the graded witt group kernel of biquadratic extensions in characteristic two baeza showed that when char f 2 if e f is the separable biquadratic extension e f 1 b1 1 b2 then ker wq f wq e wf 1 b1 wf 1 b2 here we give the analogous result for the graded witt group specifically we obtain an exact sequence f n 1 f n 1 h2n 1f h2n 1e from which the result for gwqf follows by the isomorphisms of kato applications to 2 algebras of exponent and index 4 are also given

on the brauer group complex for a multiquadratic field extension abstract let f be a field of characteristic not 2 a 1 a n f m f a 1 a n let further g be the subgroup of f f 2 generated by a 1 a n for any i 1 n put a i i i a i a 1 consider the complex i k i k i 2 g f f 2 br 2 f res br 2 m n m e f e m m e 2 br 2 e where k i f a i are all the quadratic extensions of f containing in m w i i a i n k i f w i a i z a i z the homology group of this complex at i 1 th term from the left is denoted by h i m f independently of char f we give examples of these complexes with nontrivial h 3 m f and prescribed n 3 and a 1 a n 1 also similar examples concerning the group h 1 m f are constructed

cohomology and graded witt group kernels for extensions of degree four in characteristic two abstract in earlier work the authors determined the graded witt kernel g w q e f ker g w q f g w q e when e f is a biquadratic extension in characteristic 2 by calculating the cohomological kernel h 2 e f ker h 2 f h 2 e in this paper this result is extended to the cases where e f 4 and e is either cyclic or has dihedral galois closure in addition the use of izhboldin s q groups is generalized to obtain six term exact sequences that describe the behavior of these graded rings whenever four term exact sequences and homotopies describe the arithmetic of the extension these tools are valid in characteristic p although the applications here are in characteristic 2

the behavior of differential quadratic and bilinear forms under purely inseparable field extensions abstract let f be a field of characteristic p and let e f be a purely inseparable field extension we study the group h p n 1 f coker f n f n d f n 1 of classes of differential forms under the extension e f and give a system of generators of h p n 1 e f in the case p 2 we use this to determine the kernel w q e f of the restriction map w q f w q e between the groups of nonsingular quadratic forms over f and over e we also deduce the corresponding result for the bilinear witt kernel w e f of the restriction map w f w e where e f denotes a modular purely inseparable field extension

the u invariant of the function fields of p adic curves the u invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field it is an open question whether the u invariant of function fields of p aidc curves is 8 in this paper we answer this question in the affirmative for function fields of nondyadic p adic curves

on the dec group of finite abelian galois extensions over global fields abstract if k f is a finite abelian galois extension of global fields whose galois group has exponent t we prove that there exists a short exact sequence 0 dec k f br t k f q p r q z z 0 where r q q and p is a finite set of primes of f that is empty if t is square free in particular we obtain that if t is square free then dec k f br t k f which we use to show that prime exponent division algebras over henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras finally we construct a counterexample to the result for higher exponent division algebras

r equivalence in adjoint classical groups over fields of virtual cohomological dimension 2 let f be a field of characteristic not 2 whose virtual cohomological dimension is at most 2 let g be a semisimple group of adjoint type defined over f let rg f denote the normal subgroup of g f consisting of elements r equivalent to identity we show that if g is of classical type not containing a factor of type d n g f rg f 0 if g is a simple classical adjoint group of type d n we show that if f and its multi quadratic extensions satisfy strong approximation property then g f rg f 0 this leads to a new proof of the r triviality of f rational points of adjoint classical groups defined over number fields

the witt group of a discretely valued field abstract this note provides an extension of a classical result of springer in the algebraic theory of quadratic forms for discretely valued fields to the dyadic case

