embedding theorems and existence results for nonlinear schrodinger poisson systems with unbounded and vanishing potentials abstract motivated by existence results for positive solutions of non autonomous nonlinear schrodinger poisson systems with potentials possibly unbounded or vanishing at infinity we prove embedding theorems for weighted sobolev spaces we both consider a general framework and spaces of radially symmetric functions when assuming radial symmetry of the potentials

exotic elliptic algebras of dimension 4 abstract let e be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3 let be a translation automorphism of e that is not of order 2 in a previous paper we studied an algebra a a e that depends on this data a e s e m 2 k where s e is the 4 dimensional sklyanin algebra associated to e m 2 k is the ring of 2 2 matrices over k and is z 2 z 2 acting in a particular way as automorphisms of s and m 2 k the action of on s is compatible with the translation action of the 2 torsion subgroup e 2 on e following the ideas and results in papers of artin tate van den bergh smith stafford and levasseur smith this paper examines the line modules point modules and fat point modules over a and their incidence relations the right context for the results is non commutative algebraic geometry we view a as a homogeneous coordinate ring of a non commutative analogue of p 3 that we denote by proj n c a point modules and fat point modules determine points in proj n c a line modules determine lines in proj n c a line modules for a are in bijection with certain lines in p a 1 p 3 and therefore correspond to the closed points of a certain subscheme l of the grassmannian g 1 3 shelton vancliff call l the line scheme for a we show that l is the union of 7 reduced and irreducible components 3 quartic elliptic space curves and 4 plane conics in the ambient plucker p 5 and that deg l 20 the union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in p a 1 thus the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for a

segre product of artin schelter regular algebras of dimension 2 and embeddings in quantum p3 s the segre embedding of p1 p1as a smooth quadricqin p3corresponds to the surjection of the four dimensional polynomial ring onto the segre productsof two copies of the homogeneous coordinate ring of p1 we study segre products of noncommutative algebras if in particularaandbare two copies of a quantum p1thens i ai kbi is a twisted homogeneous coordinate ring of the quadricq the main result of this paper is the classification of all embeddings of the segre product of two quantum planes into so called quantum p3 s these are the proj of artin schelter regular algebrasrof global dimension four with the hilbert series of a commutative polynomial ring and which map ontos ifris not a twist of a polynomial ring then the point scheme ofreither is the union of the quadricqwith a line or is only the quadricq in the first case ris a central extension of a three dimensional artin schelter regular algebra and a twist of an algebra mapping onto the commutative homogeneous coordinate ring ofq in the second case such an algebraris the first known example of a four dimensional artin schelter regular algebra which is not determined by its point scheme

semiprime graded algebras of dimension two semiprime noetherian connected graded k algebras r of quadratic growth are described in terms of geometric data a typical example of such a ring is obtained as follows let y be a projective variety of dimension at most one over the base field k and let e be an oy order in a finite dimensional semisimple algebra a over k k y then for any automorphism of a that restricts to an automorphism of y and any ample invertible e bimodule b van den bergh constructs a noetherian twisted homogeneous coordinate ring b h0 y b b n 1 we show that r is noetherian if and only if some veronese ring r m of r has the form k i where i is a left ideal of such a ring b and where i b at each point p y at which has finite order this allows one to give detailed information about the structure of r and its modules

existence and concentration of positive solutions for a system of coupled saturable schrodinger equations abstract it is considered a saturable system of coupled schrodinger equations with asymptotically linear nonlinearities the bound states solutions with finite energy are proved to exist and have a concentration behavior under hypotheses involving the coupling and saturation parameters the proof is based on variational methods

central singularities of quantum spaces abstract let a be a positively graded connected affine c algebra generated in degree one which is a finite module over a central subring r let a be the structure sheaf over proj r and spec z the central proj if injdim a spec z has singularities unless the ramification locus of a is pure of codimension one if gldim a a and the singular locus of spec z coincide

