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We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the generic point of the special fibre. This complements results of Gillet-Levine for K -theory, Geisser for motivic cohomology and Schmidt-Strunk and the author for étale cohomology. [ABSTRACT FROM AUTHOR]

We compute the algebraic K -theory of some classes of surfaces defined over finite fields. We achieve this by first calculating the motivic cohomology groups and then studying the motivic Atiyah-Hirzebruch spectral sequence. In an appendix, we slightly enlarge the class of surfaces for which Parshin's conjecture is known. [ABSTRACT FROM AUTHOR]

In the article The third homology of S L 2 (Q) ([7]), Hutchinson determined the structure of H 3 (SL 2 (Q) , Z [ 1 2 ]) by expressing it in terms of K 3 ind (Q) ≅ Z / 24 and the scissor congruence group of the residue field F p with p a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm. [ABSTRACT FROM AUTHOR]

We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a field. We show that the category of L -modules is equivalent to a full subcategory of quiver representations. When Γ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of L after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to L having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of L -modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, F Γ -modules) via a new colimit model for M ⊗ F Γ L. [ABSTRACT FROM AUTHOR]

Let v be a Krull valuation of a field K with valuation ring R v and K 1 , K 2 be finite separable extensions of K which are linearly disjoint over K. Assume that the integral closure of R v in the composite field K 1 K 2 is a free R v -module. For a given pair of prolongations v 1 , v 2 of v to K 1 , K 2 respectively, it is shown that there exists a unique prolongation w of v to K 1 K 2 which extends both v 1 , v 2. Moreover with S i as the integral closure of R v in K i , if the ring S 1 S 2 is integrally closed and the residue field of v is perfect, then f (w / v) = f (v 1 / v) f (v 2 / v) , where f (v ′ / v) stands for the degree of the residue field of a prolongation v ′ of v over the residue field of v. As an application, it is deduced that if K 1 , K 2 are algebraic number fields which are linearly disjoint over K = K 1 ∩ K 2 , then the number of prime ideals of the ring A K 1 K 2 of algebraic integers of K 1 K 2 lying over a given prime ideal ℘ of A K equals the product of the numbers of prime ideals of A K i lying over ℘ for i = 1 , 2. [ABSTRACT FROM AUTHOR]

We consider local Gorenstein duality for cochain spectra C ⁎ (B G ; R) on the classifying spaces of compact Lie groups G over complex orientable ring spectra R. We show that it holds systematically for a large array of examples of ring spectra R , including Lubin-Tate theories, topological K -theory, and various forms of topological modular forms. We also prove a descent result for local Gorenstein duality which allows us to access further examples. [ABSTRACT FROM AUTHOR]

Let F : A → B be an exact functor between small exact categories. We study the zeroth homotopy group K 0 ( F ) of the homotopy fiber of the map K ( A ) → K ( B ) between K -theory spectra. Under the assumption that F is a cofinal and that B is split exact, we give an explicit description of K 0 ( F ) in terms of the triangulated functor D b ( A ) → D b ( B ) between the derived categories. We apply it to the pair ( X , D ) of a scheme X and an affine closed subscheme D of X , and get a description of the relative K 0 -group K 0 ( X , D ) in terms of perfect complexes; it is generated by pairs of two perfect complexes of X together with quasi-isomorphisms along D . This description makes it possible to assign a cycle class in K 0 ( X , D ) to a cycle on X not meeting D in an intuitive way. When X is a separated regular scheme of finite type over a field and D is an affine effective Cartier divisor on X , we prove that the cycle classes induce a surjective group homomorphism from the Chow group with modulus CH ⁎ ( X | D ) defined by Binda–Saito to a suitable subquotient of K 0 ( X , D ) . [ABSTRACT FROM AUTHOR]

We show that the homotopy invariant algebraic K -theory of Weibel vanishes below the negative of the Krull dimension of a noetherian scheme. This gives evidence for a conjecture of Weibel about vanishing of negative algebraic K -groups. [ABSTRACT FROM AUTHOR]

