arXiv subjects

Guillermo Cortiñas

Publications and source records attributed to Guillermo Cortiñas.

At least 37 records · Page 2Linked to original sources

arXiv2014

Compact operators and algebraic $K$-theory for groups which act properly and isometrically on Hilbert space

We prove the $K$-theoretic Farrell-Jones conjecture for groups as in the title with coefficient rings and $C^*$-algebras which are stable with respect to compact operators. We use this and Higson-Kasparov's result that the Baum-Connes conjecture with coefficients holds for such groups, to show that if $G$ is as in the title then the algebraic and the $C^*$-crossed products of $G$ with a stable $C^*$-algebra have the same $K$-theory.

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arXiv2014

Excision, descent, and singularity in algebraic $K$-theory

Algebraic $K$-theory is a homology theory that behaves very well on sufficiently nice objects such as stable $C^*$-algebras or smooth algebraic varieties, and very badly in singular situations. This survey explains how to exploit this to detect singularity phenomena using $K$-theory and cyclic homology.

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arXiv2014

Algebraic K-theory and properly infinite C*-algebras

We show that several known results about the algebraic K-theory of tensor products of algebras with the C*-algebra of compact operators in Hilbert space remain valid for tensor products with any properly infinite C*-algebra.

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arXiv2013

Trace class operators, regulators, and assembly maps in K-theory

Let G be a group, Fin the family of its finite subgroups, and E(G,Fin) the classifying space. Let L^1 be the algebra of trace-class operators in an infinite dimensional, separable Hilbert space over the complex numbers. Consider the rational assembly map in homotopy algebraic K-theory H_p^G(E(G,Fin),KH(L^1))\otimes\Q \to KH_p(L^1[G])\otimes\Q. The rational KH-isomorphism conjecture predicts that the map above is an isomorphism; it follows from a theorem of Yu (see arXiv:1106.3796, arXiv:1202.4999) that it is always injective. In the current article we prove the following. Theorem: Assume that the map above is surjective. Let n\equiv p+1\mod 2. Then: i) The assembly map for the trivial family H_n^G(E(G,{1}),K(\Z)) \to K_n(\Z[G]) is rationally injective. ii) For every number field F, the assembly map H_n^G(E(G,Fin),K(F)) \to K_n(F[G]) is rationally injective. We remark that the K-theory Novikov conjecture asserts that part i) of the theorem above holds for all G, and that part ii) is equivalent to the rational injectivity part of the K-theory Farrell-Jones conjecture for number fields. The idea of the proof of the Theorem is to use an algebraic, equivariant version of Karoubi's multiplicative K-theory, which we introduce in this article.

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arXiv2013

Cyclic homology, tight crossed products, and small stabilizations

In \verb|arXiv:1212.5901| we associated an algebra $\Gami(\fA)$ to every bornological algebra $\fA$ and an ideal $I_{S(\fA)}\triqui\Gami(\fA)$ to every symmetric ideal $S\triqui\elli$. We showed that $I_{S(\fA)}$ has $K$-theoretical properties which are similar to those of the usual stabilization with respect to the ideal $J_S\triqui\cB$ of the algebra $\cB$ of bounded operators in Hilbert space which corresponds to $S$ under Calkin's correspondence. In the current article we compute the relative cyclic homology $HC_*(\Gami(\fA):I_{S(\fA)})$. Using these calculations, and the results of \emph{loc. cit.}, we prove that if $\fA$ is a $C^*$-algebra and $c_0$ the symmetric ideal of sequences vanishing at infinity, then $K_*(I_{c_0(\fA)})$ is homotopy invariant, and that if $*\ge 0$, it contains $K^{\top}_*(\fA)$ as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem (\cite{sw1}) that says that for the ideal $\cK=J_{c_0}$ of compact operators and the $C^*$-algebra tensor product $\fA\sotimes\cK$, we have $K_*(\fA\sotimes\cK)=K^{\top}_*(\fA)$. Similarly, we prove that if $\fA$ is a unital Banach algebra and $\ell^{\infty-}=\bigcup_{q<\infty}\ell^q$, then $K_*(I_{\ell^{\infty-}(\fA)})$ is invariant under Hölder continuous homotopies, and that for $*\ge 0$ it contains $K^{\top}_*(\fA)$ as a direct summand. These $K$-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups $HC_*(\Gami(\fA):I_{S(\fA)})$ in terms of $HC_*(\elli(\fA):S(\fA))$ for general $\fA$ and $S$. For $\fA=\C$ and general $S$, we further compute the latter groups in terms of algebraic differential forms. We prove that the map $HC_n(\Gami(\C):I_{S(\C)})\to HC_n(\cB:J_S)$ is an isomorphism in many cases.

