injective stability for k1 of the orthogonal group abstract the injective stabilization bound of l n vaserstein for k1 of the orthogonal group over an affine algebra over a perfect c 1 field is discussed

quotients of absolute galois groups which determine the entire galois cohomology for a prime power q pd and a field f containing a root of unity of order q we show that the galois cohomology ring h g f mathbb z q is determined by a quotient g f 3 of the absolute galois group gf related to its descending q central sequence conversely we show that g f 3 is determined by the lower cohomology of gf this is used to give new examples of pro p groups which do not occur as absolute galois groups of fields

conformal mappings from annuli to canonical doubly connected bell representations abstract this paper derives formulae expressed in terms of schottky klein prime functions for the conformal mapping between concentric annuli and bell representations which are canonical doubly connected planar domains new formulae relating the conformal moduli associated with the two respective canonical domains are derived and complement recent results by jeong oh and taniguchi m jeong j w oh m taniguchi equivalence problem for annuli and bell representations in the plane j math anal appl 325 2007 1295 1305

operations on orbits of unimodular vectors abstract taking the m power of an entry is a well defined operation on the unimodular vectors in an modulo addition operations if n is at least 3 for an arbitrary commutative ring a and any integer m

witt groups of curves and surfaces we study witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two in both dimensions we determine both the classical witt group and balmer s shifted witt groups in the case of curves the results are supplemented with a complete description of the shifted grothendieck witt groups in a second step we analyse the relationship of witt groups of smooth complex curves and surfaces with their real topological k groups they turn out to be surprisingly close for all curves and for all projective surfaces of geometric genus zero the witt groups may be identified with the quotients of their even ko groups by the images of their complex topological k groups under realification

generalized umbrella rings in this paper the notion of generalized umbrella rings is introduced we show that every generalized umbrella ring is a stably coherent ring as an application we show that every finitely generated projective module over r x is free if r is a generalized umbrella ring with weak global dimension two

strictly positive definite functions on the unit circle we study strictly positive definite functions on the unit circle in the euclidean space of dimension two we develop several conditions pertaining to the determination of such functions the major result is obtained by considering the set of real numbers as a vector space over the field of rational numbers and then applying the kronecker approximation theorem and weyl s criterion on equidistributions

constructing multiply connected quadrature domains multiply connected bounded quadrature domains with finite connectivity are re constructed from their quadrature data using conformal mappings that are ratios of products of schottky klein prime functions this method provides the natural generalization of the conformal maps to simply and doubly connected quadrature domains constructed by the first author in a num ber of physical applications the efficacy of the method is demonstrated by the explicit construction of a range of examples as well as by comparison with alternative constructive methods recently in troduced by crowdy r soc lond proc ser a math phys eng sci 457 2001 pp 2337 2359 and richardson european j appl math 12 2001 pp 571 599

homotopy and commutativity principle abstract we give evidence for the principle that a special linear symplectic orthogonal matrix over a commutative ring which is homotopic to the identity will commute up to an elementary matrix with all elements of the respective group

projective modules over the real algebraic sphere of dimension 3 let a be a commutative noetherian ring of krull dimension 3 we give a necessary and sufficient condition for a projective modules of rank 2 to be free using this we show that all the finitely generated projective modules over the algebraic real 3 sphere are free c 2010 elsevier inc all rights reserved

triangular witt groups part ii from usual to derived we establish that the derived witt group is isomorphic to the usual witt group when 2 is invertible this key result opens the ali baba s cave of triangular witt groups linking the abstract results of part i to classical questions for the usual witt group for commercial purposes we survey the future applications of triangular witt groups in the introduction we also establish a connection between odd indexed witt groups and formations finally we prove that over a commutative local ring in which 2 is a unit the shifted derived witt groups are all zero but the usual one

pairings in triangular witt theory abstract given a product k l m between triangulated categories with duality we show that under some conditions there exist naturally two different pairings w i k w j l w i j m where w denotes the triangulated witt functor of balmer p balmer k theory 19 2000 311 363 our main example of such a situation is the case that k l m is the bounded derived category of vector bundles over a scheme x and is the derived tensor product the derived witt groups of this scheme w x i z w i x become a graded skew commutative ring with two different product structures which are both equally natural in the last section we prove then a projection formula for our product and show as an application that a springer type theorem is true for the derived witt groups too

a nice group structure on the orbit space of unimodular rows ii abstract we establish an excision type theorem for niceness of group structure on the orbit space of unimodular rows of length n modulo elementary action this permits us to establish niceness for relative versions of results for the cases when n d 1 d being the dimension of the base algebra we then study and establish niceness for the case when n d and also establish a relative version when the base ring is a smooth affine algebra over an algebraically closed field