We show that certain isomorphisms of (twisted) KR -groups that underlie T-dualities of torus orientifold string theories have purely algebraic analogues in terms of algebraic K -theory of real varieties and equivalences of derived categories of (twisted) coherent sheaves. The most interesting conclusion is a kind of Mukai duality in which the “dual abelian variety” to a smooth projective genus-1 curve over R with no real points is (mildly) noncommutative. [ABSTRACT FROM AUTHOR]

We introduce a general framework to unify several variants of twisted topological K -theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck–Witt groups provide interesting examples of twisted K -theory. These groups are linked with the classification of algebraic vector bundles on real algebraic varieties. [ABSTRACT FROM AUTHOR]

Let Q be a commutative, Noetherian ring and Z ⊆ Spec ( Q ) a closed subset. Define K 0 Z ( Q ) to be the Grothendieck group of those bounded complexes of finitely generated projective Q -modules that have homology supported on Z . We develop “cyclic” Adams operations on K 0 Z ( Q ) and we prove these operations satisfy the four axioms used by Gillet and Soulé in [9] . From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soulé in certain cases. Our definition of the cyclic Adams operators is inspired by a formula due to Atiyah [1] . They have also been introduced and studied before by Haution [10] . [ABSTRACT FROM AUTHOR]

In this article, we prove some results on Witt, Grothendieck–Witt ( GW ) and K -theory of noetherian quasi-projective schemes X , over affine schemes Spec ( A ) . For integers k ≥ 0 , let C M k ( X ) denote the category of coherent O X -modules F , with locally free dimension dim V ( X ) ⁡ ( F ) = k = grade ( F ) . We prove that there is an equivalence D b ( C M k ( X ) ) → D k ( V ( X ) ) of the derived categories. It follows that there is a sequence of zig-zag maps K ( C M k + 1 ( X ) ) ⟶ K ( C M k ( X ) ) ⟶ ∐ x ∈ X ( k ) K ( C M k ( X x ) ) of the K -theory spectra that is a homotopy fibration. In fact, this is analogous to the homotopy fiber sequence of the G -theory spaces of Quillen (see proof of [16, Theorem 5.4] ). We also establish similar homotopy fibrations of GW -spectra and G W -bispectra, by application of the same equivalence theorem. [ABSTRACT FROM AUTHOR]

This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection? We prove that such full exceptional collection exists if and only if the geometric genus of the curve equals to 0. Moreover we can also prove that a curve with geometric genus equal or greater than 1 cannot have a categorical resolution of singularities which has a tilting object. The proofs of both results are given by a careful study of the Grothendieck group and the Picard group of that curve. [ABSTRACT FROM AUTHOR]

Let ℓ be a commutative ring with involution ⁎ containing an element λ such that λ + λ ⁎ = 1 and let Alg ℓ ⁎ be the category of ℓ -algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category k k h and a functor j h : Alg ℓ ⁎ → k k h that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in k k h. By the universal property of the latter, this implies that any functor H : Alg ℓ ⁎ → T with values in a triangulated category which is homotopy invariant, matricially and hermitian stable and excisive satisfies the fundamental theorem. We also prove a bivariant version of Karoubi's 12-term exact sequence. [ABSTRACT FROM AUTHOR]

We develop the theory of a category C A which is a generalisation to non-restricted g -modules of a category famously studied by Andersen, Jantzen and Soergel for restricted g -modules, where g is the Lie algebra of a reductive group G over an algebraically closed field K of characteristic p > 0. Its objects are certain graded bimodules. On the left, they are graded modules over an algebra U χ associated to g and to χ ∈ g ⁎ in standard Levi form. On the right, they are modules over a commutative Noetherian S (h) -algebra A , where h is the Lie algebra of a maximal torus of G. We define here certain important modules Z A , χ (λ) , Q A , χ I (λ) and Q A , χ (λ) in C A which generalise familiar objects when A = K , and we prove some key structural results regarding them. [ABSTRACT FROM AUTHOR]