witt rings and brauer groups under multiquadratic extensions ii let f be a field of characteristic not 2 and suppose m is a multiquadratic extension of f that is m f is a finite abelian extension of exponent 2 so that m f for some finite subgroup g e f f where f 0 in studying brauer groups and products of quaternion algebras the second author was led to consider the homology groups n m f of a certain complex gm f associated to the extension m f this complex appears in 11 3 1 301 and in 1 1 below the purpose of the present work is to investigate the first homology group n m f and to exhibit its close connections with quadratic form theory the higher homology groups are examined in 11 301 the field f is said to have property pi n if n m f 1 for every multiquadratic extension m f with m f q 2 properties p l and p 2 always hold but there are examples 29 of fields f for which p 3 fails these examples are generalized in section 5

squeeze flow of a piezoviscous fluid we investigate squeeze flow of piezoviscous fluids presence of contact line leads to strong pressure variations strong pressure variations induce strong viscosity variations in piezoviscous fluid squeeze flow characteristics are influenced by piezoviscosity of the fluid we investigate the squeeze flow of a piezoviscous fluid the classical navier stokes fluid model and the no slip boundary condition on the plate fluid interface implies the presence of the pressure singularity at the contact line it can be conjectured that the same behaviour is present even for piezoviscous fluids consequently the pressure variation in the fluid could be large and the pressure dependent viscosity could be important even if the piezoviscous coefficient is apparently small we investigate this conjecture by means of numerical simulation based on a variant of the spectral collocation method the numerical simulations indicate that the effects due to pressure dependent viscosity are for certain range of parameter values significant

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

flows of incompressible fluids subject to navier s slip on the boundary we establish closed form analytical solutions for the flows of a generalized fluid of complexity two which includes the navier stokes fluid power law fluid and the second grade fluid as special subclasses in special geometries under the assumption that the flows meet navier slip conditions at the boundary the boundary conditions allow the extremes of no slipping and no sticking we also allow for different boundary conditions at different parts of the boundary which leads to interesting consequences with regard to the solutions the dependence of the form of the solutions on the boundary conditions is discussed in some detail

on some invariants of fields of characteristic p 0 abstract let k be a field of characteristic p 0 k n the k vector space of n differential forms and k n k n d k n 1 the artin schreier operator given by x d x 1 x 1 d x n x n x p x d x 1 x 1 d x n x n let h p n 1 k coker and define p k min m h p m 1 k 0 we show that for any finite extension l k p k p l p k 1

galois groups of maximal p extensions let p be an odd prime and f a field of characteristic different from p containing a primitive pth root of unity assume that the galois group g of the maximal p extension of f has a finite normal series with abelian factor groups then the commutator subgroup of g is abelian moreover g has a normal abelian subgroup with pro cyclic factor group if in addition f contains a primitive p 2 th root of unity then g has generators x y i i i with relations y i y j y j y i and xy i x 1 y i q 1 where q 0 or q p n for some n 1

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

a new construction of noncrossed product algebras new examples of noncrossed product division algebras are obtained using methods different from all previous noncrossed product constructions the examples are division algebras over intersections of henselian valued fields and they have schur index and exponent p for any prime number p and any integers m n 2 n 3if 2 the basic tools used in the construction are valuation theory and galois cohomology no generic methods are applied and there is no p i theory along the way local global principles are proved for central simple algebras over intersections of p henselian valued fields the first examples of central simple division algebras which are not crossed products were obtained by amitsur in am in 1972 amitsur thereby settled a question that had been one of the outstanding open problems in the theory of algebras for at least thirty years his examples were the generic division algebras ud q n of index n over q the rational numbers for any natural number n such that p2 n p an odd prime or 8 all subsequent constructions of noncrossed products in ss 83 883 r r an ti have heretofore been based on amitsur s specialization argument and they are all generic division algebras or extensions of generic division algebras the centers of these noncrossed product algebras are not known nor are the brauer groups nor the absolute galois groups of the centers we present here a new method of constructing noncrossed product algebras in our approach the noncrossed product is realized as the underlying division algebra d of a tensor product of suitably chosen cyclic algebras over a field f l n l2 where each l is a pth root henselian valued field we prove local global principles relating the splitting fields of d to those of d f lt i 1 2 it is shown that the part of the brauer group of f is completely determined by that of ll and l2 computations for central simple f algebras thus become very tractable for exam ple we not only show that d is not a crossed product but also calculate exactly how large r must be so that the matrix ring mr d is a crossed product and how large s must be so that ms d is a tensor product of cyclic algebras indeed the structure of the brauer group of f is so nice that we were somewhat surprised that noncrossed products could possibly exist over f