projectively simple rings abstract an infinite dimensional n graded k algebra a is called projectively simple if dim k a i for every nonzero two sided ideal i a we show that if a projectively simple ring a is strongly noetherian is generated in degree 1 and has a point module then a is equal in large degree to a twisted homogeneous coordinate ring b b x l here x is a smooth projective variety is an automorphism of x with no proper invariant subvariety we call such automorphisms wild and l is a ample line bundle we conjecture that if x admits a wild automorphism then every irreducible component of x is an abelian variety we prove several results in support of this conjecture in particular we show that the conjecture is true if dim x 2 in the case where x is an abelian variety we describe all wild automorphisms of x finally we show that if a is projectively simple and admits a balanced dualizing complex then proj a is cohen macaulay and gorenstein

involutions of varieties and rost s degree formula to an algebraic variety equipped with an involution we associate a cycle class in the modulo two chow group of its fixed locus this association is functorial with respect to proper morphisms having a degree and preserving the involutions specialising to the exchange involution of the square of a complete variety we obtain rost s degree formula in arbitrary characteristic this formula was proved by rost and merkurjev in characteristic not two

cocycle twists of 4 dimensional sklyanin algebras abstract we study cocycle twists of a 4 dimensional sklyanin algebra a and a factor ring b which is a twisted homogeneous coordinate ring twisting such algebras by the klein four group g we show that the twists a g and b g have very different geometric properties to their untwisted counterparts for example a g has only 20 point modules and infinitely many fat point modules of multiplicity 2 the ring b g falls under the purview of artin and stafford s classification of noncommutative curves and we describe it using a sheaf of orders over an elliptic curve

large mass boundary condensation patterns in the stationary keller segel system abstract we consider the boundary value problem u u e u in u 0 on where is a bounded smooth domain in r 2 0 and is the inner normal derivative at this problem is equivalent to the stationary keller segel system from chemotaxis we establish the existence of a solution u which exhibits a sharp boundary layer along the entire boundary as 0 these solutions have large mass in the sense that e u log

on the first witt index of quadratic forms we prove hoffmann s conjecture determining the possible values of the first witt index of anisotropic quadratic forms of any given dimension the proof makes use of the steenrod type operations on the modulo 2 chow groups constructed by p brosnan

isotropie dʼune forme bilineaire dʼalbert sur le corps de fonctions dʼune quadrique en caracteristique 2 abstract in this paper we give a complete answer to the isotropy of an albert bilinear form over the function field of a quadric in characteristic 2 as a consequence we complete the classification of nongood bilinear forms of height and degree 2 given in laghribi and rehmann 2009 8

graded witt kernels of the compositum of multiquadratic extensions with the function fields of pfister forms abstract let f be a field of characteristic 2 and w q f be the witt group of nonsingular quadratic forms over f let be a bilinear pfister form over f and l be a multiquadratic extension of f of separability degree less than of equal to 2 in this paper we compute the kernel of the natural homomorphism h 2 m 1 f h 2 m 1 l where h 2 m 1 f is the cokernel of the artin schreier operator f m f m d f m 1 given by x d x 1 x 1 d x m x m x 2 x d x 1 x 1 d x m x m where f m is the space of m differential forms over f and f is the function field of the affine quadric given by the diagonal quadratic form associated to the bilinear form as a consequence we deduce the kernel of the natural homomorphisms i q m 1 f i q m 1 l and i q m 1 f i q m 1 l where i q m 1 f denotes the quotient i q m 1 f i q m 2 f such that i q m 1 f i m f w q f and i m f is the m th power of the fundamental ideal if of the witt ring of f bilinear forms we also include some results concerning the case where is replaced by a bilinear pfister neighbor or a quadratic pfister form