stably free modules in su prob 3 suslin had asked the following question let a be any affine algebra of dimension n over an algebraically closed field what is the smallest integer m such that all stably free projective modules of rank bigger than m are free all the examples in the literature of stably free non free modules have rank less than or equal to n 1 2 the aim of this note is to construct examples of such modules of large rank we construct rank p stably free non free modules over p 2 dimensional affine algebras over algebraically closed fields wherep is any prime these varieties are actually smooth and rational over c these are trivial as holomorphic vector bundles forp 2 this is classical forp 2 see ms so these are strictly algebraic examples i had described this construction in mk 1 and proved the result for p 2 we will reproduce the construction with necessary modifications in this note let p be any prime number and k any field letf x be any polynomial of degree p over k letf 0 a e k and fi xo xl f xo x1 xp f x0 x1 also let

a cohomological classification of vector bundles on smooth affine threefolds we give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2 as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties the proofs of these results involve three main ingredients first we give a description of the first nonstable a1 homotopy sheaf of the symplectic group second these computations can be used in concert with f morel s a1 homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques stemming from a version of the postnikov tower in a1 homotopy theory to reduce the classification results to cohomology vanishing statements third we prove the required vanishing statements

on completing unimodular polynomial vectors of length three it is shown that if r is a local ring of dimension three with r then a polynomial three vector vq x vx x v2 x over r x can be completed to an invertible matrix if and only if it is unimodular in particular if 1 3 j then every stably free projective r xx y module is free

zero cycles and projective modules let a be a reduced affine k algebra of dimension n over an algebraically closed field k let fnko a denote the sub group of ko a generated by the images of all the residue fields of all smooth maximal ideals of height n for a module of finite projective dimension m let m denote the image of m in ko a for a projective a module p of rank n we define the nth chern class of p to be cn p e l i aip where p is the dual of p suppose fnko a has no n 1 torsion then our main result th 3 8 is that p has a free direct summand of rank one if and only if cn p 0 when the characteristic of k is zero or a is normal and n 3 it is known that fnko a is torsion free le sr hence our theorem is applicable in these cases also when a is regular fnko a coincides with chn x the chow group of zero cycles of spec a and cn p coincides with the usual nth chern class as defined by grothendieck see fu when n 3 and a is regular this result was proved in mkm when n 3 and the characteristic of k is not equal to two this result in mkm was extended to the singular case by m levine in this paper we extend the results of mkm to all dimensions with the assumption stated above on torsion in fnko a we first show that cor 3 4 if i c a is a local complete intersection of height n then a i is zero in ko a if and only if i is a complete intersection corollary 3 4 together with a result of mohan kumar mk cor 1 9 here at once gives theorem 3 8 in section 1 we give some preliminaries and generalize results in mk2 these results are crucially used in the rest of the paper the basic ideas in section 1 are all taken from mk2 in section 2 we prove theorem 2 2 which strengthens a result of boratynski bo for local complete intersections with trivial co normal bundle theorem 2 2 is one of the crucial ingredients in the proofs of the results in section 3 as an amusing application of theorem 2 2 we give a new proof of a theorem of srinivas sr about torsion in zero cycles

on triviality of the euler class group of a deleted neighbourhood of a smooth local scheme let k be an infinite perfect field and let x specr where r is the localization of a k algebra b of finite type with respect to a regular prime ideal p specb letd be a divisor onx with a decompositiond i 1di into its irreducible components di such that each di is nonsingular and they intersect transversally consider y x d a deleted neighbourhood of the closed point x0 of x let e be a vector bundle on y in this set up e is trivial in the following cases i dim x 3 gabber ga1 ii t 1 bhatwadekar rao br iii ranke min dimr 2 t rao ra1 the above exposition is borrowed from a paper of nisnevich ni which contains some beautiful applications of these purity theorems in the same set up now let y y be a closed point and let my be the corresponding maximal ideal of oy y it is natural to ask whether my is a complete intersection observing that the conormal bundle my my is trivial we consider a more general situation let i be the defining ideal of a closed subscheme z of y such that the conormal bundle i i2 defined on z is trivial is i a complete intersection the questions we are addressing in this paper are a little stronger in nature can a given basis of i i2 be lifted to a minimal set of generators of i we remark that with the hypothesis on x each component di is defined by a single equation fi which vanishes at the closed point x0 of x let m be the maximal ideal at x0 the condition of transversality of intersection of di s is equivalent to saying that the images of f1 ft in the a m vector space m m2 are linearly independent keeping this discussion in mind we now give an excerpt of our main results in commutative algebraic terms see 4 2 4 3 4 4

the mayer vietoris principle for grothendieck witt groups of schemes we prove localization and zariski mayer vietoris for higher gro thendieck witt groups alias hermitian k groups of schemes admitting an ample family of line bundles no assumption on the characteristic is needed and our schemes can be singular along the way we prove additivity fibration and approximation theorems for the hermitian k theory of exact categories with weak equivalences and duality

