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Let K be a finite extension of the p -adic numbers Q p with ring of integers O K and residue field κ. Let X a regular scheme, proper, flat, and geometrically irreducible over O K of dimension d , and X K its generic fiber. We show, under some assumptions on X , that there is a reciprocity isomorphism of locally compact groups H a r 2 d − 1 (X K , Z (d)) ≃ π 1 a b (X K) W from the cohomology theory defined in [10] to an integral model π 1 a b (X K) W of the abelianized fundamental group π 1 a b (X K). After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups. The key ingredient is the duality result in [10]. [ABSTRACT FROM AUTHOR]

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]

For a field E of characteristic different from 2 and cohomological 2-dimension one, quadratic forms over the rational function field E (X) are studied. A characterisation in terms of polynomials in E [ X ] is obtained for having that quadratic forms over E (X) satisfy a local-global principle with respect to discrete valuations that are trivial on E. In this way new elementary proofs for the local-global principle are achieved in the cases where E is finite or pseudo-algebraically closed. The study is complemented by various examples. [ABSTRACT FROM AUTHOR]

For two unital Kirchberg algebras with finitely generated K-groups, we introduce a property, called reciprocality, which is proved to be closely related to the homotopy theory of Kirchberg algebras. We show the Spanier–Whitehead duality for bundles of separable nuclear UCT C*-algebras with finitely generated K-groups and conclude that two reciprocal Kirchberg algebras share the same structure of their bundles. [ABSTRACT FROM AUTHOR]

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric. [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]

In this paper, we introduce Kasparov's bivariant K-theory that is equivariant under symmetries of a C*-tensor category. It is motivated by some dualities in quantum group equivariant KK-theory, and the classification theory of inclusions of C*-algebras. The fundamental properties of the KK-theory, i.e., the existence of the Kasparov product, Cuntz's picture, universality, and triangulated category structure, hold in this generalization as well. Moreover, we further prove a new property specific to this theory; the invariance of KK-theory under weak Morita equivalence of the tensor categories. As an example, we study the Baum–Connes type property for 3-cocycle twists of discrete groups. [ABSTRACT FROM AUTHOR]

We discuss a categorical approach to the theory of discriminants in the combinatorial language introduced by Gelfand, Kapranov and Zelevinsky. Our point of view is inspired by homological mirror symmetry and provides K -theoretic evidence for a conjecture presented by Paul Aspinwall in a conference talk in Banff in March 2016 and later in a joint paper with Plesser and Wang. [ABSTRACT FROM AUTHOR]

The set of T -invariant curves in a Schubert variety through a T -fixed point is relatively easy to characterize in terms of its weights, but the tangent space is more difficult. We prove that the weights of the tangent space are contained in the rational cone generated by the weights of the T -invariant curves. In simply laced types, this remains true if "rational" is replaced by "integral". We also obtain conditions under which every weight of the tangent space is the weight of a T -invariant curve, as well as a smoothness criterion. The results rely on equivariant K -theory, as well as the study of different notions of decomposability of roots. [ABSTRACT FROM AUTHOR]

We investigate ⁎-homomorphisms with nuclear dimension equal to zero. In the framework of classification of ⁎-homo-morphisms, we characterise such maps as those that can be approximately factorised through an AF-algebra. Along the way, we obtain various obstructions for the total invariant of zero-dimensional morphisms and show that in the presence of real rank zero, nuclear dimension zero can be completely determined at the level of the total invariant. We end by characterising when unital embeddings of Z have nuclear dimension equal to zero. [ABSTRACT FROM AUTHOR]

We give an explicit formula for the duality, previously conjectured by Horja and Borisov, of two systems of GKZ hypergeometric PDEs. We prove that in the appropriate limit this duality can be identified with the inverse of the Euler characteristics pairing on cohomology of certain toric Deligne-Mumford stacks, by way of Γ-series cohomology valued solutions to the equations. [ABSTRACT FROM AUTHOR]

We give a description of operator algebras of free wreath products in terms of fundamental algebras of graphs of operator algebras as well as an explicit formula for the Haar state. This allows us to deduce stability properties for certain approximation properties such as exactness, Haagerup property, hyperlinearity and K-amenability. We study qualitative properties of the associated von Neumann algebra: factoriality, fullness, primeness and absence of Cartan subalgebra and we give a formula for Connes' T -invariant and τ -invariant. We also study maximal amenable von Neumann subalgebras. Finally, we give some explicit computations of K-theory groups for C*-algebras of free wreath products. As an application we show that the reduced C*-algebras of quantum reflection groups are pairwise non-isomorphic. [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]

The Shortest Vector Problem and the computation of the tame kernel in algebraic K-theory are two different but very important problems, and up to now no one has found any manipulated relations between them. In this paper, we are successful for the first time to reduce the computation of tame kernel, in particular the computation of elements of the set C m , the key part in the method proposed by Tate, to the Shortest Vector Problem of some lattice. Then, with the help of PARI library we implement the above idea and develop a program for computing the tame kernels of the cyclotomic fields with class number 1, and as an example, by running the program we obtain the tame kernel of the cyclotomic field Q (ζ 7). [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]

