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In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in [3] that R ζ = c ζ 2 where R ζ is their map on K 3 defined using the cyclic quantum dilogarithm and c ζ is the Chern class map on K 3. [ABSTRACT FROM AUTHOR]

Let A be a local domain of characteristic 2 such that its residue field has more than 64 elements. Then we find an exact relation between the third integral homology of the group SL 2 (A) and Hutchinson's refined Bloch group RB (A). [ABSTRACT FROM AUTHOR]

We derive cancellation-free Chevalley-type multiplication formulas for the T -equivariant quantum K -theory ring of Grassmannians of type A and C , and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley formula in the T -equivariant quantum K -theory ring of arbitrary flag manifolds G / B , which was derived earlier in terms of the quantum alcove model, by the last three authors. [ABSTRACT FROM AUTHOR]

We give a brief review of the cohomological Hall algebra CoHA H and the K-theoretical Hall algebra KHA R associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) R → H ˆ σ ˜ where H ˆ σ ˜ is a Zhang twist of the completion of H. Moreover, we establish the equivalence of categories of "locally finite" graded modules H - Mod l f ≃ R Q - Mod l f. Examples of locally finite H ˆ -, resp. R Q -modules appear naturally as the cohomology, resp. K-theory, of framed moduli spaces of quivers. [ABSTRACT FROM AUTHOR]

We show that any pointed, preordered module map BF gr (E) → BF gr (F) between Bowen-Franks modules of finite graphs can be lifted to a unital, graded, diagonal preserving ⁎-homomorphism L ℓ (E) → L ℓ (F) between the corresponding Leavitt path algebras over any commutative unital ring with involution ℓ. Specializing to the case when ℓ is a field, we establish the fullness part of Hazrat's conjecture about the functor from Leavitt path ℓ -algebras of finite graphs to preordered modules with order unit that maps L ℓ (E) to its graded Grothendieck group. Our construction of lifts is of combinatorial nature; we characterize the maps arising from this construction as the scalar extensions along ℓ of unital, graded ⁎-homomorphisms L Z (E) → L Z (F) that preserve a sub-⁎-semiring introduced here. [ABSTRACT FROM AUTHOR]

Proto-exact categories , introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B -modules as proto-exact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfield K , a well-known relation between finite K -modules and finite incidence geometries yields a combinatorial interpretation of exact sequences. [ABSTRACT FROM AUTHOR]

The definition of Milnor-Witt cycle modules in [5] can easily be adapted over general regular base schemes. However, there are simple examples (see (2.9)) to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying an extra axiom, called here as R5, the expected properties of exactness of Gersten complex and A 1 -invariance hold. Moreover R5 is vacuously satisfied when the base is a field and it is also satisfied by K MW over any base. As a corollary, we obtain the strict A 1 -invariance and the exactness of Gersten complex for K MW over an excellent DVR. [ABSTRACT FROM AUTHOR]

We develop the K -theory of sets with an action of a pointed monoid (or monoid scheme), analogous to the K -theory of modules over a ring (or scheme). In order to form localization sequences, we construct the quotient category of a nice regular category by a Serre subcategory. [ABSTRACT FROM AUTHOR]

Owing to the difference in K -theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for the Frobenius pair of faithful functors between two abelian categories to be a Quillen equivalence, which is also equivalent to that the Frobenius functors induce mutually inverse equivalences between stable categories of Gorenstein projective objects. We show that the category of Gorenstein projective objects is a Waldhausen category, then Gorenstein K -groups are introduced and characterized. As applications, we show that stable equivalences of Morita type preserve Gorenstein K -groups, CM-finiteness and CM-freeness. Two specific examples of path algebras are presented to illustrate the results, for which the Gorenstein K 0 and K 1 -groups are calculated. [ABSTRACT FROM AUTHOR]

We introduce a ring of Z -valued functions on a complete fan Δ called Grothendieck weights to describe the ordinary operational K -theory of the associated toric variety X. These functions satisfy a K -theoretic analogue of the balancing condition for Minkowski weights, which is induced by a presentation of the Grothendieck group of X. We explicitly give a combinatorial presentation in low dimensions, and relate Grothendieck weights to other fan-based invariants such as piecewise exponential functions and Minkowski weights. As an application, we give an example of a projective toric surface X such that the forgetful map K T ∘ (X) → K ∘ (X) is not surjective. [ABSTRACT FROM AUTHOR]