cohomological invariants for quadratic forms over local rings let a be local ring in which 2 is invertible and let n be a non negative integer we show that the nth cohomological invariant of quadratic forms is a well defined homomorphism from the nth power of the fundamental ideal in the witt ring of a to the degree n etale cohomology of a with mod 2 coefficients which is surjective and has kernel the hbox n 1 th power of the fundamental ideal this is obtained by proving the gersten conjecture for witt groups in an important mixed characteristic case

the coefficient body of bell representations of finitely connected planar domains in this note we determine the homotopy type of the coecient body of bell representations of non degenerate n connected planar domains with n 3 also by considering the isomorphism classes of rational functions we get the precise number of those corresponding to bell representations wth the same set of critical values further the case of those with the same set of critical points is discussed

strictly positive definite kernels on subsets of the complex plane let z w x zw be a positive definite kernel and b a subset of in this paper we seek conditions in order that the restriction z w b x b zw be strictly positive definite since this problem has been solved recently in the cases in which b is either or the unit circle in our purpose here is twofold to present some results we obtained when attempting to solve the problem for the above and other choices of b and to acquaint the audience with some other questions that remain for two different classes of subsets we completely characterize the strict positive definiteness of the kernel we include a complete discussion of the case in which b is the unit circle of making a comparison with the classical problem of strict positive definiteness on the real circle

the euler class group of a polynomial algebra in this paper we continue our study on the theory of the euler class group of a polynomial algebra a t where a is a commutative noetherian ring containing q of dimension n for such a ring a in 9 we defined the notion of the nth euler class group en a t of a t for simplicity let us call it e a t in 9 we also studied the relations between e a t and e a where e a is the nth euler class group of a for example there is canonical map e a e a t which is an injective group homomorphism and it is an isomorphism when a is a smooth affine domain 9 proposition 5 7 in general these two groups are not isomorphic see discussion preceding 9 proposition 5 7 in this context the following question is natural

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

decomposability for division algebras of exponent two and associated forms this article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated witt ring is torsion free

finitely generated function fields and complexity in potential theory in the plane we prove that the bergman kernel function associated to a finitely connected domain in the plane is given as a rational combination of only three basic functions of one complex variable an alhfors map its derivative and one other function whose existence is deduced by means of the field of meromorphic functions on the double of because many other functions of conformal mapping and potential theory can be expressed in terms of the bergman kernel our results shed light on the complexity of these objects we also prove that the bergman kernel is an algebraic function of a single ahlfors map and its derivative it follows that many objects of potential theory associated to a multiply connected domain are algebraic if and only if the domain is a finite branched cover of the unit disc via an algebraic holomorphic mapping

the homotopy limit problem for hermitian k theory equivariant motivic homotopy theory and motivic real cobordism abstract the homotopy limit problem for karoubi s hermitian k theory karoubi 1980 26 was posed by thomason 1983 44 there is a canonical map from algebraic hermitian k theory to the z 2 homotopy fixed points of algebraic k theory the problem asks roughly how close this map is to being an isomorphism specifically after completion at 2 in this paper we solve this problem completely for fields of characteristic 0 theorems 16 20 we show that the 2 completed map is an isomorphism for fields f of characteristic 0 which satisfy cd 2 f i but not in general

complexity in complex analysis abstract we show that the classical kernel and domain functions associated to an n connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable we characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable such domains turn out to have the property that their classical domain functions all extend to be meromorphic functions on a compact riemann surface and this condition will be shown to be equivalent to the condition that an ahlfors map and its derivative are algebraically dependent we also show how many of these results can be generalized to finite riemann surfaces

catalan numbers and branched coverings by the riemann sphere abstract we prove that the number of pgl 2 c equivalence classes of degree d rational maps with a fixed branch set is generically equal to the catalan number d 1 d 2d 2 d 1 several previously known applications of catalan numbers to combinatorics and algebraic geometry are discussed throughout the course of the article

stably free modules over of rank dim are free we prove that for any finite dimensional ring r and n dim r 2 the group e n r x acts transitively on um n r x in particular we obtain that for any finite dimensional ring r all finitely generated stably free modules over r x of rank dim r are free this result was only known for noetherian rings the proof we give is short simple and constructive

the euler class group of a polynomial algebra with coefficients in a line bundle let a be a commutative noetherian ring and p be a projective a module of rank text dim a 1 an intriguing open question is to find the precise obstruction for p to split as p simeq q oplus a for some a module q in this paper we settle this question when a r t for some ring r containing the field of rationals and p is a projective a module of rank text dim r