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

regularity of algebras related to the sklyanin algebra this paper continues the research of ss by finding further examples of artin schelter regular rings of dimension four unlike the threedimensional case studied in atv1 2 these examples show that the fourdimensional regular rings are not uniquely determined by their associated geometric data indeed we find a one parameter family of regular algebras associated to this data

on the generic splitting of quadratic forms in characteristic 2 in 9 and 10 knebusch established the basic facts of generic splitting theory of quadratic forms over a field of characteristic different from 2 this paper is related to 11 and 13 where knebusch and rehmann generalized partially this theory to a field of characteristic 2 more precisely we begin with a complete characterization of quadratic forms of height 1 we don t exclude anisotropic quadratic forms with quasi linear part of dimension at least 1 this allows us to extend the notion of degree to characteristic 2 we prove some results on excellent forms and splitting tower of a quadratic form some results on quadratic forms of height 2 and degree 1 or 2 are given

classification of quadratic forms over skew fields of characteristic 2 quadratic forms over division algebras over local or global fields of characteristic 2 are classified by an invariant derived from the clifford algebra construction

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

wild pfister forms over henselian fields k theory and conic division algebras the epicenter of this paper concerns pfister quadratic forms over a field f with a henselian discrete valuation all characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but f does not we also prove results about round quadratic forms composition algebras generalizations of composition algebras we call conic algebras and central simple associative symbol algebras finally we give relationships between these objects and kato s filtration on the milnor k groups of f

essential dimension of quadrics let x be an anisotropic projective quadric over a field f of characteristic not 2 the essential dimension dimes x of x a s def ined by oleg izhboldin is dimes x dim x i x 1 where i x is the first witt index of x i e the witt index of x over its function field let y be a complete possibly singular algebraic variety over f with all closed points of even degree and such that y has a closed point of odd degree over f x our main theorem states that dimes x dim y and that in the case dimes x dim y the quadric x is isotropic over f y applying the main theorem to a projective quadric y we get a proof of izhboldin s conjecture stated as follows if an anisotropic quadric y becomes isotropic over f x t hen dim es x dimes y and the equality holds if and only if x is isotropic over f y we also solve knebusch s problem by proving that the smallest transcendence degree of a generic splitting field of a quadric x is equal to dimes x let v be a non degenerate quadratic form of dimension at least 2 over a field f of characteristic not 2 and let x q be the quadric hypersurface given by the equation x 0 in the projective space p v we say that the quadric x is anisotropic if is an anisotropic quadratic form by springer s theorem every closed point of an anisotropic quadric x has

constructing clifford quantum p3s with finitely many points abstract we present an algebro geometric technique for constructing regular clifford algebras a of global dimension four with associated point scheme consisting of a prespecified finite number of points in particular if a has more than one point in its point scheme then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors these cubic divisors correspond to regular clifford subalgebras of a of global dimension three if a has exactly a finite number n of distinct points in its point scheme then n 1 3 4 5 13 14 16 18 20 and all these possibilities occur we also prove that if a regular clifford algebra r of global dimension d 2 has exactly a finite number n of distinct isomorphism classes of point modules then n is odd if and only if r is an ore extension of a regular clifford subalgebra of r of global dimension d 1

positive solutions of asymptotically linear equations via pohozaev manifold abstract we present a new method for finding positive solutions of nonlinear elliptic equations which are non homogeneous and asymptotically linear at infinity by using projections on a pohozaev manifold rather than the nehari manifold associated with the problem

singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities abstract we consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities the dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect we investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function necessary conditions to locate the possible concentration points are also given

homologically homogeneous rings in this paper we study the structure of a right noetherian ring r of finite right global dimesion integral over a central subring c and satisfying the following condition if v w are irreducible right r modules with rc v rc w then pr dim v

on koszul algebras and a new construction of artin schelter regular algebras abstract we prove the simple fact that the factor ring of a koszul algebra by a regular normal quadratic element is a koszul algebra this fact leads to a new construction of quadratic artin schelter regular algebras this construction generalizes the construction of artin schelter regular clifford algebras