We consider Bott-Samelson varieties BS c (s) for a semisimple compact Lie group C corresponding to sequences of (not necessarily simple) reflections s. Let n be the length of s , K be a maximal torus in C and W be the Weyl group of C. For any set R of not overlapping integer pairs (i , j) such that 1 ⩽ i ⩽ j ⩽ n and a function v : R → W , we consider the subspace BS c (s , v) ⊂ BS c (s) of solutions of the equations in C / K requiring that the K -orbit of the product of coordinates counted from i to j be equal to the K -orbit of v evaluated at (i , j) ∈ R. We decompose BS c (s , v) into a twisted product (in the sense of iterated fibre bundles) of smaller Bott-Samelson varieties BS c (t) and the fibres of the canonical projections from BS c (t) to the flag variety. Finally, we prove the tensor product decomposition for the K -equivariant cohomology of BS c (s , v). [ABSTRACT FROM AUTHOR]

Green-Griffiths observed that we could eliminate obstructions to deforming divisors. Motivated by recent work of Bloch-Esnault-Kerz on deformation of algebraic cycle classes, we use Chern character to generalize Green-Griffiths' observation and to show how to eliminate obstructions to deforming cycles of codimension p. [ABSTRACT FROM AUTHOR]

We develop a version of Bass-Serre theory for Lie algebras (over a field k) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to N -graded Lie algebras: for example, we show that one relator N -graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras. [ABSTRACT FROM AUTHOR]

We calculate the structure of H 3 (SL 2 (Q) , Z [ 1 2 ]). Let H 3 (SL 2 (Q) , Z) 0 denote the kernel of the (split) surjective homomorphism H 3 (SL 2 (Q) , Z) → K 3 ind (Q). Each prime number p determines an operator 〈 p 〉 on H 3 (SL 2 (Q) , Z) with square the identity. We prove that H 3 (SL 2 (Q) , Z [ 1 2 ]) 0 is the direct sum of the (− 1) -eigenspaces of these operators. The (− 1) -eigenspace of 〈 p 〉 is the scissors congruence group, over Z [ 1 2 ] , of the field F p , which is a cyclic group whose order is the odd part of p + 1. [ABSTRACT FROM AUTHOR]

Let X be a smooth algebraic variety over an arbitrary field. Let φ be the canonical surjective homomorphism of the Chow ring of X onto the ring associated with the Chow filtration on the Grothendieck ring K (X). We remark that φ is injective if and only if the connective K-theory CK (X) coincides with the terms of the Chow filtration on K (X). As a consequence, CK (X) turns out to be computed for numerous flag varieties (under semisimple algebraic groups) for which the injectivity of φ had already been established. This especially applies to the so-called generic flag varieties X of many different types, identifying for them CK (X) with the terms of the explicit Chern filtration on K (X). Besides, for arbitrary X , we compare CK (X) with the fibered product of the Chow ring of X and the graded ring formed by the terms of the Chow filtration on K (X). [ABSTRACT FROM AUTHOR]

We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme X , the graded ring that our functor associates to X is related to the associated graded ring of the γ -filtration on the Grothendieck ring of finite rank locally free sheaves on X via a Grothendieck-Riemann-Roch type theorem. [ABSTRACT FROM AUTHOR]

The aim of the paper is to define an infinitesimal analog of the Bloch regulator, which attaches to a pair of meromorphic functions on a Riemann surface, a line bundle with connection on the punctured surface. In the infinitesimal context, we consider a pair (X , X _) of schemes over a field of characteristic 0, such that the regular scheme X _ is defined in X by a square-zero sheaf of ideals which is locally free on X _. We propose a definition of the weight two motivic cohomology of X based on the Bloch group, which is defined in terms of the functional equation of the dilogarithm. The analog of the Bloch regulator is a map from a subspace of the infinitesimal part of H M 2 (X , Q (2)) to the first cohomology group of the Zariski sheaf associated to an André-Quillen homology group. Using Goodwillie's theorem, we deduce that this map is an isomorphism, which is an infinitesimal analog of the injectivity conjecture for the Bloch regulator. [ABSTRACT FROM AUTHOR]

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic two. Any non-trivial self-dual irreducible K G -module W admits a non-degenerate G -invariant alternating bilinear form, thus giving a representation f : G → Sp (W). In the case where G = SL n (K) and W has highest weight ϖ 1 + ϖ n − 1 , and in the case where G = Sp 2 n (K) and W has highest weight ϖ 2 , we determine for every unipotent element u ∈ G the conjugacy class of f (u) in Sp (W). As a part of this result, we describe the conjugacy classes of unipotent elements of Sp (V 1) ⊗ Sp (V 2) in Sp (V 1 ⊗ V 2). [ABSTRACT FROM AUTHOR]

