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Irakli Patchkoria

Publications and source records attributed to Irakli Patchkoria.

26 records · Page 2Linked to original sources
arXiv2016

On exotic equivalences and a theorem of Franke

Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum $R$ whose graded homotopy ring $π_*R$ is concentrated in dimensions divisible by a natural number $N \geq 5$ and has homological dimension at most three, the homotopy category of $R$-modules is equivalent to the derived category of $π_*R$. The Johnson-Wilson spectrum $E(3)$ and the truncated Brown-Peterson spectrum $BP\langle 2 \rangle$ for any prime $p \geq 5$ are our main examples. If additionally the homological dimension of $π_*R$ is equal to two, then the homotopy category of $R$-modules and the derived category of $π_*R$ are triangulated equivalent. Here the main examples are $E(2)$ and $BP \langle 1 \rangle$ at $p \geq 5$. The last part of the paper discusses a triangulated equivalence between the homotopy category of $E(1)$-local spectra at a prime $p \geq 5$ and the derived category of Franke's model. This is a theorem of Franke and we fill a gap in the proof.

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arXiv2016

Stable finiteness properties of infinite discrete groups

Let $G$ be an infinite discrete group. A classifying space for proper actions of $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are contractible for all finite subgroups $H$ of $G$. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper $G$-spectra and study its finiteness properties. We investigate when $G$ admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper $G$-spectra and to classical finiteness properties of the Weyl groups of finite subgroups of $G$. Finally, if the group $G$ is virtually torsion-free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of $G$, thus providing the first geometric interpretation of the virtual cohomological dimension of a group.

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arXiv2016

The derived category of complex periodic K-theory localized at an odd prime

We prove that for an odd prime $p$, the derived category $\mathcal{D}(KU_{(p)})$ of the $p$-local complex periodic $K$-theory spectrum $KU_{(p)}$ is triangulated equivalent to the derived category of its homotopy ring $π_*KU_{(p)}$. This implies that if $p$ is an odd prime, the triangulated category $\mathcal{D}(KU_{(p)})$ is algebraic.

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arXiv2014

Rigidity in Equivariant Stable Homotopy Theory

For any finite group G, we show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all "higher order structure" of the 2-local G-equivariant stable homotopy category, such as the equivariant homotopy types of function G-spaces. The theorem can be seen as an equivariant version of Schwede's rigidity theorem at the prime 2.

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arXiv2011

On the Algebraic Classification of Module Spectra

Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $π_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $π_*R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of $R$-modules. The main examples of ring spectra to which our result applies are the $p$-local real connective $K$-theory spectrum $ko_{(p)}$, the Johnson-Wilson spectrum E(2), and the truncated Brown-Peterson spectrum $BP<1>$, for an odd prime $p$.

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arXiv2010

Comparison of Cubical and Simplicial Derived Functors

In this note we prove that the simplicial derived functors introduced by Tierney and Vogel [TV69] are naturally isomorphic to the cubical derived functors introduced by the author in [P09]. We also explain how this result generalizes the well-known fact that the simplicial and cubical singular homologies of a topological space are naturally isomorphic.

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arXiv2009

Cubical Resolutions and Derived Functors

We introduce pseudocubical objects with pseudoconnections in an arbitrary category, obtained from the Brown-Higgins structure of a cubical object with connections by suitably relaxing their identities, and construct a cubical analog of the Tierney-Vogel theory of simplicial derived functors. The crucial point in the construction is that projective precubical resolutions which are naturally used to define our cubical derived functors possess pseudodegeneracies and pseudoconnections. The same fact is essentially used for proving that in the case of an additive functor between abelian categories, our theory coincides with the classical relative theory of derived functors by Eilenberg-Moore.

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