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arXiv2013

Homotopy invariance through small stabilizations

We associate an algebra $\Gami(\fA)$ to each bornological algebra $\fA$. The algebra $\Gami(\fA)$ contains a two-sided ideal $I_{S(\fA)}$ for each symmetric ideal $S\triqui\elli$ of bounded sequences of complex numbers. In the case of $\Gami=\Gami(\C)$, these are all the two-sided ideals, and $I_S\mapsto J_S=\cB I_S\cB$ gives a bijection between the two-sided ideals of $\Gami$ and those of $\cB=\cB(\ell^2)$. We prove that Weibel's $K$-theory groups $KH_*(I_{S(\fA)})$ are homotopy invariant for certain ideals $S$ including $c_0$ and $\ell^p$. Moreover, if either $S=c_0$ and $\fA$ is a local $C^*$-algebra or $S=\ell^p,\ell^{p\pm}$ and $\fA$ is a local Banach algebra, then $KH_*(I_{S(\fA)})$ contains $K_*^{\top}(\fA)$ as a direct summand. Furthermore, we prove that for $S\in\{c_0,\ell^p,\ell^{p\pm}\}$ the map $K_*(Γ^\infty(\fA):I_{S(\fA)})\to KH_*(I_{S(\fA)})$ fits into a long exact sequence with the relative cyclic homology groups $HC_*(Γ^\infty(\fA):I_{S(\fA)})$. Thus the latter groups measure the failure of the former map to be an isomorphism.

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arXiv2012

Operator ideals and assembly maps in $K$-theory

Let $\cB$ be the ring of bounded operators in a complex, separable Hilbert space. For $p>0$ consider the Schatten ideal $\cL^p$ consisting of those operators whose sequence of singular values is $p$-summable; put $\cS=\bigcup_p\cL^p$. Let $G$ be a group and $\vcyc$ the family of virtually cyclic subgroups. Guoliang Yu proved that the $K$-theory assembly map \[ H_*^G(\cE(G,\vcyc),K(\cS))\to K_*(\cS[G]) \] is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients $\cS$ and the use of some results about algebraic $K$-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu's result. Our proof uses the usual Chern character to cyclic homology. Like Yu's, it relies on results on algebraic $K$-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy $K$-theory. We prove that the rational assembly map \[ H_*^G(\cE(G,\fin),KH(\cL^p))\otimes\Q\to KH_*(\cL^p[G])\otimes\Q \] is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.

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arXiv2012

Toric varieties, monoid schemes and $cdh$ descent

We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.

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arXiv2012

The K-theory of toric varieties in positive characteristic

We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic 0. The affine case of our result was conjectured by Gubeladze.

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arXiv2011

Tensor products of Leavitt path algebras

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, $L_\infty$ and $L_\infty\otimes L_\infty$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_*(L_2)=K_*(L_2\otimes L_2)=0$ and $K_*(L_\infty)=K_*(L_\infty\otimes L_\infty)=K_*(k)$.

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arXiv2011

Isomorphism conjectures with proper coefficients

Let $G$ be a group and let $E$ be a functor from small $\Z$-linear categories to spectra. Also let $A$ be a ring with a $G$-action. Under mild conditions on $E$ and $A$ one can define an equivariant homology theory of $G$-simplicial sets $H^G(-,E(A))$ with the property that if $H\subset G$ is a subgroup, then \[ H^G_*(G/H,E(A))=E_*(A\rtimes H) \] If now $\cF$ is a nonempty family of subgroups of $G$, closed under conjugation and under subgroups, then there is a model category structure on $G$-simplicial sets such that a map $X\to Y$ is a weak equivalence (resp. a fibration) if and only if $X^H\to Y^H$ is an equivalence (resp. a fibration) for all $H\in\cF$. The strong isomorphism conjecture for the quadruple $(G,\cF,E,A)$ asserts that if $cX\to X$ is the $(G,\cF)$-cofibrant replacement then \[ H^G(cX,E(A))\to H^G(X,E(A)) \] is an equivalence. The isomorphism conjecture says that this holds when $X$ is the one point space, in which case $cX$ is the classifying space $\cE(G,\cF)$. In this paper we introduce an algebraic notion of $(G,\cF)$-properness for $G$-rings, modelled on the analogous notion for $G$-$C^*$-algebras, and show that the strong $(G,\cF,E,P)$ isomorphism conjecture for $(G,\cF)$-proper $P$ is true in several cases of interest in the algebraic $K$-theory context. Thus we give a purely algebraic, discrete counterpart to a result of Guentner, Higson and Trout in the $C^*$-algebraic case. We apply this to show that under rather general hypothesis, the assembly map $H_*^G(\cE(G,\cF),E(A))\to E_*(A\rtimes G)$ can be identified with the boundary map in the long exact sequence of $E$-groups associated to certain exact sequence of rings. Along the way we prove several results on excision in algebraic $K$-theory and cyclic homology which are of independent interest.