We introduce preordered semi-orthogonal decompositions (psod-s) of dg-categories. We show that homotopy limits of dg-categories equipped with compatible psod-s carry a natural psod. This gives a way to glue semi-orthogonal decompositions along faithfully flat covers, extending the main result of [4]. As applications we will construct semi-orthogonal decompositions for root stacks of log pairs (X , D) where D is a (not necessarily simple) normal crossing divisor, generalizing results from [17] and [3]. Further we will compute the Kummer flat K-theory of general log pairs (X , D) , generalizing earlier results of Hagihara and Nizioł in the simple normal crossing case [15] , [23]. [ABSTRACT FROM AUTHOR]

Let ℓ be a commutative ring with unit. Garkusha constructed a functor from the category of ℓ -algebras into a triangulated category D , that is a universal excisive and homotopy invariant homology theory. Later on, he provided different descriptions of D , as an application of his motivic homotopy theory of algebras. Using these, it can be shown that D is triangulated equivalent to a category, denote it by K , whose objects are pairs (A , m) with A an ℓ -algebra and m an integer, and whose Hom-sets can be described in terms of homotopy classes of morphisms. All these computations, however, require a heavy machinery of homotopy theory. In this paper, we give a more explicit construction of the triangulated category K and prove its universal property, avoiding the homotopy-theoretic methods and using instead the ones developed by Cortiñas-Thom for defining kk -theory. Moreover, we give a new description of the composition law in K , mimicking the one in the suspension-stable homotopy category of bornological algebras defined by Cuntz-Meyer-Rosenberg. We also prove that the triangulated structure in K can be defined using either extension or mapping path triangles. [ABSTRACT FROM AUTHOR]

Given a quasi-compact and quasi-separated (qcqs) scheme X of finite type over a Noetherian ring R having an ample family of line bundles, we construct a closed embedding of X into a smooth (qcqs) scheme over R having an ample family of line bundles. Such a smooth scheme arises as an open subscheme of the multihomogeneous Proj of a Z n -graded ring. [ABSTRACT FROM AUTHOR]

There is a tight relation between the geometry of a directed graph and the algebraic structure of a Leavitt path algebra associated to it. In this note, we show a similar connection between the geometry of the graph and the structure of a certain monoid associated to it. This monoid is isomorphic to the positive cone of the graded K 0 -group of the Leavitt path algebra which is naturally equipped with a Z -action. As an example, we show that a graph has a cycle without an exit if and only if the monoid has a periodic element. Consequently a graph has Condition (L) if and only if the group Z acts freely on the monoid. We go on to show that the algebraic structure of Leavitt path algebras (such as simplicity, purely infinite simplicity, or the lattice of ideals) can be described completely via this monoid. Therefore an isomorphism between the monoids (or graded K 0 's) of two Leavitt path algebras implies that the algebras have similar algebraic structures. These all bolster the claim that the graded Grothendieck group could be a sought-after complete invariant for the classification of Leavitt path algebras. [ABSTRACT FROM AUTHOR]

We study the double Grothendieck polynomials of Kirillov–Naruse for the symplectic and odd orthogonal Grassmannians. These functions are explicitly written as Pfaffian sum form and are identified with the stable limits of fundamental classes of the Schubert varieties in torus equivariant connective K -theory of these isotropic Grassmannians. We also provide a combinatorial description of the ring formally spanned be the double Grothendieck polynomials. [ABSTRACT FROM AUTHOR]

