deformation theory of objects in homotopy and derived categories iii abelian categories abstract this is the third paper in a series in part i we developed a deformation theory of objects in homotopy and derived categories of dg categories here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories then we consider examples from noncommutative algebraic geometry in particular we study noncommutative grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces

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

maps between non commutative spaces let j be a graded ideal in a not necessarily commutative graded k algebra a a 0 a 1 in which dim k a i for all i we show that the map a a j induces a closed immersion i proj nc a j proj nc a between the non commutative projective spaces with homogeneous coordinate rings a and a j we also examine two other kinds of maps between non commutative spaces first a homomorphism a b between not necessarily commutative n graded rings induces an affine map proj nc b u proj nc a from a non empty open subspace u c proj nc b second if a is a right noetherian connected graded algebra not necessarily generated in degree one and a n is a veronese subalgebra of a there is a map proj nc a proj nc a n we identify open subspaces on which this map is an isomorphism applying these general results when a is a quotient of a weighted polynomial ring produces a non commutative resolution of a closed subscheme of a weighted projective space

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

the geometry of arithmetic noncommutative projective lines abstract let k be a perfect field and let k k be a finite extension of fields an arithmetic noncommutative projective line is a noncommutative space of the form proj s k v where v be a k central two sided vector space over k of rank two and s k v is the noncommutative symmetric algebra generated by v over k defined by m van den bergh 26 we study the geometry of these spaces more precisely we prove they are integral we classify vector bundles over them we classify them up to isomorphism and we classify isomorphisms between them using the classification of isomorphisms we compute the automorphism group of an arithmetic noncommutative projective line

betti numbers of mcm modules over the cone of an elliptic normal curve abstract we apply orlov s equivalence to derive formulas for the betti numbers of maximal cohen macaulay modules over the cone an elliptic curve e x embedded into p n 1 by the full linear system o n x for n 3 the answers are given in terms of recursive sequences these results are applied to give a criterion of co koszulity in the last two sections of the paper we apply our methods to study the cases n 1 2 geometrically these correspond to the embedding of an elliptic curve into a weighted projective space the singularities of the corresponding cones are called minimal elliptic they were studied by k saito 13 where he introduced the notation e 8 for n 1 e 7 for n 2 and e 6 for the cone over a smooth cubic that is for the case n 3 for the singularities e 7 and e 8 we obtain formulas for the betti numbers and the numerical invariants of mcm modules analogous to the case of a plane cubic

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

deformations of global symmetries in the extended antifield formalism it is outlined how deformations of field theoretical rigid symmetries can be constructed and classified by cohomological means in the extended antifield formalism special attention is devoted to deformations referring only to a subset of the rigid symmetries of a given model and leading to a nontrivial extension of the graded lie algebra associated with that subset the method is illustrated for a d 4 n 2 supersymmetric model where the central extension of the supersymmetry algebra emerges via a deformation deformations of gauge fixed actions with a brst symmetry are discussed too and illustrated by the curci ferrari model

noetherian connected graded algebras of global dimension 3 abstract let a be a noetherian connected graded algebra of global dimension 3 we show that a is regular in the sense of artin and schelter if one of the following conditions holds 1 a is generated by two elements 2 the graded simple module has a standard resolution 3 the degrees of minimal relations are the same this includes the quadratic case some general properties of a are studied without assuming regularity

algebras associated to elliptic curves this paper completes the classification of artin schelter regular algebras of global dimension three for algebras generated by elements of degree one this has been achieved by artin schelter tate and van den bergh we are therefore concerned with algebras which are not generated in degree one we show that there exist some exceptional algebras each of which has geometric data consisting of an elliptic curve together with an automorphism just as in the case where the algebras are assumed to be generated in degree one in particular we study the elliptic algebras a a and a a where a p2 which were first defined in an earlier paper we omit a set s p2 consisting of 11 specified points where the algebras a a become too degenerate to be regular theorem let a represent a a or a a where a p2 s then a is an artin schelter regular algebra of global dimension three moreover a is a noetherian domain with the same hilbert series as the appropriately graded commutative polynomial ring in three variables this combined with our earlier results completes the classification