Let K be an imaginary quadratic field of discriminant d K , and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let Q N (d K) be the set of primitive positive definite binary quadratic forms of discriminant d K whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼ n on Q N (d K) so that the set of equivalence classes Q N (d K) / ∼ n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of Q N (d K) / ∼ n onto Gal (K n / K) in terms of Fricke invariants, where K n denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet. [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]

We suspect that the "genus part" of the class number of a number field K may be an obstruction for an "easy and unconditional proof" of the classical p -rank ε -conjecture for p -class groups and, a fortiori, for a proof of the "strong ε -conjecture": # (C ℓ K ⊗ Z p) ≪ d , p , ε (| D K | ) ε for all K of degree d. We analyze the weight of genus theory in this inequality by means of an infinite family of degree p cyclic fields with many ramified primes, then we prove the p -rank ε -conjecture: # (C ℓ K ⊗ F p) ≪ d , p , ε (| D K | ) ε , for d = p and the family of degree p cyclic extensions (Theorem 2.5) then sketch the case of arbitrary base fields. The possible obstruction for the strong form, in the degree p cyclic case, is the order of magnitude of the set of "exceptional" p -classes given by a well-known non-predictible algorithm, but largely controlled thanks to recent density results due to Koymans–Pagano. Then we compare the ε -conjectures with some p -adic conjectures, of Brauer–Siegel type, about the torsion group T K of the Galois group of the maximal abelian p -ramified pro- p -extension of totally real number fields K. We give numerical computations with the corresponding PARI/GP programs. [ABSTRACT FROM AUTHOR]

We give a definition of the E -theory groups for real graded C ⁎ -algebras and further we formulate a proof of 8-fold Bott Periodicity by using a method of Guentner and Higson. This method involves formulating Bott periodicity in real graded K -theory and then transferring it into the E -theoretic framework using the product of E -theory. [ABSTRACT FROM AUTHOR]

We develop a new method for studying the Baum–Connes conjecture, which we call the direct splitting method. We introduce what we call property (γ) for G -equivariant Kasparov cycles. We show that the existence of a G -equivariant Kasparov cycle with property (γ) implies the split-injectivity of the assembly map μ A G for any separable G - C ⁎ -algebra A. We also show that if such a cycle exists, the assembly map μ A G is an isomorphism if and only if the cycle acts as the identity on the right-hand side group K ⁎ (A ⋊ r G) of the Baum–Connes conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, we use this method to give a finite-dimensional proof of the Baum–Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space. [ABSTRACT FROM AUTHOR]

Alain Connes and Nigel Higson pointed out in the 1990s that the Connes-Kasparov "conjecture" for the K-theory of reduced group C ⁎ -algebras seemed, in the case of reductive Lie groups, to be a cohomological echo of a conjecture of George Mackey concerning the rigidity of representation theory along the deformation from a real reductive group to its Cartan motion group. For complex semisimple groups, Nigel Higson established in 2008 that Mackey's analogy is a real phenomenon, and does lead to a simple proof of the Connes-Kasparov isomorphism. We here turn to more general reductive groups and use our recent work on Mackey's proposal, together with Higson's work, to obtain a new proof of the Connes-Kasparov isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we apply quantitative operator K -theory to develop an algorithm for computing K -theory for the class of filtered C ⁎ -algebras with asymptotic finite nuclear decomposition. As a consequence, we prove the Künneth formula for C ⁎ -algebras in this class. Our main technical tool is a quantitative Mayer–Vietoris sequence for K -theory of filtered C ⁎ -algebras. [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]

In this paper we continue the work of using the recent advances in algebraic K -theory to extend computations done in characteristic p to the mixed characteristic setting using perfectoid rings. We extend the work of Hesselholt-Nikolaus in [18] on the algebraic K -Theory of cuspidal curves. We consider both cuspidal curves and the p -completion of cuspidal curves. Along the way we also study the K -theory of the p -completed affine line over a perfectoid ring. [ABSTRACT FROM AUTHOR]

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1 → N → G → Q → 1 , we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G. [ABSTRACT FROM AUTHOR]

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive. [ABSTRACT FROM AUTHOR]

In this paper, we settle an open conjecture regarding the assertion that the Euler-characteristic of G / N G (T) for a split reductive group scheme G and the normalizer of a split maximal torus N G (T) over a field is 1 in the Grothendieck-Witt ring with the characteristic exponent of the field inverted, under the assumption that the base field contains a − 1. Numerous applications of this to splittings in the motivic stable homotopy category and to Algebraic K-Theory are worked out in several related papers by Gunnar Carlsson and the authors. [ABSTRACT FROM AUTHOR]

We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many features formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C⁎-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C⁎-algebras. Finally, we show how to use the bivariant Cuntz semigroup of the present work to classify unital and stably finite C⁎-algebras. [ABSTRACT FROM AUTHOR]

We construct a relative version of topological K -theory, in the sense of [11] , over an arbitrary quasi-compact, quasi-separated C -scheme X. This has as input a Perf (X) -linear stable ∞-category and output a sheaf of spectra on X (C) , the space of complex points of X. We then characterize the values of this functor on inputs of the form Perf (X , A) , for A a derived Azumaya algebra over X. In such cases we show that this coincides with the α -twisted topological K -theory of X (C) for some appropriately defined twist of K -theory. We use this to provide a topological analogue of a classical result of Quillen's on the algebraic K -theory of Severi-Brauer varieties. [ABSTRACT FROM AUTHOR]