We give a topological construction of graded vertex F -algebras by generalizing Joyce's vertex algebra construction to complex-oriented homology. Given an H-space X with a B U (1) -action, a choice of K-theory class, and a complex oriented homology theory E , we build a graded vertex F -algebra structure on E ⁎ (X) where F is the formal group law associated with E. [ABSTRACT FROM AUTHOR]

We study the algebraic K -theory and Grothendieck–Witt theory of proto-exact categories, with a particular focus on classes of examples of F 1 -linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the K -theory or Grothendieck–Witt theory spaces of proto-exact categories defined using the (hermitian) Q -construction and group completion. [ABSTRACT FROM AUTHOR]

Given a finite group G acting on a ring R , Merling constructed an equivariant algebraic K -theory G -spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction and have therefore not yet been calculated. In this work, we provide an interpretation of the homotopy Mackey functors of equivariant algebraic K -theory in terms of a purely algebraic construction. In particular, we construct Mackey functors out of the n th algebraic K -groups of group rings whose multiplication is twisted by the group action. Restrictions and transfers for these functors admit a tractable algebraic description in that they arise from restriction and extension of scalars along homomorphisms of twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980's which constructs Mackey functors out of the algebraic K -theory of group rings. We develop many families of examples of Mackey functors, both new and old, including K -theory of endomorphism rings, the K -theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings. [ABSTRACT FROM AUTHOR]

We provide descriptions of the Whitehead groups, and the algebraic K -theory groups, of the fundamental group of a connected, oriented, closed 3-manifold in terms of Whitehead groups of their finite subgroups and certain Nil-groups. The main tools we use are: the K-theoretic Farrell-Jones isomorphism conjecture, the construction of models for the universal space for the family of virtually cyclic subgroups in 3-manifold groups, and both the prime and JSJ-decompositions together with the well-known geometrization theorem. [ABSTRACT FROM AUTHOR]

The purpose of this article is to show that the bivariant algebraic A -cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work. [ABSTRACT FROM AUTHOR]

We construct the topological Fukaya category of a surface with genus greater than one, making this model intrinsic to the topology of the surface. Instead of using the area form of the surface, we use an admissibility condition borrowed from Heegaard-Floer theory which ensures invariance under isotopy. In this paper we show finiteness of the moduli space using purely topological means and compute the Grothendieck group of the topological Fukaya category. We also show the faithfulness of MCG-action on the topological Fukaya category in this setup. [ABSTRACT FROM AUTHOR]

Grothendieck-Witt spectra represent higher Grothendieck-Witt groups and higher Hermitian K-theory in particular. A description of the Grothendieck-Witt spectrum of a finite dimensional projective bundle P (E) over a base scheme X is given in terms of the Grothendieck-Witt spectra of the base, using the dg category of strictly perfect complexes, provided that X is a scheme over Spec Z [ 1 / 2 ] and satisfies the resolution property, e.g. if X has an ample family of line bundles. [ABSTRACT FROM AUTHOR]

Abstract: We show that, for a complete simplicial toric variety , we can determine its KH-theory entirely in terms of the torus pieces of open sets forming an open cover of . We then construct conditions under which, given two complete simplicial toric varieties, the two spectra and are weakly equivalent. We apply this result to determine the rational KH-theory of weighted projective spaces. We next examine K-regularity for complete toric surfaces; in particular, we show that complete toric surfaces are K0-regular. We then determine conditions under which our approach for dimension 2 works in arbitrary dimensions, before demonstrating that weighted projective spaces are not K1-regular, and for dimensions bigger than 2 are also not in general K0-regular. [Copyright &y& Elsevier]

Abstract: We formulate a “correct” version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory “homological symbols algebra”. As a byproduct, we prove the Quillen conjecture in homological degree two for the rank two and the prime 5. [Copyright &y& Elsevier]

Abstract: Given a homomorphism we show that the natural map from the Whitehead group of to the Whitehead group of the Novikov ring is surjective. The group is of interest for the simple chain homotopy type of the Novikov complex. It also contains the Latour obstruction for the existence of a nonsingular closed 1-form within a fixed cohomology class , where is a closed connected smooth manifold. [Copyright &y& Elsevier]

