Mathematics - K-Theory and Homology Publications (50)

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Mathematics - K-Theory and Homology Publications

We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $\mathbb{S} \to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $\pi_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra. Read More


From a genuine $\mathbb{Z}/2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure we produce a genuine $\mathbb{Z}/2$-equivariant spectrum $KR(A)$. This construction extends the real $K$-theory framework of Hesselholt-Madsen for discrete rings and the Hermitian $K$-theory framework of Burghelea-Fiedorowicz for simplicial rings. We construct a natural trace map of $\mathbb{Z}/2$-spectra $tr\colon KR(A)\to THR(A)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of B\"{o}kstedt-Hsiang-Madsen. Read More


We relate the old and new cohomology monoids of an arbitrary monoid $M$ with coefficients in semimodules over $M$, introduced in the author's previous papers, to monoid and group extensions. More precisely, the old and new second cohomology monoids describe Schreier extensions of semimodules by monoids, and the new third cohomology monoid is related to a certain group extension problem. Read More


Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin $L$-series at $s=r$ to the compact support cohomology of the \'etale $p$-adic sheaf $\mathbb Z_p(r)$. Read More


Let $G,H$ be groups, $\phi: G \rightarrow H$ a group morphism, and $A$ a $G$-graded algebra. The morphism $\phi$ induces an $H$-grading on $A$, and on any $G$-graded $A$-module, which thus becomes an $H$-graded $A$-module. Given an injective $G$-graded $A$-module, we give bounds for its injective dimension when seen as $H$-graded $A$-module. Read More


Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmanian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. Read More


We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. Read More


We define a functorial "Artin map" attached to any small $\bf{Z}$-linear stable $\infty$-category, which in the case of perfect complexes over a global field recovers the usual Artin map from the idele class group to the abelianized absolute Galois group. In particular, this gives a new proof of the Artin reciprocity law. Read More


We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic K-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$--homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy groups. Read More


Let S be a finitely generated subsemigroup of Z^2. We derive a general formula for the K-theory of the left regular C*-algebra for S. Read More


We prove that if a smooth projective algebraic variety of dimension less or equal to three has a unit type integral $K$-motive, then its integral Chow motive is of Lefschetz type. As a consequence, the integral Chow motive is of Lefschetz type for a smooth projective variety of dimension less or equal to three that admits a full exceptional collection. Read More


We compute the Hochschild cohomology ring of the algebras $A= k\langle X, Y\rangle/ (X^a, XY-qYX, Y^a)$ over a field $k$ where $a\geq 2$ and where $q\in k$ is a primitive $a$-th root of unity. We find the the dimension of $\mathrm{HH}^n(A)$ and show that it is independent of $a$. We compute explicitly the ring structure of the even part of the Hochschild cohomology modulo homogeneous nilpotent elements. Read More


We show that the triviality of sections of the sheaf of A^1-chain connected components of a space over finitely generated separable field extensions of the base field is not sufficient to ensure the triviality of the sheaf of its A^1-chain connected components, contrary to the situation with genuine A^1-connected components. As a consequence, we show that there exists an A^1-connected scheme for which the Morel-Voevodsky singular construction is not A^1-local. Read More


For every connected manifold with corners we introduce a very computable homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), $\chi_{cn}:=\chi_0-\chi_1$, is given by the alternated sum of the number of (open) faces of a given codimension. The main result of the present paper is that for a compact connected manifold with corners $X$ given as a finite product of manifolds with corners of codimension less or equal to three we have that 1) If $X$ satisfies the Fredholm Perturbation property (every elliptic pseudodifferential b-operator on $X$ can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of $X$ vanishes, i. Read More


Building upon Hovey's work on Smith ideals for monoids, we develop a homotopy theory of Smith ideals for general operads in a symmetric monoidal category. For a sufficiently nice stable monoidal model category and an operad satisfying a cofibrancy condition, we show that there is a Quillen equivalence between a model structure on Smith ideals and a model structure on algebra maps induced by the cokernel and the kernel. For symmetric spectra this applies to the commutative operad and all Sigma-cofibrant operads. Read More


We compute the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with $\mathbf{I}^j$-coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. Read More


We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This result confirms a belief expressed by Hovey in his work on Smith ideals. As illustrations we include examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, and topological spaces. Read More