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arXiv2011

Algebraic Geometry of Topological Spaces I

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parametrized version of a theorem of Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, seminormal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case when M=N^n gives a parametrized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case when M=Z^n. We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C*-algebras, and for a homology theory of commutative algebras to vanish on C*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil-K-theory implies that commutative C*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild-Kostant-Rosenberg and Loday-Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beilinson-Soule and Farrell-Jones are also given.

math.RA↗

arXiv2010

$K$-theory of cones of smooth varieties

Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $\k$ of characteristic~0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a curve then we calculate $K_0(R)$ and $K_1(R)$, and prove that $K_{-1}(R)=\oplus H^1(C,\cO(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted Kähler differentials on the variety.

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arXiv2009

Algebraic v. topological K-theory: a friendly match

These notes evolved from the lecture notes of a minicourse given in Swisk, the Sedano Winter School on K-theory held in Sedano, Spain, during the week January 22--27 of 2007, and from those of a longer course given in the University of Buenos Aires, during the second half of 2006. They intend to be an introduction to $K$-theory, with emphasis in the comparison between its algebraic and topological variants. We have tried to keep as elementary as possible.

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arXiv2009

$K$-theory of Leavitt path algebras

Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacency matrix $N'_E=(n_{ij})\in\Z^{(E_0\times E_0)}$, $n_{ij}=#\{$ arrows from $i$ to $j\}$. Write $N^t_E$ and 1 for the matrices $\in \Z^{(E_0\times E_0\setminus\Sink(E))}$ which result from $N'^t_E$ and from the identity matrix after removing the columns corresponding to sinks. We consider the $K$-theory of the Leavitt algebra $L_R(E)=L_\Z(E)\otimes R$. We show that if $R$ is either a Noetherian regular ring or a stable $C^*$-algebra, then there is an exact sequence ($n\in\Z$) \[ K_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow} K_n(R)^{(E_0)}\to K_n(L_R(E))\to K_{n-1}(R)^{(E_0\setminus\Sink(E))} \] We also show that for general $R$, the obstruction for having a sequence as above is measured by twisted nil-$K$-groups. If we replace $K$-theory by homotopy algebraic $K$-theory, the obstructions dissapear, and we get, for every ring $R$, a long exact sequence \[ KH_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow}KH_n(R)^{(E_0)}\to KH_n(L_R(E))\to KH_{n-1}(R)^{(E_0\setminus\Sink(E))} \] We also compare, for a $C^*$-algebra $\fA$, the algebraic $K$-theory of $L_\fA(E)$ with the topological $K$-theory of the Cuntz-Krieger algebra $C^*_\fA(E)$. We show that the map \[ K_n(L_\fA(E))\to K^{\top}_n(C^*_\fA(E)) \] is an isomorphism if $\fA$ is stable and $n\in\Z$, and also if $\fA=\C$, $n\ge 0$, $E$ is finite with no sinks, and $\det(1-N_E^t)\ne 0$.

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arXiv2008

Relative Chern characters for nilpotent ideals

If I is a nilpotent ideal in a $\mathbb{Q}$-algebra $A$, Goodwillie defined two isomorphisms from $K_*(A,I)$ to negative cyclic homology, $HN_*(A,I)$. One is the relative version of the absolute Chern character, and the other is defined using rational homotopy theory. The question of whether they agree was implicit in Goodwillie's 1986 Annals paper. In this paper, we show that the two isomorphisms agree. Here are three applications. 1.Cathelineau proved that the rational homotopy character is compatible with the $λ$-filtration. It follows that the relative Chern character is also compatible with this filtration for nilpotent ideals. 2.This agreement, together with Cathelineau's result, was used by the authors and Haesemeyer to show that the absolute Chern character, from $K(A)$ to $HN(A)$, is compatible with the $λ$-filtration for every commutative $\mathbb{Q}$-algebra. This is the main result of Infinitesimal cohomology and the Chern character to negative cyclic homology, arXiv:math/0703133v1. 3.This agreement can be used to strengthen Ginot's results in "Formules explicites pour le charactere de Chern en $K$-théorie algébrique", Ann. Inst. Fourier 54 (2004).

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arXiv2007

Comparison between algebraic and topological K-theory of locally convex algebras

This paper is concerned with the algebraic K-theory of locally convex algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that the obstruction for the comparison map between algebraic and topological K-theory to be an isomorphism is (absolute) algebraic cyclic homology and prove the existence of an 6-term exact sequence. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Frechet algebra with bounded app. unit. This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper "Algebraic K-theory and functional analysis", First European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr. Math., 120, Birkhäuser, Basel, 1994. We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. We study the algebraic K-theory of the tensor product of a sub-harmonic ideal with an arbitrary complex algebra and prove that the obstruction for the periodicity of algebraic K-theory is again cyclic homology. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize), and the excision theorem for infinitesimal K-theory, due to the first author.

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arXiv2007

The $K$-theory of toric varieties

Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data.

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