a stable category of lagrangian cobordisms abstract given an exact symplectic manifold m and a support lagrangian m we construct an category lag m which we conjecture to be equivalent after specialization of the coefficients to the partially wrapped fukaya category of m relative to roughly speaking the objects of lag m are lagrangian branes inside of m t r n for large n and the morphisms are lagrangian cobordisms that are non characteristic with respect to the main theorem of this paper is that lag m is a stable category and in particular its homotopy category is triangulated with mapping cones given by an elementary construction the shift functor is equivalent to the familiar shift of grading for lagrangian branes

a characterization of c fields via galois groups abstract it is known that those fields f which are of class c in the sense of r ware can be characterized by their witt ring structure as well as by the behavior of their binary quadratic forms and the structure of their galois 2 extensions in this paper we show that c fields which have at least 8 square classes are characterized by the absence of a certain group dc of order 16 as a galois group over them moreover we show that the level of such a field is determined by the presence or absence of the cyclic group of order 4 and the dihedral group of order 8 as galois groups over it these results provide an interesting continuation of the work of ware in the appendix we give an explicit construction of field extensions with galois group dc

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

symmetric bilinear forms quadratic forms and milnor k theory in characteristic two is bijective for any field f of characteristic two and for any n 0 here k f is the k group of milnor defined in 11 and i is the unique maximal ideal of the witt ring w f of non degenerate symmetric bilinear forms over f on the other hand sah studied quadratic forms in characteristic two in 15 let f be a field of characteristic two wq f the witt group of non degenerate quadratic forms over f and define i wq f n o by the canonical w f module structure on wq f then he proved that wq f iwq f is isomorphic to the group of quadratic cyclic extensions of f and iwq f i2wq f is isomorphic to the subgroup br f 2 of the brauer group br f of f consisting of elements annihilated by 2 in this paper we prove the above milnor s conjecture and describe the precise structures of w f and wq f generalizing the result of sah for any field f of characteristic p 0 let be the n th exterior power over f of the absolute differential module n t g21 let d 2 1 be the f z f fp image of the exterior derivation d f2 t f2 and let v n v resp h 1 f be the kernel resp cokernel of the homomorphism

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

stability of plane couette flow for the compressible navier stokes equations with navier slip boundary abstract this paper is devoted to the stability analysis of the plane couette flow for the 3d compressible navier stokes equations with navier slip boundary condition at the bottom boundary it is shown that the plane couette flow is asymptotically stable for small perturbation provided that the slip length reynolds and mach numbers satisfy 3 1 2 0 1 and 2 1 for some constant 0 0 in particular the reynolds number 1 can be large if the slip length is suitably small this means that the constraint required in 11 on the reynolds number to guarantee the stability of the plane couette flow can be relaxed and improved so long as the slip effect at the boundary is involved

quadratic forms over c t1 t2 abstract we show that dimension 4 quadratic forms over c t1 t2 ti transcendental do not satisfy what might be naturally taken as a strong hasse principle though a weak hasse principle exists we reduced solvability to a question of determining the quadratic reciprocity group for the discriminant d the classes modulo d of elements x such that d is a quadratic residue modulo x we determine this group in terms of the class group of a function field we prove that the weak hasse principle does not hold for forms over c t1 d therefore witt and brauer groups of 2 dimensional function fields over c do not restrict monomorphically to their localizations