There are two $\mathbb Z_2$ orbifolds of the Podle\'s quantum two-sphere, one being the quantum two-disc $D_q$ and other the quantum two-dimensional real projective space $\mathbb RP^2_q$ . In this article we calculate the Hochschild and cyclic homology and cohomology groups of these orbifolds and also the corresponding Chern-Connes indices. Read More


Let K be a non-empty set of ideals of the commutative ring R, closed under taking smaller ideals. A subset X of the group ring R[Z^s] is called a K-set if the ideal generated by the coefficients of the elements of X is in K. For X not a K-set we investigate the set of those homomorphisms p from Z^s to Z^t such that p_*(X) is a K-set. Read More


Let $(Q, q)$ be a quadratic space over a commutative ring $R$ in which $2$ is invertible, and consider the Dickson--Siegel--Eichler--Roy's subgroup $EO_{R}(Q, H(R)^{m})$ of the orthogonal group $O_R(Q \perp H(R)^m)$, with rank $Q= n \geq 1$ and $m\geq 2$. We show that $EO_{R}(Q, H(R)^{m})$ is a normal subgroup of $O_R(Q \perp H(R)^m)$, for all $m\geq 2$. We also prove that the DSER group $EO_{R}(Q, H(P))$ is a normal subgroup of $O_{R}(Q \perp H(P))$, where $Q$ and $H(P)$ are quadratic spaces over a commutative ring $R$, with rank $(Q) \ge 1$ and rank $(P) \ge 2$. Read More


Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gerstenhaber bracket, on $\mathcal{D}^*(A,A)$. Read More


We compute the group of $K_1$-zero-cycles on the second generalized involution variety for an algebra of index 4 with symplectic involution. Our method utilizes the framework of Chernousov and Merkurjev for computing $K_1$-zero-cycles in terms of $R$-equivalence classes of prescribed algebraic groups, which yields our result for algebras of degree 4. We then use Rost's spectral sequence associated to a morphism of schemes to obtain our main result. Read More


In this paper, we study representation homology of topological spaces, that is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology in terms of classical (abelian) homological algebra. Our construction is parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by showing that the representation homology of the (reduced) suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. Read More


In this note, we prove a comparision theorem between equivariant higher Chow groups of an algebraic variety and ordinary higher Chow groups of its fixed points. As a consequence, we show that the equivariant motivic spectral sequence constructed by Levine-Serp\'e degenerates rationally which gives a Riemann-Roch Theorem for equivariant K-theory. Read More


By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic K-theory. We deduce Suslin's result from an exact sequence of categories of perfect modules. Our approach applies more generally to connective ring spectra, and yields excision for any localizing invariant. Read More


We prove that the projective model structure on the category of unbounded cochain complexes extends naturally to the category of contractions. The proof is completely elementary and we do not assume familiarity with model categories. Read More


Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra $\Lambda$ is triangular with respect to a system of non necessarily primitive idempotents, and if the algebras at the idempotents belong to $\mathcal H$, then $\Lambda$ is in $\mathcal H$. Read More


In this note we show that Waldhausen's K-theory functor from Waldhausen categories to spaces has a universal property: It is the target of the "universal global Euler characteristic", in other words, the "additivization" of the functor sending a Waldhausen category C to obj(C) . We also show that a large class of functors possesses such an additivization. Read More


We study the twisted Hochschild homology of quantum full flag manifolds, with the twist being the modular automorphism of the Haar state. We show that non-trivial 2-cycles can be constructed from appropriate invariant projections. The main result is that $HH_2^\theta(\mathbb{C}_q[G / T])$ is infinite-dimensional when $\mathrm{rank}(\mathfrak{g}) > 1$. Read More


We construct a triangulated monoidal Karoubi closed category with the Grothendieck ring naturally isomorphic to the ring of integers localized at two. Read More


The goal of this note is to describe a class of formal deformations of a symplectic manifold $M$ in the case when the base ring of the deformation problem involves parameters of non-positive degrees. The interesting feature of such deformations is that these are deformations "in $A_{\infty}$-direction" and, in general, their description involves all cohomology classes of $M$ of degrees $\ge 2$. Read More


