singularity category mathsf d sg underline mathcal x mathsf d b mathsf mod underline mathcal x mathsf k b mathsf proj mathsf mod underline mathcal x of the stable category underline mathcal x of mathcal x we consider when the singularity category is triangle equivalent to the stable category of gorenstein projective objects and when the stable categories of two resolving subcategories have triangle equivalent singularity categories applying this to the module category of a gorenstein ring we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type mathsf a 1 we also generalize several results of yoshino on totally reflexive 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

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

the sources reconstruction method for amplitude only field measurements an extension of the sources reconstruction method srm for antenna diagnostics using amplitude only field measurements is presented while previous works limit the application to canonical domain cases the combination of the srm capabilities for handling with arbitrary geometry domains jointly with phase retrieval techniques allows antenna diagnostics extension to complex geometry antenna and field acquisition domains the consideration of the radiation inverse problem with a general integral equation formulation using arbitrary geometry field and currents domains and field phaseless information supposes a challenging ill posed problem that is solved using iterative minimization techniques for non linear problems different examples are presented from simple to general antenna diagnostics cases

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

symmetry theorems for ext vanishing abstract it was proved by avramov and buchweitz that if a is a commutative local complete intersection ring with finitely generated modules m and n then the ext groups between m and n vanish from some step if and only if the ext groups between n and m vanish from some step this paper shows that the same is true under the weaker conditions that a is gorenstein and that m and n have finite complete intersection dimension the result is also proved if a is gorenstein and has finite cohen macaulay type similar results are given for non commutative complete semi local algebras

radial field retrieval in spherical scanning for current reconstruction and nf ff transformation a near field to far field nf ff transformation is addressed for the case of spherical scanning using equivalent magnetic currents emcs and matrix methods it is based on the decoupling of the field components and the iterative retrieval of the radial component of the electric field the technique is applied for far field calculation as well as for the estimation of the current distribution of the antenna under test aut using spherical near field facilities results from measured near field data of several antennas are presented and compared to those of the analytical solution via a spherical wave mode expansion method

comparing complexities of pairs of modules abstract let r be a local ring and m n be finitely generated r modules the complexity of m n denoted by cx r m n measures the polynomial growth rate of the number of generators of the modules ext r n m n in this paper we study several basic equalities and inequalities involving complexities of different pairs of modules

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

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