a method for the numerical inversion of laplace transforms abstract in this paper a numerical inversion method for laplace transforms based on a fourier series expansion developed by durbin 5 is presented the disadvantage of the inversion methods of that type the encountered dependence of discretization and truncation error on the free parameters is removed by the simultaneous application of a procedure for the reduction of the discretization error a method for accelerating the convergence of the fourier series and a procedure that computes approximately the best choice of the free parameters suitable for a given problem the inversion method allows the adequate application of these procedures therefore in a big range of applications a high accuracy can be achieved with only a few function evaluations of the laplace transform the inversion method is implemented as a fortran subroutine

non commutative mori contractions and p1 bundles abstract we give a method for constructing maps from a non commutative scheme to a commutative projective curve with the aid of artin zhang s abstract hilbert schemes this is used to construct analogues of the extremal contraction of a k negative curve with self intersection zero on a smooth projective surface this result will hopefully be useful in studying artin s conjecture on the birational classification of non commutative surfaces as a non trivial example of the theory developed we look at non commutative ruled surfaces and more generally at non commutative p 1 bundles we show in particular that non commutative p 1 bundles are smooth have well behaved hilbert schemes and we compute its serre functor we then show that non commutative ruled surfaces give examples of the aforementioned non commutative mori contractions

basic properties of generalized down up algebras abstract we introduce a large class of infinite dimensional associative algebras which generalize down up algebras let k be a field and fix f k x and r s k define l l f r s to be the algebra generated by d u and h with defining relations d h r d 0 h u r u 0 d u s f h 0 included in this family are smith s class of algebras similar to u sl 2 le bruyn s conformal sl 2 enveloping algebras and the algebras studied by rueda the algebras l have gelfand kirillov dimension 3 and are noetherian domains if and only if rs 0 we calculate the global dimension of l and for rs 0 classify the simple weight modules for l including all finite dimensional simple modules simple weight modules need not be classical highest weight modules

quantum planes of weight 1 1 n 1 in this article we study the categories tails a arising from artin schelter regular algebras of global dimension 3 of weight 1 1 n where n is an integer greater than 1 we determine generators and relations for all such algebras and this information is used to classify the points and lines in tails a

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

homological perturbation theory and the algebraic structure of the antifield antibracket formalism for gauge theories the algebraic structure of the antifield antibracket formalism for both reducible and irreducible gauge theories is clarified this is done by using the methods of homological perturbation theory hpt a crucial ingredient of the construction is the koszul tate complex associated with the stationary surface of the classical extremals the koszul tate differential acts on the antifields and is graded by the antighost number it provides a resolution of the algebraa of functions defined on the stationary surface namely it is acyclic except at degree zero where its homology group reduces toa acyclicity only holds because of the introduction of the ghosts of ghosts and provides an alternative criterion for what is meant by a proper solution of the master equation the existence of the brst symmetry follows from the techniques of hpt the classical lagrangian brst cohomology is completely worked out and shown to be isomorphic with the cohomology of the exterior derivative along the gauge orbits on the stationary surface the algebraic structure of the formalism is identical with the structure of the hamiltonian brst construction the role played there by the constraint surface is played here by the stationary surface only elementary quantum questions general properties of the measure are addressed

blowing up quantum weighted projective planes abstract quantum weighted projective planes of degree 1 1 n were classified by stephenson we show that the blowups of some of these spaces are quantum ruled surfaces as defined by patrick phd thesis mit 1997 this is a quantum version of the fact that projective cones over rational normal curves blowup to rational ruled surfaces the class of algebras which we can exhibit as blowdowns of quantum ruled surfaces are those which can be twisted to ore extensions of k x y

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

generalized serre duality abstract we introduce a notion of generalized serre duality on a hom finite krull schmidt triangulated category t this duality induces the generalized serre functor on t which is a linear triangle equivalence between two thick triangulated subcategories of t moreover the domain of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the third term of an auslander reiten triangle in t dually the range of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the first term of an auslander reiten triangle in t we compute explicitly the generalized serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of artin and zhang we obtain a characterization of gorenstein algebras an artin algebra a is gorenstein if and only if the bounded homotopy category of finitely generated projective a modules has serre duality in the sense of bondal and kapranov

geometric algebras on projective surfaces abstract let x be a projective surface let aut x and let l be a ample invertible sheaf on x we study the properties of a family of subrings parameterized by geometric data of the twisted homogeneous coordinate ring b x l in particular we find necessary and sufficient conditions for these subrings to be noetherian we also study their homological properties their associated noncommutative projective schemes and when they are maximal orders in the process we produce new examples of maximal orders these are graded and have the property that no veronese subring is generated in degree 1 our results are used in the companion paper s j sierra 2009 sie09 to give defining data for a large class of noncommutative projective surfaces

species and non commutative p1 s over non algebraic bimodules abstract we study non commutative projective lines over not necessarily algebraic bimodules in particular we give a complete description of their categories of coherent sheaves and show they are derived equivalent to certain bimodule species this allows us to classify modules over these species and thus generalize and give a geometric interpretation for results of c ringel 21