The purpose of this article is to clarify the question what makes motives $\mathbb{A}^1$-homotopy invariance. we give construction of the stable model category of nilpotent invariant motives $\mathcal{M}ot_{\operatorname{dg}}^{\operatorname{nilp}}$ and define the nilpotent invriant motives associated with schemes and relative exact categories. For a noetherian scheme $X$, there are two kind of motives associated with $X$ in the homotopy category $\operatorname{Ho}(\mathcal{M}ot^{\operatorname{nilp}}_{\operatorname{dg}})$, namely $M_{\operatorname{nilp}}(X)$ and $M_{\operatorname{nilp}}'(X)$. Read More


These notes follows the articles \cite{kamel, Cam, cam-cubique} which show how powerful can be the method of \textit{Stretchings} initiated with the \textit{Globular Geometry} by Jacques Penon in \cite{penon} , to weakened \textit{strict higher structures}. Here we adapt this method to weakened strict multiple $\infty$-categories, strict multiple $(\infty,m)$-categories, and in particular we obtain algebraic models of weak multiple $\infty$-groupoids. Read More


In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension. Read More


We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective resolution once the generating degrees for the first term of some particular projective resolution are known. Read More


Using the concept of ring diadic range 1 we proved that a commutative Bezout ring is an elementary divisor ring iff it is a ring diadic range 1. Read More


We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws. Read More


The aim of this work is to give an algebraic weak version of the Atiyah-Singer index theorem. We compute then a few small examples with the elliptic differential operator of order $\leq 1$ coming from the Atiyah class in $\mathrm{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_X,\Omega^1_{X/k})$, where $X \longrightarrow \mathrm{Spec}(k)$ is a smooth projective scheme over a perfect field $k$. Read More


We prove the Banach strong Novikov conjecture for groups having polynomially bounded higher-order combinatorial functions. This includes all automatic groups. Read More


We give a concise introduction to the Farrell-Jones Conjecture in algebraic $K$-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods. Read More


Motivated by the topological classification of hamiltonians in condensed matter physics (topological insulators) we study the relations between chiral Dirac operators coupled to an abelian vector potential on a torus in 3 and 1 space dimensions. We find that a large class of these hamiltonians in three dimensions is equivalent, in K theory, to a family of hamiltonians in just one space dimension but with a different abelian gauge group. The moduli space of U(1) gauge connections over a torus with a fixed Chern class is again a torus up to a homotopy. Read More


Let $A$ be a commutative noetherian ring, let $\mathfrak{a} \subseteq A$ be an ideal, and let $J$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module $\Gamma_{\mathfrak{a}}(J)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Equivalently, the torsion theory associated to $\mathfrak{a}$ is stable. Read More


In this article we show how to build main aspects of our paper on globular weak $(\infty,n)$-categories, but now for the cubical geometry. Thus we define a monad on the category $\mathbb{C}\mathbb{S}ets$ of cubical sets which algebras are models of cubical weak $\infty$-categories. Also for each $n\in\mathbb{N}$ we define a monad on $\mathbb{C}\mathbb{S}ets$ which algebras are models of cubical weak $(\infty,n)$-categories. Read More


In this article, we study the relative negative K-groups $K_{-n}(f)$ of a map $f: X \to S $ of schemes. We prove a relative version of the Weibel conjecture i.e. Read More


The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula. Other special cases include an index theorem by Connes and Moscovici for homogeneous spaces, and an earlier index theorem by the second author, both in cases where the group acting is connected and semisimple. Read More


We lift to equivariant algebra three closely related classical algebraic concepts: abelian group objects in augmented commutative algebras, derivations, and K\"ahler differentials. We define Mackey functor objects in the category of Tambara functors augmented to a fixed Tambara functor $\underline{R}$, and we show that the usual square-zero extension gives an equivalence of categories between these Mackey functor objects and ordinary modules over $\underline{R}$. We then describe the natural generalization to Tambara functors of a derivation, building on the intuition that a Tambara functor has products twisted by arbitrary finite $G$-sets, and we connect this to square-zero extensions in the expected way. Read More


We solve affirmatively the homotopy limit problem for $K$-theory over fields of finite virtual cohomological dimension. Our solution employs the motivic slice filtration and the first motivic Hopf map. Read More


We prove a {\Gamma}-equivariant version of the algebraic index theorem, where {\Gamma} is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Read More


We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any elliptic curve $A$ in any characteristic. Read More