on the center of 3 dimensional and 4 dimensional sklyanin algebras abstract in this article a new proof is given of the description of the center of quadratic sklyanin algebras of global dimension three and four and the center of cubic sklyanin algebras of global dimension three the representation theory of the heisenberg groups h 2 h 3 and h 4 will play an important role in addition a new proof is given of van den bergh s result regarding noncommutative quadrics

quantum deformations of projective three space abstract we describe the possible noncommutative deformations of complex projective three space by exhibiting the calabi yau algebras that serve as their homogeneous coordinate rings we prove that the space parametrizing such deformations has exactly six irreducible components and we give explicit presentations for the generic members of each family in terms of generators and relations the proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic poisson structures in four dimensions which we extract from cerveau and lins neto s classification of degree two foliations on projective space corresponding to the exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical schwarzenberger bundles

the behavior of quadratic and differential forms under function field extensions in characteristic two abstract let f be a field of characteristic 2 let n f be the f space of absolute differential forms over f there is a homomorphism n f n f d n 1 f given by x d x 1 x 1 d x n x n x 2 x d x 1 x 1 d x n x n mod d f n 1 let h n 1 f coker we study the behavior of hn 1 f under the function field f f where b1 bn is an n fold pfister form and f is the function field of the quadric 0 over f we show that ker h n 1 f h n 1 f f d b 1 b 1 d b n b n using kato s isomorphism of hn 1 f with the quotient inwq f in 1wq f where wq f is the witt group of quadratic forms over f and i w f is the maximal ideal of even dimensional bilinear forms over f we deduce from the above result the analogue in characteristic 2 of knebusch s degree conjecture i e inwq f is the set of all classes q with deg q n

quadratic forms and pfister neighbors in characteristic 2 we study pfister neighbors and their characterization over fields of characteristic 2 where we include the case of singular forms we give a somewhat simplified proof of a theorem of fitzgerald which provides a criterion for when a nonsingular quadratic form q is similar to a pfister form in terms of the hyperbolicity of this form over the function field of a form which is dominated by q from this we derive an analogue in characteristic 2 of a result by knebusch saying that in characteristic 2 a form is a pfister neighbor if its anisotropic part over its own function field is defined over the base field our result includes certain cases of singular forms but we also give examples which show that knebusch s result generally fails in characteristic 2 for singular forms as an application we characterize certain forms of height 1 in the sense of knebusch whose quasi linear parts are of small dimension we also develop some of the basics of a theory of totally singular quadratic forms this is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper

indecomposable p algebras and galois subfields in generic abelian crossed products abstract let f be a henselian valued field with char f p and d a semi ramified not strongly degenerate p algebra we show that all galois subfields of d are inertial using this as a tool we study generic abelian crossed product p algebras proving among other things that the noncyclic generic abelian crossed product p algebras defined by non degenerate matrices are indecomposable p algebras to construct examples of these indecomposable p algebras with exponent p and large index we study the relationship between degeneracy in matrices defining abelian crossed products and torsion in ch 2 of severi brauer varieties

powers of the fundamental ideal in the witt ring abstract let k be a field of characteristic different from 2 in the algebraic theory of quadratic forms one studies the witt ring w k of equivalence classes of non degenerate quadratic forms the witt ring has a filtration given by the powers in k of the fundamental ideal i k of even dimensional forms the ideal in k is generated by the set pn k of n fold pfister forms a1 an ni 1 1 ai where ai k k 0 many questions about this filtration and its quotients have arisen in the study of w k the spectacular work of voevodsky together with his collaborative work with orlov and vishik allows one to answer many old questions the purpose of this paper is to indicate some of these solutions

down up algebras and their representation theory abstract a class of algebras called down up algebras was introduced by g benkart and t roby 1998 j algebra 209 305 344 we classify the finite dimensional simple modules over noetherian down up algebras and show that in some cases every finite dimensional module is semisimple we also study the question of when two down up algebras are isomorphic

on a class of asymptotically linear elliptic problems in rn under these conditions p is an asymptotically linear problem when this equation is considered in a bounded domain 0 r with say the dirichlet boundary condition there is a large literature on existence and multiplicity results of particular interest is then the case of resonance where s and s is the asymptotic linearization of the problem in other words s d s l 0 l 0 is the operator given by