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

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

a half space problem in the theory of generalized thermoelastic diffusion abstract in this work we consider the problem of a thermoelastic half space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time the bounding surface of the half space is taken to be traction free and subjected to a time dependent thermal shock the chemical potential is also assumed to be a known function of time on the bounding plane laplace transform techniques are used the solution is obtained in the laplace transform domain by using a direct approach the solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the laplace transform based on fourier expansion techniques the temperature displacement stress and concentration as well as the chemical potential are obtained numerical computations are carried out and represented graphically

fractional order theory of thermoelasticity in this work a new theory of thermoelasticity is derived using the methodology of fractional calculus the theories of coupled thermoelasticity and of generalized thermoelasticity with one relaxation time follow as limit cases a uniqueness theorem for this model is proved a variational principle and a reciprocity theorem are derived

the dynamics of leavitt path algebras abstract recently it was shown that the notion of flow equivalence of shifts of finite type in symbolic dynamics is related to the morita theory and the grothendieck group in the theory of leavitt path algebras abrams et al 2011 4 in this paper we show that the notion of the conjugacy of shifts of finite type is closely related to the graded morita theory and consequently the graded grothendieck group this fits into the general framework we have in these two theories conjugacy yields the flow equivalence and the graded morita equivalence can be lifted to the morita equivalence starting from a finite directed graph the observation that the graded grothendieck group of the leavitt path algebra associated to e coincides with the krieger dimension group of the shift of finite type associated to e provides a link between the theory of leavitt path algebras and symbolic dynamics it has been conjectured that the ordered graded grothendieck group as z x x 1 module we call this the graded dimension group classifies the unital leavitt path algebras completely hazrat 2013 20 via the above correspondence utilising the results from symbolic dynamics we prove that for two purely infinite simple unital leavitt path algebras if their graded dimension groups are isomorphic then the algebras are isomorphic

subspaces of non commutative spaces this paper concerns the closed points closed subspaces open subspaces weakly closed and weakly open subspaces and effective divisors on a non commutative space

a problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion in this work we study a one dimensional problem in a generalized thermoelastic diffusion in infinite medium with a spherical cavity subjected to a time dependent thermal shock of its internal boundary which is assumed to be traction free the chemical potential is also assumed to be a known function of time on the bounding cavity laplace transform techniques are used the solution of the problem in the transformed domain is obtained by using a direct approach without the customary use of potential functions by means of numerical laplace inversion the problem is solved in the physical domain numerical results predict finite speeds of propagation for thermoelastic and diffusive waves to investigate the diffusions effects a comparison is made with the results obtained in the thermoelastic problem

hopf dense galois extensions with applications abstract let h be a finite dimensional hopf algebra and let a be a left h module algebra motivated by the study of the isolated singularities of a h and the endomorphism ring end a h a we introduce the concept of hopf dense galois extensions in this paper hopf dense galois extensions yield certain equivalences between the quotient categories over a and a h a special class of hopf dense galois extensions consists of the so called densely group graded algebras which are weaker versions of strongly graded algebras a weaker version of dade s theorem holds for densely group graded algebras as applications we recover the classical equivalence of the noncommutative projective scheme over a noetherian n graded algebra a and its d th veronese subalgebra a d respectively hopf dense galois extensions are also applied to the study of noncommutative graded isolated singularities

integral non commutative spaces abstract a non commutative space x is a grothendieck category mod x we say x is integral if there is an indecomposable injective x module e x such that its endomorphism ring is a division ring and every x module is a subquotient of a direct sum of copies of e x a noetherian scheme is integral in this sense if and only if it is integral in the usual sense we show that several classes of non commutative spaces are integral we also define the function field and generic point of an integral space and show that these notions behave as one might expect