finiteness theorems for the shifted witt and higher grothendieck witt groups of arithmetic schemes abstract for smooth varieties over finite fields we prove that the shifted aka derived witt groups of surfaces are finite and the higher grothendieck witt groups aka hermitian k theory of curves are finitely generated for more general arithmetic schemes we give conditional results for example finite generation of the motivic cohomology groups implies finite generation of the grothendieck witt groups

euler class groups and the homology of elementary and special linear groups abstract we improve homology stability ranges for elementary and special linear groups over rings with many units our result implies stability for unstable quillen k groups and proves a conjecture of bass for commutative local rings with infinite residue fields we show that the obstruction to further stability is given by milnor witt k theory as an application we construct euler classes of projective modules with values in the cohomology of the milnor witt k theory sheaf for d dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the euler class is necessary and sufficient for an oriented projective module p of rank d to split off a rank 1 free direct summand along the way we obtain a new presentation of milnor witt k theory and of symplectic k 2 simplifying the classical matsumoto moore presentation

equivalence problem for annuli and bell representations in the plane in this paper we solve the problem of equivalence between annuli and bell representations which are canonical planar domains of connectivity 2 we give a necessary and sufficient condition for equivalence as an explicit formula in parameters of the domains

strictly positive definite functions on spheres in euclidean spaces in this paper we study strictly positive definite functions on the unit sphere of the m dimensional euclidean space such functions can be used for solving a scattered data interpolation problem on spheres since positive definite functions on the sphere were already characterized by schoenberg some fifty years ago the issue here is to determine what kind of positive definite functions are actually strictly positive definite the study of this problem was initiated recently by xu and cheney proc amer math soc 116 1992 977 981 where certain sufficient conditions were derived a new approach which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres is presented here the relevant interpolation problem is subsequently analyzed by three different complementary methods the first is based on the de boor ron general least solution for the multivariate polynomial interpolation problem the second which is suitable only for m 2 is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials and reduces the problem to the structure of the integer zeros of bounded univariate exponentials finally the last method invokes the realization of harmonic polynomials as the polynomial kernel of the laplacian thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels

on maximal ideals in polynomial and laurent polynomial rings abstract when r is a complete local ring containing a field of characteristic zero we show that any maximal ideal in r t t 1 is projectively generated if r is a regular ring containing a field of characteristic zero we show that any maximal ideal m of r t such that ht m dim r is efficiently generated

generalized ahlfors functions let be a bordered riemann surface with genus g and m boundary components let z z be a smooth family of smooth jordan curves in c which all contain the point 0 in their interior let p and let f be the family of all bounded holomorphic functions f on s such that f p 0 and f z z for almost every z then there exists a smooth up to the boundary holomorphic function f 0 f with at most 2g m 1 zeros on s so that f 0 z z for every z and such that f 0 p f p for every f f if in addition all the curves z z are strictly convex then f 0 is unique among all the functions from the family f

the hermite ring conjecture in dimension one abstract we prove constructively that for any ring r of krull dimension 1 and n 3 the group e n r x acts transitively on um n r x in particular we obtain that for any ring r with krull dimension 1 all finitely generated stably free modules over r x are free this settles the long standing hermite ring conjecture for rings of krull dimension 1

mathbb p 1 p 1 gluing for local complete intersections we prove an analogue of the affine horrocks theorem for local complete intersection ideals of height n in r t where r is a regular domain of dimension d which is essentially of finite type over an infinite perfect field of characteristic unequal to 2 and 2n ge d 3

chow witt groups and grothendieck witt groups of regular schemes let a be a noetherian commutative z 1 2 algebra of krull dimension d and let p be a projective a module of rank d we use derived grothendieck witt groups and euler classes to detect some obstructions for p to split off a free factor of rank one if d 3 we show that the vanishing of its euler class in the corresponding grothendieck witt group is a necessary and sufficient condition for p to have a free factor of rank one if d is odd we also get some results in that direction if a is regular we show that the chow witt groups defined by morel and barge appear naturally as some homology groups of a gersten type complex in grothendieck witt theory from this we deduce that if d 3 then the vanishing of the euler class of p in the corresponding chow witt group is a necessary and sufficient condition for p to have a free factor of rank one crown copyright c 2008 published by elsevier inc all rights reserved

odd orthogonal matrices and the non injectivity of the vaserstein symbol abstract r a rao w van der kallen showed that the vaserstein symbol v s r 3 from the orbit space of unimodular rows of length three over the coordinate ring of the real three sphere s r 3 modulo elementary action to the elementary symplectic witt group w e s r 3 is not injective dhvanita r rao neena gupta gave an uncountable family of singular real threefolds a for which the vaserstein symbol v a is not injective in this paper we give a countable family of smooth real birationally equivalent threefolds a n for which the vaserstein symbol v a n is not injective