Abstract: Let be the nth Morava K-theory spectrum. Let be the Lubin–Tate spectrum, which plays a central role in understanding , the -local sphere. For any spectrum X, define to be . Let G be a closed subgroup of the profinite group , the group of ring spectrum automorphisms of in the stable homotopy category. We show that is a continuous G-spectrum, with homotopy fixed point spectrum . Also, we construct a descent spectral sequence with abutment . [Copyright &y& Elsevier]

Abstract: We prove descent theorems for K-theories of quasi-compact quasi-separated schemes of finite Krull dimension. [Copyright &y& Elsevier]

Abstract: The higher wild kernels are finite subgroups of the even K-groups of a number field F, generalizing Tate''s wild kernel for . Each wild kernel contains the subgroup of divisible elements, as a subgroup of index at most two. We determine when they are equal, i.e., when the wild kernel is divisible in K-theory. [Copyright &y& Elsevier]

Abstract: We define a total Chern class map for the K-theory of a variety X twisted by a central simple algebra A. This includes defining a suitable notion of the motivic cohomology of X twisted by A to serve as the target for such a map. Our twisted motivic groups turn out to be different than those defined and studied by Kahn and Levine. [Copyright &y& Elsevier]

Abstract: Extending ideas of twisted equivariant K-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective -graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups. As an application, we consider the Lie supergroup , obtained by taking the cotangent bundle of a compact Lie group and reversing the parity of its fibers. An inclusion induces a homomorphism from the twisted representation ring of to the twisted representation ring of , which pulls back via an algebraic version of the Thom isomorphism to give an additive homomorphism from to (possibly with twistings). We then show that this homomorphism is in fact Dirac induction, which takes an H-module U to the G-equivariant index of the Dirac operator on the homogeneous space with values in the homogeneous bundle induced by U. [Copyright &y& Elsevier]

Abstract: For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute , where X is an isotropic projective homogeneous variety and A means algebraic K-theory, motivic cohomology or algebraic cobordism MGL. [Copyright &y& Elsevier]

Abstract: Following the introduction of an algebraic K-theory of special groups in [Dickmann and Miraglia, Algebra Colloq. 10 (2003) 149–176], generalizing Milnor''s mod 2 K-theory for fields, the aim of this paper is to compute the K-theory of Boolean algebras, inductive limits, finite products, extensions, SG-sums and (finitely) filtered Boolean powers of special groups. A parallel theme is the preservation by these constructions of property [SMC], an analog for the K-theory of special groups of the property “multiplication by is injective” in Milnor''s mod 2 K-theory (see [Milnor, Invent. Math. 9 (1970) 318–344]). [Copyright &y& Elsevier]

We derive an explicit formula, with no cancellations, for expanding in the basis of Grothendieck polynomials the product of two such polynomials, one of which is indexed by an arbitrary permutation, and the other by a simple transposition; hence, this is a Monk-type formula, expressing the hyperplane section of a Schubert variety in K-theory. Our formula is in terms of increasing chains in the k-Bruhat order on the symmetric group with certain labels on its covers. An intermediate result concerns the multiplication of a Grothendieck polynomial by a single variable. As applications, we rederive some known results, such as Lascoux''s transition formula for Grothendieck polynomials. Our results are reformulated in the context of recently introduced Pieri operators on posets and combinatorial Hopf algebras. In this context, we derive an inverse formula to the Monk-type one, which immediately implies a new formula for the restriction of a dominant line bundle to a Schubert variety. [Copyright &y& Elsevier]

Let p be a prime number, and let A be a ring in which p is nilpotent. In this paper, we consider the maps K q + 1 (A [ x ] / (x m) , (x)) → K q + 1 (A [ x ] / (x m n) , (x)) , induced by the ring homomorphism A [ x ] / (x m) → A [ x ] / (x m n) , x ↦ x n. We evaluate these maps, up to extension, for general A in terms of topological Hochschild homology, and for regular F p -algebras A , in terms of groups of de Rham-Witt forms. After the evaluation, we give a calculation of the relative K -group of O K / p O K for certain perfectoid fields K. [ABSTRACT FROM AUTHOR]

