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Guillermo Cortiñas

Publications and source records attributed to Guillermo Cortiñas.

49 records · Page 3Linked to original sources
arXiv2007

Bivariant algebraic K-Theory

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, matrix-stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, kk_*(A,B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel's homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk_*(H,A)=KH_*(A). We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.

math.KT↗
arXiv2005

The obstruction to excision in K-theory and in cyclic homology

Let $f:A \to B$ be a ring homomorphism of not necessarily unital rings and $I\triangleleft A$ an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups $K_*(A:I) \to K_*(B:f(I))$ to be an isomorphism; it is measured by the birelative groups $K_*(A,B:I)$. We show that these are rationally isomorphic to the corresponding birelative groups for cyclic homology up to a dimension shift. In the particular case when A and B are $\Q$-algebras we obtain an integral isomorphism.

math.KT↗
arXiv2005

Cyclic homology of $H$-unital (pro-) algebras, Lie algebra homology of matrices, and a paper of Hanlon's

We consider algebras over a field $k$ of characteristic zero. The article is concerned with the isomorphism of graded vectorspaces \[ H(\gl(A))\iso\wedge (HC(A)[-1]) \] between the Lie algebra homology of matrices and the free graded commutative algebra on the cyclic homology of the $k$-algebra $A$, shifted down one degree. For unital algebras this isomorphism is a classical result obtained by Loday and Quillen and independently by Tsygan. For $H$-unital algebras, it is known to hold too, as is that the proof follows from results of Hanlon's. However, to our knowledge, the proof is not immediate, and has not been published. In this paper we fill this gap in the literature by offering a detailed proof. Moreover we establish the isomorphism in the general setting of ($H$-unital) pro-algebras.

math.KT↗
arXiv2000

Interchanging homotopy limits and colimits in CAT

Let $I$,$J$ be small categories and $C:I\times J@>>>\CAT$ a functor to the category of small categories. We show that if $I$ has a final object then the canonical map $\operatorname{hocolim}_J\operatorname{holim}_IC@>>>\operatorname{holim}_I \operatorname{hocolim}_JC$ is a strong homotopy equivalence.

math.AT↗
arXiv2000

The structure of smooth algebras in Kapranov's framework for noncommutative geometry

In Kapranov, M. {\it Noncommutative geometry based on commutator expansions,} J. reine angew. Math {\bf 505} (1998) 73-118, a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the noncommutative coordinate rings of affine open subsets of such of those varieties which are smooth (Theorem 3.4). The theorem describes the local ring of a point as a truncation of a quantization of the enveloping Poisson algebra of a smooth commutative local algebra. An explicit descripition of this quantization is given in Theorem 2.5. A description of the $A$- module structure of the Poisson envelope of a smooth commutative algebra $A$ was given in loc. cit., Theorem 4.1.3. However the proof given in loc. cit. has a gap. We fix this gap for $A$ local (Theorem 1.4) and prove a weaker global result (Theorem 1.6).

math.RA↗
arXiv2000

An explicit formula for PBW quantization

Let $k$ be a field of characteristic zero, $\g$ a $k$-Lie algebra, $e:S\g@>>>U\g$ the symmetrization map. The PBW quantization is the one parameter family of associative products: $$ x\star_t y=\sum_{p=0}^\infty B_p(x,y)t^p\qquad (t\in k) $$ where $B_p$ is the homogeneous component of degree $-p$ of the map $B:S\g\otimes_kS\g@>>>S\g$, $B(x,y)=e^{-1}(exey)$. In this paper we give an explicit formula for $B$. As an application, we prove that for each $p\ge 0$, $B_p$ is a bidifferential operator of order $\le p$.

math.QA↗
arXiv2000

On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology

Cuntz and Quillen have shown that for algebras over a field $k$ with $char(k)=0$, periodic cyclic homology may be regarded, in some sense, as the derived functor of (non-commutative) de Rham (co-)homology. The purpose of this paper is to formalize this derived functor analogy. We show that the localization ${Def}^{-1}\Cal{PA}$ of the category $\Cal{PA}$ of countable pro-algebras at the class of (infinitesimal) deformations exists (in any characteristic) (Theorem 3.2) and that, in characteristic zero, periodic cyclic homology is the derived functor of de Rham cohomology with respect to this localization (Corollary 5.4). We also compute the derived functor of rational $K$-theory for algebras over $\Bbb Q$, which we show is essentially the fiber of the Chern character to negative cyclic homology (Theorem 6.2).