• Introduces the notion of an opposite to a (skew left) brace. • (Known) Skew left braces provide solutions to the Yang-Baxter equation; (new) opposite braces provide inverse solutions. • The inverse solution to the YBE gives information about the coalgebra structure of a Hopf-Galois structure. • The opposite brace gives information on the intermediate fields found through the associated Hopf-Galois correspondence. Given a skew left brace B , we introduce the notion of an "opposite" skew left brace B ′ , which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by B ′ is the inverse to the solution given by B. Every Hopf-Galois structure on a Galois field extension L / K gives rise to a skew left brace B ; if the underlying Hopf algebra is not commutative, then one can construct an additional "opposite" Hopf-Galois structure (see [1] , which relates the Hopf-Galois module structures of each, and refers to the structures as "commuting"); the corresponding skew left brace to this second structure is precisely B ′. We show how left ideals (and a newly introduced family of quasi-ideals) of B ′ allow us to identify the intermediate fields of L / K which occur as fixed fields of sub-Hopf algebras under this correspondence and to identify which of these are Galois, or Hopf-Galois, over K. Finally, we use the opposite to connect the inverse solution to the YBE and the structure of the Hopf algebra H acting on L / K ; this allows us to identify the group-like elements of H. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an infinite-dimensional Lie algebra D S for any abelian group S. If S is the additive group of integers, D S reduces to the q -Virasoro algebra D q introduced by Belov and Chaltikian in the study of lattice conformal theories. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra g S with S as an automorphism group and we prove that D S is isomorphic to the S -covariant algebra of the affine Lie algebra g S ˆ. We then relate restricted D S -modules of level ℓ ∈ C to equivariant quasi modules for the vertex algebra V g S ˆ (ℓ , 0) associated to g S ˆ with level ℓ. Furthermore, we establish an intrinsic connection between the q -Virasoro algebra D q and affine Kac-Moody Lie algebras. More specifically, we show that if S is a finite abelian group of order 2 l + 1 , D S is isomorphic to the affine Kac-Moody algebra of type B l (1). [ABSTRACT FROM AUTHOR]

Abstract We prove versions of the Suslin and Gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic K -theory, presheaves with equivariant transfers, equivariant Suslin homology, and Bredon motivic cohomology. [ABSTRACT FROM AUTHOR]

Let G be a reductive group over an algebraically closed field F of characteristic zero such that the Dynkin diagram of G is the disjoint union of diagrams of type A . We prove that the third unramified cohomology group of the classifying space of G with coefficients in Q / Z is trivial. [ABSTRACT FROM AUTHOR]

We prove a special case of a conjecture of Dao–Kurano concerning the vanishing of Hochster's theta pairing. The proof uses Adams operations on both topological K -theory and perfect complexes with support. [ABSTRACT FROM AUTHOR]

Ulf Rehmann and Jun Morita, in their 1989 paper A Matsumoto Type Theorem for Kac–Moody Groups , gave a presentation of K 2 ( A , F ) for any generalised Cartan matrix A and field F . The purpose of this paper is to use this presentation to compute K 2 ( A , F ) more explicitly in the case when A is hyperbolic. In particular, we shall show that these K 2 ( A , F ) can always be expressed as a product of quotients of K 2 ( F ) and K 2 ( 2 , F ) . Along the way, we shall also prove a similar result in the case when A has an odd entry in each column. [ABSTRACT FROM AUTHOR]

Graph monoids arise naturally in the study of non-stable K-theory of graph C*-algebras and Leavitt path algebras. They play also an important role in the current approaches to the realization problem for von Neumann regular rings. In this paper, we characterize when a finitely generated conical refinement monoid can be represented as a graph monoid. The characterization is expressed in terms of the behavior of the structural maps of the associated I -system at the free primes of the monoid. [ABSTRACT FROM AUTHOR]

We show that under favorable circumstances, one can construct an intersection product on the Chow groups of a tensor triangulated category T (as defined in [5] ) which generalizes the usual intersection product on a non-singular algebraic variety. Our construction depends on the choice of an algebraic model for T (a tensor Frobenius pair ), which has to satisfy a K-theoretic regularity condition analogous to the Gersten conjecture from algebraic geometry. In this situation, we are able to prove an analogue of the Bloch formula and use it to define an intersection product similar to Grayson's construction from [14] . We then recover the usual intersection product on a non-singular algebraic variety assuming a K-theoretic compatibility condition. [ABSTRACT FROM AUTHOR]

We classify central extensions of a reductive group G by K 3 and B K 3 , the sheaf of third Quillen K-theory groups and its classifying stack. These turn out to be parametrized by the group of Weyl-invariant quadratic forms on the cocharacter lattice valued in k × and the group of integral Weyl-invariant cubic forms on the cocharacter lattice respectively. [ABSTRACT FROM AUTHOR]

We show that the Loday assembly map for the algebraic K-theory for a group algebra of a finite group with coefficient ring a finite field is in general not injective. [ABSTRACT FROM AUTHOR]

We show that the long exact sequence for K -groups of Leavitt path algebras deduced by Ara, Brustenga, and Cortiñas extends to Leavitt path algebras of countable graphs with infinite emitters in the obvious way. Using this long exact sequence, we compute explicit formulas for the higher algebraic K -groups of Leavitt path algebras over certain fields, including all finite fields and all algebraically closed fields. We also examine classification of Leavitt path algebras using K -theory. It is known that the K 0 -group and K 1 -group do not suffice to classify purely infinite simple unital Leavitt path algebras of infinite graphs up to Morita equivalence when the underlying field is the rational numbers. We prove for these Leavitt path algebras, if the underlying field is a number field (which includes the case when the field is the rational numbers), then the pair consisting of the K 0 -group and the K 6 -group does suffice to classify these Leavitt path algebras up to Morita equivalence. [ABSTRACT FROM AUTHOR]