In this paper, we extend Waldhausen's results on algebraic K-theory of generalized free products in a more general setting and we give some properties of the Nil functors. As a consequence, we get new groups with trivial Whitehead groups. [ABSTRACT FROM AUTHOR]

We present a method for computing A 1 -homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type A 2 n quivers. [ABSTRACT FROM AUTHOR]

A Laurent polynomial ring R 0 t , t − 1 with coefficients in a unital ring determines a category of quasi-coherent sheaves on the projective line over R 0 ; its K -theory is known to split into a direct sum of two copies of the K -theory of R 0. In this paper, the result is generalised to the case of an arbitrary strongly Z -graded ring R in place of the Laurent polynomial ring. The projective line associated with R is indirectly defined by specifying the corresponding category of quasi-coherent sheaves. Notions from algebraic geometry like sheaf cohomology and twisting sheaves are transferred to the new setting, and the K -theoretical splitting is established. [ABSTRACT FROM AUTHOR]

We show that a complete hereditary cotorsion pair (C , C ⊥) in an exact category E , together with a subcategory Z ⊆ E containing C ⊥ , determines a Waldhausen category structure on the exact category C , in which Z is the class of acyclic objects. This allows us to prove a new version of Quillen's Localization Theorem, relating the K -theory of exact categories A ⊆ B to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require A to be a Serre subcategory, which produces new examples. Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences. [ABSTRACT FROM AUTHOR]

In this paper we express the class of the structure sheaves of the closures of Deligne–Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved by viewing such closures as degeneracy loci of morphisms of vector bundles. [ABSTRACT FROM AUTHOR]

We generalize Rost's theory of cycle modules [20] using the Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a (quadratic) setting to study general cycle complexes and their (co)homology groups. The standard constructions are developed: proper pushfoward, (essentially) smooth pullback, long exact sequences, spectral sequences and products, as well as the homotopy invariance property; in addition, Gysin morphisms for lci morphisms are constructed. We prove an adjunction theorem linking our theory to Rost's. This work extends Schmid's thesis [22]. [ABSTRACT FROM AUTHOR]

Let E be a field of characteristic different from 2 which is the centre of a quaternion division algebra and which is not euclidean. Then there exists a biquaternion division algebra over the rational function field E (t) which does not contain any quaternion algebra defined over E. The proof is based on the study of Bezoutian forms developed in [1]. [ABSTRACT FROM AUTHOR]

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K -theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on h than here. [ABSTRACT FROM AUTHOR]

Let Φ → Γ → Σ be a conormal extension of Hopf algebras over a commutative ring k , and let M be a Γ-comodule. The Cartan-Eilenberg spectral sequence E 2 = Ext Φ (k , Ext Σ (k , M)) ⇒ Ext Γ (k , M) is a standard tool for computing the Hopf algebra cohomology of Γ with coefficients in M in terms of the cohomology of Φ and Σ. We construct a generalization of the Cartan-Eilenberg spectral sequence converging to Ext Γ (k , M) that can be defined when Φ = Γ □ Σ k is compatibly an algebra and a Γ-comodule; this is related to a construction independently developed by Bruner and Rognes. We show that this spectral sequence is isomorphic, starting at the E 1 page, to both the Adams spectral sequence in the stable category of Γ-comodules as studied by Margolis and Palmieri, and to a filtration spectral sequence on the cobar complex for Γ originally due to Adams. We obtain a description of the E 2 term under an additional flatness assumption. We discuss applications to computing localizations of the Adams spectral sequence E 2 page. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Hermitian K -theory of schemes with involution, for which we construct a transfer morphism and prove a version of the dévissage theorem. This theorem is then used to compute the Hermitian K -theory of P 1 with involution given by [ X : Y ] ↦ [ Y : X ]. We also prove the C 2 -equivariant A 1 -invariance of Hermitian K -theory, which confirms the representability of Hermitian K -theory in the C 2 -equivariant motivic homotopy category of Heller, Krishna and Østvær [14]. [ABSTRACT FROM AUTHOR]