demuskin fields with valuations we study the arithmetic structure of fields f of characteristic neq p containing a primitive pth root of unity for which the maximal pro p galois group of f is a finitely generated demuskin group

versions of springer s theorem for quadratic forms in characteristic 2 whenever f is a henselian valued field whose residue class field inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i has characteristic different from 2 a theorem of springer completely reduces the quadratic form theory of f to the quadratic form theory of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i and the structure of the value group this paper presents analogues of this result in the case where f is a maximally complete field of characteristic 2 two different results are given the first determines the witt group and the second classifies the anisotropic quadratic forms

witt rings and almost free pro 2 groups let f be a field of characteristic not 2 wf the witt ring of quadratic forms over f if the maximal ideal consisting of even dimensional forms and i f its square let g 2 gal f 2 f where f 2 is the maximal 2 extension quadratic closure of f in this note we prove a decomposition theorem for wf in the case that z f is torsion free and use it to obtain a new proof of a theorem of ershov on the structure of almost free pro a groups e theorem 41 in what follows w f denotes the torsion subgroup of wf and wredf wf w f is the reduced witt ring the set of orderings on f will be denoted by x and will be given the harrison topology with subbasis consisting of the sets h a e x 1 0 a f is called a sap field if every open and closed clopen subset of x is equal to some h a if r r are abstract witt rings then r x r denotes their product in the category of witt rings ml all unexplained notations and terminology about quadratic forms and witt rings can be found in l and m for elementary facts about profinite groups the reader is referred to ri following kijima and nishi kn the field f is called quasi pythugorean if z f is torsion free i e u f d 2 equivalently if the kaplansky radical of f consists of all nonzero sums of two squares ell corollary 2 91 compare el2 classification theorem 23 examples of these fields include formally real generalized hilbert fields k pseudo real closed fields p and fields satisfying the strong hasse principle h4 every totally indefinite quadratic form of dimension 4 over f is isotropic i e ii f 2 a formally real field satisfies h if and only if it is a quasi pythagorean sap field elp in ko koziol gives a criterion characterizing the class of quasi pythagorean algebraic extensions of q such fields necessarily satisfy h the best known examples of fields satisfying h are formally real

nonexcellence of multiquadratic field extensions abstract let k0 be a field chark0 2 n 2 a b 1 b n k 0 such that the elements a b 1 b n k 0 k 0 2 are linearly independent we show that there exists a field extension f k0 and an anisotropic 4 dimensional form over f such that d a the form f b 1 b n is isotropic and the form f b 1 b n an is not defined over f

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

a generalization of marshall s equivalence relation for p prime and for a field f containing a root of unity of order p we generalize marshall s equivalence relation on orderings to arbitrary subgroups of f of index p the equivalence classes then correspond to free pro p factors of the maximal pro p galois group of f we generalize to this setting results of jacob on the maximal pro 2 galois group of a pythagorean field

demuskin groups galois modules and the elementary type conjecture abstract let p be a prime and f p the maximal p extension of a field f containing a primitive pth root of unity we give a new characterization of demuskin groups among galois groups gal f p f when p 2 and assuming the elementary type conjecture when p 2 as well this characterization is in terms of the structure as galois modules of the galois cohomology of index p subgroups of gal f p f

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

a recursive description of pro p galois groups abstract in this note we study a modified version of the elementary type conjecture for pro p galois groups to be precise let g p f be the galois group of the maximal galois p extension of a field f containing a primitive p th root of unity under some natural assumptions concerning valuation rings of f and also orderings of f when p 2 we prove that g p f can be obtained from suitable closed subgroups using a finite number of free pro p products and semidirect products