math.KT↗
arXiv2000

Cyclic homology of commutative algebras over general ground rings

We consider commutative algebras and chain DG algebras over a fixed commutative ground ring $k$ as in the title. We are concerned with the problem of computing the cyclic (and Hochschild) homology of such algebras via free DG-resolutions $ΛV @>>> A$. We find spectral sequences $$E^2_{p,q}=H_p(ΛV\otimesΓ^q(dV))\Rightarrow HH_{p+q}(ΛV)$$ and $${E'}^2_{\pq}=H_p(ΛV\otimesΓ^{\le q}(dV)) \Rightarrow HC_{p+q}(ΛV)$$ The algebra $ΛV\otimesΓ(dV)$ is a divided power version of the de Rham algebra; in the particular case when $k$ is a field of characteristic zero, the spectral sequences above agree with those found by Burghelea and Vigué (Cyclic homology of commutative algebras I, Lecture Notes in Math. {\bf 1318} (1988) 51-72), where it is shown they degenerate at the $E^2$ term. For arbitrary ground rings we prove here (Theorem 2.3) that if $V_n=0$ for $n\ge 2$ then $E^2=E^\infty$. From this we derive a formula for the Hochschild homology of flat complete intersections in terms of a filtration of the complex for crystalline cohomology, and find a description of ${E'}^2$ also in terms of crystalline cohomology (theorem 3.0). The latter spectral sequence degenerates for complete intersections of embedding dimension $\le 2$ (Corollary 3.1). Without flatness assumptions, our results can be viewed as the computation Shukla (cyclic) homology (T. Pirashvili, F. Waldhausen; Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra{\bf 82} (1992) 81-98).

math.KT↗
arXiv2000

Artinian algebras and differential forms

This article concerns commutative algebras over a field $k$ of characteristic zero which are finite dimensional as vectorspaces, and particularly those of such algebras which are graded. Here the term graded is applied to non-negatively graded algebras $A$ with $A_0$ reduced and finite dimensional. Thus the trivial grading $A=A_0$ is only allowed if $A$ is a product of finite field extensions of $k$. It has been conjectured (G. Cortiñas, S. Geller, C. Weibel; The Artinian Berger Conjecture. Math. Zeitschrift {\bf 228} 3 (1998) 569-588) that for all finite dimensional algebras $A$ which are not principal ideal algebras (i.e. have at least one nonprincipal ideal), the following submodule of the Kähler differentials is nonzero: $$\bigcap{\ker(Ω_A @>>>Ω_B)}$$ Here the intersection is taken over all principal ideal algebras $B$ and all homomorphisms $A @>>>B$. In this paper we prove that the conjecture holds for both Gorenstein graded and standard graded algebras.

math.AG↗
arXiv2000

Infinitesimal K-theory

In this paper we study the fiber F of the rational Jones-Goodwillie character $$ F:=\hofiber(ch:K^\rat(A)@>>>HN^\rat(A)) $$ going from K-theory to negative cyclic homology of associative rings. We describe this fiber F in terms of sheaf cohomology. We prove that, for $n\ge 1$, there is an isomorphism: $$ π_n(F)\cong H^{-n}_{inf}(A,K^\rat) $$ between the homotopy of the fiber and the hypercohomology groups of $K^\rat$ on a non-commutative version of Grothendieck's infinitesimal site.

math.KT↗
arXiv1998

Periodic cyclic homology as sheaf cohomology

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example we show that hypercohomology with coefficients in $K$-theory gives the fiber of the Jones-Goodwillie character which goes from $K$-theory to negative cyclic homology.

math.KT↗
arXiv1995

The Artinian Berger Conjecture

We propose an Artinian version of Berger's Conjecture for curves, concerning the module of Kähler differentials of an algebra. Our version implies Berger's Conjecture in characteristic 0. We establish our Artinian Berger Conjecture in a number of cases, and prove that Berger's Conjecture holds for curve singularities whose conductor ideal contains the cube of a maximal ideal.

math.AC↗