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

variational elliptic problems which are nonquadratic at infinity p co if n 1 2 then it is well known that the weak solutions u e h 2 of p are the critical points of the c functional j u ulvul dx jcf x u dx where f x s jif x t dt in this variational setting the literature usually distinguishes between two situations for prob lem p the subquadratic situation where the potential f satisfies lim suplsl f x s s2 i c m and the superquadratic situation where f satisfies limlsl m f x s s2 00 the main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on f more precisely our hypotheses are as follows

existence of solutions for a class of nonlinear schrodinger equations with potential vanishing at infinity abstract in this paper we investigate the existence of positive ground state solution for the following class of elliptic equations u v x u k x f u in r n where n 3 v k are nonnegative continuous functions and f is a continuous function with a quasicritical growth here we prove a hardy type inequality and use it together with variational method to get a ground state solution

quasilinear quadratic forms and function fields of quadrics let p and q be anisotropic quadratic forms of dimension ge 2 over a field f in a recent article we formulated a conjecture describing a general constraint which the dimensions of p and q impose on the isotropy index of q after scalar extension to the function field of p this can be viewed as a generalization of hoffmann s separation theorem which simultaneously incorporates and refines some well known classical results on the witt kernels of function fields of quadrics using algebro geometric methods it was shown that large parts of this conjecture hold in the case where the characteristic of f is not 2 in the present article we prove similar in fact somewhat sharper results in the case where f has characteristic 2 and q is a so called quasilinear form in contrast to the situation where mathrm char f ne 2 the methods used to treat this case are purely algebraic

homological properties of sklyanin algebras to a pair consisting of an elliptic curve and a point on it odeskii and feigin associate certain quadratic algebras sklyanin algebras having the hilbert series of a polynomial algebra in this paper we show that sklyanin algebras have good homological properties and we obtain some information about their so called linear modules we also show how the construction by odeskii and feigin may be generalized so as to yield other sklyanin type algebras

irreducible representations of the 4 dimensional sklyanin algebra at points of infinite order abstract in 1982 sklyanin funct anal appl 16 1982 27 34 defined a family of graded algebras a e depending on an elliptic curve e and a point e which is not 4 torsion basic properties of these algebras were established in smith and stafford compositio math 83 1992 259 289 and a study of their representation theory was begun in levasseur and smith bull soc math france121 1993 35 90 the present paper classifies the finite dimensional simple a modules when is a point of infinite order sklyanin funct anal appl 17 1983 273 284 defines for each k n a representation of a in a certain k dimensional subspace of theta functions of order 2 k 1 we prove that these are irreducible representations and that any other simple module is obtained by twisting one of these by an automorphism of a the automorphism group of a is explicitly computed the method of proof relies on results in levasseur and smith in particular it is proved that every finite dimensional simple module is a quotient of a line module an important part of the analysis is a determination of the 1 critical a modules and the fact that such a module is equivalent to a quotient of a line module by a shifted line module

noncommutative graded domains with quadratic growth letk be an algebraically closed field and letr be a finitely generated connected gradedk algebra which is a domain of gelfand kirillov dimension two write the graded quotient ringq r ofr asd z z 1 for some automorphism of the division ringd we prove thatd is a finitely generated field extension ofk of transcendence degree one moreover we describer in terms of geometric data ifr is generated in degree one then up to a finite dimensional vector space r is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curvey this implies in particular thatr is noetherian thatr is primitive when and thatr is a finite module over its centre when ifr is not generated in degree one thenr will still be noetherian and primitive if has infinite order butr need not be noetherian when has finite order