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

relations in in and inwq in characteristic 2 abstract let k be a field of characteristic 2 we give natural presentations of the subgroups i n k of the witt ring w k of k and the subgroups i n w q k of the witt group w q k of k our results generalize the results of arason and elman in j k arason r elman powers of the fundamental ideal in the witt ring j algebra 239 2001 150 160 where the characteristic is assumed to be different from 2

classification theorems for hermitian forms the rost kernel and hasse principle over fields with cd2 k 3 abstract in this paper we prove classification theorems for hermitian forms over some central simple algebras with involution over a field k with cd 2 k 3 we apply these results to show the triviality of the kernel of the rost invariant for the classical algebraic groups associated to such hermitian forms over k we also deduce a hasse principle for algebraic groups defined over function fields of curves over p adic fields thus proving a conjecture due to colliot thelene parimala suresh for a large class of groups

the transfer ideal of quadratic forms and a hasse norm theorem mod squares any finite degree field extension k f determines an ideal jk f of the witt ring wf of f called the transfer ideal which is the image of any nonzero transfer map wk wf the ideal 59k f is computed for certain field extensions concentrating on the case where k has the form f va a ai e f when f and k are global fields we investigate whether there is a local global principle for membership in tkifthis is shown to be equivalent to the existence of a hasse norm theorem mod squares i e a local global principle for the image of the norm map nk f k k 2 f f 2 it is shown that such a hasse norm theorem holds whenever k f aj va although it does not always hold for more general extensions of global fields even some galois extensions with group z 2z x z 4z

the kummerian property and maximal pro p galois groups abstract for a prime number p we give a new restriction on pro p groups g which are realizable as the maximal pro p galois group g f p for a field f containing a root of unity of order p this restriction arises from kummer theory and the structure of the maximal p radical extension of f we study it in the abstract context of pro p groups g with a continuous homomorphism g 1 p z p and characterize it cohomologically and in terms of 1 cocycles on g this is used to produce new examples of pro p groups which do not occur as maximal pro p galois groups of fields as above

differential forms and bilinear forms under field extensions abstract let f be a field of characteristic p 0 let n f be the f vector space of n differentials of f over f p let k f g be the function field of an irreducible polynomial g in m 1 variables over f we derive an explicit description of the kernel of the restriction map n f n k as an application in the case p 2 we determine the kernel of the restriction map when passing from the witt ring resp graded witt ring of symmetric bilinear forms over f to that over such a function field extension k

unramified cohomology and witt groups of anisotropic pfister quadrics the unramified witt group of an anisotropic conic over a field k with char k 0 2 defined by the form 1 a b is known to be a quotient of the witt group w k of k and isomorphic to w k 1 a b ab w k we compute the unramified cohomology group h43rk c where c is the three dimensional anisotropic quadric defined by the quadratic form 1 a b ab c over k we use these computations to study the unramified witt group of c let k be a field of characteristic not two and k k be a finitely generated field extension the n th unramified cohomology group of k over k with coefficients in z 2 cf ct is denoted hnnr k k or just hnnr k when the ground field being considered is clear it is a subgroup of the galois cohomology group hn k z 2 abbreviated to hn k and is defined as hnnr k k n h6nt spec ov u2 vev k here v k is the set of all rank one discrete valuations on k that are trivial on k for v e v k ov denotes the corresponding discrete valuation ring it is well known that the group hnt spec ov j2 injects into hn k if x k is a smooth projective geometrically integral variety and k x is its function field then by abuse of terminology we refer to the groups hnrn k x as the unramified cohomology groups of x for a field f let w f be the witt group of quadratic forms over f sc for k and k k as above we can similarly define the unramified witt group of k k denoted wtr k k as the subgroup of w k defined by wnr k k n w ov vev k here w ov is the witt group of the discrete valuation ring ov which is a subgroup of w k sc chapter vi if x k is smooth projective then we refer to wnr k x as the unramified witt group of x recall that a variety x is said to be k rational if x is k birational to the projective space then the natural maps hn k hn k x resp w k w k x induce isomorphisms htn k hnrr k x resp w k wnr k x thus the unramified cohomology groups of a k rational variety resp the unramified witt received by the editors november 7 1995 1991 mathematics subject classification primary iie70 secondary 13k05 12g05