This is the first out of two papers on independent resolutions for totally disconnected dynamical systems. In the present paper, we discuss independent resolutions from an algebraic point of view. We also present applications to group homology and cohomology. This first paper sets the stage for our second paper, where we explain how to use independent resolutions in K-theory computations for crossed products attached to totally disconnected dynamical systems. [ABSTRACT FROM AUTHOR]

I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev-Panin's computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneous varieties. As a second application, we show that the noncommutative motive of a twisted projective homogeneous variety is trivial if and only if the Brauer classes of the associated central simple algebras are trivial. Along the way we construct a fully-faithful ⊗-functor from Merkurjev-Panin's motivic category to Kontsevich's category of noncommutative Chow motives, which is of independent interest. [ABSTRACT FROM AUTHOR]

Abstract: We show that of “sufficiently regular” localisations of local rings (e.g. inverting a sequence of regular parameters) can be described by the Steinberg presentation. The proof is inductive on the number of irreducible elements being inverted, successively using a generalisation of a co-Cartesian square first exploited by Dennis and Stein. [Copyright &y& Elsevier]

Abstract: We introduce an equivariant algebraic kk-theory for G-algebras and G-graded algebras. We study some adjointness theorems related with crossed product, trivial action, induction and restriction. In particular we obtain an algebraic version of the Green–Julg Theorem which gives us a computational tool. [Copyright &y& Elsevier]

Abstract: Let R be a clean ring with an ideal I such that is semiperfect and is torsion-free. We prove that, under some mild conditions, units in can be lifted to units in R. This implies that the matrix ring over Bergmanʼs example of a non-clean exchange ring R is not clean, for every n. We also obtain some other results concerning lifting units in clean rings. [Copyright &y& Elsevier]

Abstract: We show that a Pythagorean field (more generally, a reduced abstract Witt ring) has finite stability index if and only if it has finite 2-symbol length. We give explicit bounds for the two invariants in terms of one another. To approach the question whether those bounds are optimal we consider examples of Pythagorean fields. [Copyright &y& Elsevier]

Abstract: 1. Let be central simple disjoint algebras over a field F. Let also , , , and for each , let and have the same sets of prime divisors. Then there exists a field extension such that and , . 2. Let be a central simple algebra over a field K with an involution τ of the second kind. We prove that there exists a regular field extension preserving indices of central simple K-algebras such that is cyclic and has an involution of the second kind extending τ. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

Abstract: For a commutative ring R with many units, we describe the kernel of . Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable part of . [Copyright &y& Elsevier]

Abstract: We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations induce Morita equivalence of the corresponding Leavitt path algebras. As a consequence we obtain our two main results: the first gives sufficient conditions for which the Leavitt path algebras in a certain class are Morita equivalent, while the second gives sufficient conditions which yield isomorphisms. We discuss a possible approach to establishing whether or not these conditions are also in fact necessary. In the final section we present many additional operations on graphs which preserve Morita equivalence (resp. isomorphism) of the corresponding Leavitt path algebras. [Copyright &y& Elsevier]

Abstract: Let C be a cyclic 2-group that acts fixed-point-freely on a group K, and let be the subgroup of index 2. The main result of this paper is that the square-free parts of the degrees of the T-invariant irreducible characters of K are never divisible by primes mod . [ABSTRACT FROM AUTHOR]

Abstract: For any , we show that the cone of -secant lines of a closed subscheme over an algebraically closed field K running through a closed point is defined by the k-th partial elimination ideal of Z with respect to p. We use this fact to give an algorithm for computing secant cones. Also, we show that under certain conditions partial elimination ideals describe the length of the fibres of a multiple projection in a way similar to the way they do for simple projections. Finally, we study some examples illustrating these results, computed by means of Singular. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We study the K-group K1 of the group ring for a finite group over a coefficient ring which is p-adically complete and admits a lift of Frobenius. In this paper, we consider the image of K1 under the determinant map; the central tool is the group logarithm which we can define using the Frobenius lift. Using this we prove a fixed point theorem for the determinantal image of K1. [ABSTRACT FROM AUTHOR]

Abstract: 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. [ABSTRACT FROM AUTHOR]