symmetry results for semilinear elliptic systems in the whole space in this paper we obtain symmetry results of gidas ni nirenberg type for cooperative nonlinear elliptic systems in the whole space

on totally decomposable algebras with involution in characteristic two abstract a necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution in terms of its frobenius subalgebras is given it is also proved that a bilinear pfister form recently introduced by a dolphin can classify totally decomposable central simple algebras of orthogonal type

on the noncommutative geometry of square superpotential algebras a superpotential algebra is square if its quiver admits an embedding into a two torus such that the image of its underlying graph is a square grid possibly with diagonal edges in the unit squares examples are provided by dimer models in physics such an embedding reveals much of the algebras representation theory through a device we introduce called an impression let a be a square superpotential algebra z its center and m the maximal ideal at the origin of spec z using an impression we give a classification of all simple a modules up to isomorphism and give algebraic and homological characterizations of the simple a modules of maximal k dimension show that z is a 3 dimensional normal toric domain and zm is gorenstein by determining transcendence bases and z regular sequences and show that am is a noncommutative crepant resolution of zm and thus a local calabi yau algebra a particular class of square superpotential algebras the yp q algebras is considered in detail we show that the azumaya and smooth loci of the centers coincide and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size

on involutions of type o q k over a field of characteristic two abstract in this article we study the involutions of o v q an orthogonal group for a vector space v with quadratic form q over a field of characteristic two the classification proceeds by discussing conjugacy classes of involutions arising as a product of transvections involutions with respect to a hyperbolic space and involutions acting nontrivially in the radical of v we achieve a complete classification of the conjugacy classes of involutions when the quadratic space v q is non defective and conclude with a discussion of the defective case

on bilinear forms of height 2 and degree 1 or 2 in characteristic 2 our aim in this paper is to complete some results given in a laghribi sur le deploiement des formes bilineaires en caracteristique 2 pacific j math 232 2007 207 232 on the classification of symmetric bilinear forms of height 2 and degree d 1 or 2 over fields of characteristic 2 i e those whose anisotropic parts over their own function fields are similar to d fold bilinear pfister forms

isotropy of quadratic forms over the function field of a quadric in characteristic 2 abstract we extend to characteristic 2 a theorem by the first author which states that if and are anisotropic quadratic forms over a field f such that dim 2 n dim for some nonnegative integer n then stays anisotropic over the function field f of the case of singular forms is systematically included we give applications to the characterization of quadratic forms with maximal splitting we also prove a characteristic 2 version of a theorem by izhboldin on the isotropy of over f in the case dim 2 n 1 dim

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

generating graded central simple algebras abstract we solve the generator problem for z graded central simple algebras applications are given to automorphisms of trace rings of generic matrices and to periodic fat point modules

les formes bilineaires et quadratiques bonnes de hauteur 2 en caracteristique 2 our aim is to give a complete classification of bilinear and quadratic forms which are good of height 2 over a field of characteristic 2 i e those whose anisotropic parts over their function fields are similar to bilinear pfister forms which are definable over the ground field we include other related results

totally decomposable quadratic pairs in this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic pair is adjoint to a quadratic pfister form this result is new in characteristic two otherwise it is equivalent to the pfister factor conjecture proven in becher invent math 173 1 6 2008

generalised quadratic forms and the u invariant abstract the u invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field we define corresponding u invariants for hermitian and generalised quadratic forms over a division algebra with involution in characteristic 2 and investigate the relationships between them we investigate these invariants in particular in the case of a quaternion algebra and further when this quaternion algebra is the unique quaternion division algebra over a field

canonical maps to twisted rings if a is a strongly noetherian graded algebra generated in degree one then there is a canonically constructed graded ring homomorphism from a to a twisted homogeneous coordinate ring open image in new window which is surjective in large degree this result is a key step in the study of projectively simple rings the proof relies on some results concerning the growth of graded rings which are of independent interest