The affine part of the Picard scheme
We describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field.
math.AG↗
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Results for “math.KT”
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On the KK-theory of strongly self-absorbing C*-algebras
Let $\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\Dh$ be strongly self-absorbing. It is known that any two unital $^*$-homomorphisms from $\Dh$ to $A \otimes \Dh$ are approximately unitarily equivalent. We show that, if $\Dh$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\Dh$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\Aut(\Dh)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\Dh$. As an application, we give a description of the Kasparov group $KK(\Dh, A\ot \Dh)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\Dh, A\ot \Dh)$ is isomorphic to $K_0(A\ot \Dh)$.
math.OA↗
K_0-theory of n-potents in rings and algebras
Let $n \geq 2$ be an integer. An \emph{$n$-potent} is an element $e$ of a ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a complex algebra, there is a group isomorphism $K_0^n(A) \cong \bigl(K_0(A)\bigr)^{n-1}$ for all $n \geq 2$. However, for algebras over cyclotomic fields, this is not true in general. We consider $K_0^n$ as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an \emph{$n$-homomorphism}.
math.KT↗
Explicit HRS-Tilting
For an abelian category $A$ equipped with a torsion pair, we give an explicit description for the abelian category $B$ introduced by Happel-Reiten-Smalo, and also for the category of chain complexes $Ch(B)$ and the derived category $D(B)$ of $B$. We also describe the DG structure on $Ch(B)$. As a consequence, we find new proofs of certain results of Happel-Reiten-Smalo. The main ingredient is the category of {\em decorated} complexes.
math.RT↗
Lectures on derived and triangulated categories
These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.
math.KT↗
Cyclic cohomology of certain nuclear Fréchet and DF algebras
We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain topological algebras. To this end we show that, for a continuous morphism $ϕ: \X\to \Y$ of complexes of complete nuclear $DF$-spaces, the isomorphism of cohomology groups $H^n(ϕ): H^n(\X) \to H^n(\Y)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $\hat{\otimes}$-algebras: the tensor algebra $E \hat{\otimes} F$ generated by the duality $(E, F, < \cdot, \cdot >)$ for nuclear Fréchet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$; nuclear biprojective Köthe algebras $λ(P)$ which are Fréchet spaces or $DF$-spaces; the algebra of distributions $\mathcal{E}^*(G)$ on a compact Lie group $G$.
math.KT↗
Arithmetic homology and an integral version of Katos conjecture
We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles.
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L-theory of groups with unstable derived series
In this short note we prove that the Farrell-Jones Fibered Isomorphism Conjecture in L-theory, after inverting 2, is true for a group whose some derived subgroup is free.
math.KT↗
Gersten's conjecture
The purpose of this article is to prove that Gersten's conjecture for a commutative regular local ring is true. As its applications, we will prove the vanishing conjecture for certain Chow groups, generator conjecture for certain $K$-groups and Bloch's formula for absolute case.
math.KT↗
Parallel computation of the rank of large sparse matrices from algebraic K-theory
This paper deals with the computation of the rank and of some integer Smith forms of a series of sparse matrices arising in algebraic K-theory. The number of non zero entries in the considered matrices ranges from 8 to 37 millions. The largest rank computation took more than 35 days on 50 processors. We report on the actual algorithms we used to build the matrices, their link to the motivic cohomology and the linear algebra and parallelizations required to perform such huge computations. In particular, these results are part of the first computation of the cohomology of the linear group GL_7(Z).
math.KT↗
A groupoid approach to noncommutative T-duality
Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the construction may be of independent interest. These are a Pontryagin type duality that interchanges commutative principal bundles with gerbes, a nonabelian Takai type duality for groupoids, and the computation of certain equivariant Brauer groups.
math.QA↗
The E-theoretic descent functor for groupoids
The paper establishes, for a wide class of locally compact groupoids $Γ$, the E-theoretic descent functor at the $C^{*}$-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. The second section shows that $Γ$-actions on a $C_{0}(X)$-algebra $B$, where $X$ is the unit space of $Γ$, can be usefully formulated in terms of an action on the associated bundle $B^{\sharp}$. The third section shows that the functor $B\to C^{*}(Γ,B)$ is continuous and exact, and uses the disintegration theory of J. Renault. The last section establishes the existence of the descent functor under a very mild condition on $Γ$, the main technical difficulty involved being that of finding a $Γ$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the group case.
math.OA↗
The equivariant analytic index for proper groupoid actions
The paper constructs the analytic index for an elliptic pseudodifferential family of $L^{m}_{ρ,\de}$-operators invariant under the proper action of a continuous family groupoid on a $G$-compact, $C^{\infty,0}$ $G$-space.
math.OA↗
The analytic index for proper, Lie groupoid actions
Many index theorems (both classical and in noncommutative geometry) can be interpreted in terms of a Lie groupoid acting properly on a manifold and leaving an elliptic family of pseudodifferential operators invariant. Alain Connes in his book raised the question of an index theorem in this general context. In this paper, an analytic index for many such situations is constructed. The approach is inspired by the classical families theorem of Atiyah and Singer, and the proof generalizes, to the case of proper Lie groupoid actions, some of the results proved for proper locally compact group actions by N. C. Phillips.
math.OA↗
Continuous family groupoids
In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant Atiyah-Singer index theorem for families, and is also required in noncommutative geometry. The class includes that of Lie groupoids, and the paper shows that, like Lie groupoids, continuous family groupoids always admit (an essentially unique) continuous left Haar system of smooth measures. We also show that the action of a continuous family groupoid $G$ on a continuous family $G$-space fibered over another continuous family $G$-space $Y$ can always be regarded as an action of the continuous family groupoid $G*Y$ on an ordinary $G*Y$-space.
math.KT↗
Characterizing group $C^\ast$-algebras through their unitary groups: the Abelian case
We study to what extent group $C^\ast$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^\ast$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of $C^\ast(Γ)$, such as the $K$-theoretic $K_1(C^\ast(Γ))$ and find that $C^\ast$-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic $K_1$-groups, in sharp contrast with the well-known fact that $C^\ast(Γ)$ (even $Γ$) is characterized by the topological group structure of its unitary group when $Γ$ is torsion-free and Abelian.
math.OA↗
Mittag-Leffler conditions on modules
We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.
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A converse to the Second Whitehead Lemma
We show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple.
math.RA↗