decomposable and indecomposable algebras of degree 8 and exponent 2 we study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras we consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension let b be a biquaternion algebra over f a with trivial corestriction a degree 3 cohomological invariant is defined and we show that it determines whether b has a descent to f this invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2 cohomological dimension 3 and over a field m t where the u invariant of m is 8 and t is an indeterminate the construction of these indecomposable algebras uses chow group computations provided by merkurjev in appendix

annihilators of quadratic and bilinear forms over fields of characteristic two abstract let f be a field with 2 0 w f the witt ring of symmetric bilinear forms over f and w q f the w f module of quadratic forms over f let i f w f be the maximal ideal we compute explicitly in i f m and i m w q f the annihilators of n fold bilinear and quadratic pfister forms thereby answering positively in the case 2 0 certain conjectures stated by kruskemper in m kruskemper on annihilators in graded witt rings and in milnor s k theory in b jacob et al eds recent advances in real algebraic geometry and quadratic forms in contemp math vol 155 1994 pp 307 320

cohomological kernels of mixed extensions in characteristic 2 abstract let f be a field of characteristic 2 f m the space of m differential forms over f d f m 1 f m the differential operator and h 2 m 1 f the cokernel of the artin schreier operator f m f m d f m 1 given on generators 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 d f m 1 it was shown in 8 that given a multiquadratic purely inseparable extension l f and a separable quadratic extension k f then the kernel of the natural homomorphism h 2 m 1 f h 2 m 1 k l is equal to the sum of the kernels of the extensions l f and k f in this paper we extend this result to the compositum of a finite purely inseparable multiquadratic extension with a separable biquadratic extension and we prove that this splitting property of kernels is no longer true by computing the kernel of the compositum of a separable quadratic or biquadratic extension with a simple purely inseparable extension of arbitrary degree

sheaf quantization of hamiltonian isotopies and applications to nondisplaceability problems let i be an open interval m be a real manifold t m its cotangent bundle and phi phi t t in i a homogeneous hamiltonian isotopy of t m defined outside the zero section let lambda be the conic lagrangian submanifold associated with phi lambda is a subset of t m x t m x t i we prove the existence and unicity of a sheaf k on mxmxi whose microsupport is contained in the union of lambda and the zero section and whose restriction to t 0 is the constant sheaf on the diagonal of mxm we give applications of this result to problems of non displaceability in contact and symplectic topology in particular we prove that some strong morse inequalities are stable by hamiltonian isotopies and we also give results of non displaceability for positive isotopies in the contact setting in this new version we suppress one hypothesis in the main theorem and we extend the result of non displaceability for positive isotopies

period index and u invariant questions for function fields over complete discretely valued fields let k be a complete discretely valued field with residue field and f the function field of a curve over k let p be the characteristic of and l a prime not equal to p if the brauer l dimensions of all finite extensions of are bounded by d and the brauer l dimensions of all extensions of of transcendence degree at most 1 are bounded by d 1 then it is known that the brauer l dimension of f is at most d 2 lieblich in j reine angew math 659 1 41 2011 saltman in j ramanujan math soc 12 25 47 1997 harbater et al in invent math 178 231 263 2009 in this paper we give a bound for the brauer p dimension of f in terms of the p rank of as an application we show that if is a perfect field of characteristic 2 then any quadratic form over f in at least 9 variables is isotropic this leads to the fact that every element in h 3 f mathbb z 2 mathbb z is a symbol if is a finite field of characteristic 2 u f 8 is a result of heath brown leep heath brown in compos math 146 271 287 2010 leep in j reine angew math 2013 to appear

compatible valuations and generalized milnor k theory given a field and a subgroup of there is a minimal group for which there exists an compatible valuation whose units are contained in assuming that has finite index in and contains for prime we describe in computable theoretic terms