arXiv subjects

Tobias Barthel

Publications and source records attributed to Tobias Barthel.

At least 37 records · Page 2Linked to original sources

arXiv2023

Lattices over finite group schemes and stratification

This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full subcategory of all $G$-modules projective over $R$ generated by the lattices. The stable category of such $G$-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of $G$. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

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arXiv2023

Stratification in tensor triangular geometry with applications to spectral Mackey functors

We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our theory of stratification is based on the approach of Stevenson which uses the Balmer-Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-to-global principle and establish that the Balmer-Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson-Iyengar-Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite étale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria-Sanders-Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height $0$ and height $\infty$ chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of $E(n)$-local spectral Mackey functors noting that it bijects onto the height $\le n$ chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.

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arXiv2023

Descent in tensor triangular geometry

We investigate to what extent we can descend the classification of localizing, smashing and thick ideals in a presentably symmetric monoidal stable $\infty$-category $\mathscr{C}$ along a descendable commutative algebra $A$. We establish equalizer diagrams relating the lattices of localizing and smashing ideals of $\mathscr{C}$ to those of $\mathrm{Mod}_{A}(\mathscr{C})$ and $\mathrm{Mod}_{A\otimes A}(\mathscr{C})$. If $A$ is compact, we obtain a similar equalizer for the lattices of thick ideals which, via Stone duality, yields a coequalizer diagram of Balmer spectra in the category of spectral spaces. We then give conditions under which the telescope conjecture and stratification descend from $\mathrm{Mod}_{A}(\mathscr{C})$ to $\mathscr{C}$. The utility of these results is demonstrated in the case of faithful Galois extensions in tensor triangular geometry.

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arXiv2023

Cosupport in tensor triangular geometry

We develop a theory of cosupport and costratification in tensor triangular geometry. We study the geometric relationship between support and cosupport, provide a conceptual foundation for cosupport as categorically dual to support, and discover surprising relations between the theory of costratification and the theory of stratification. We prove that many categories in algebra, topology and geometry are costratified by developing and applying descent techniques. An overarching theme is that cosupport is relevant for diverse questions in tensor triangular geometry and that a full understanding of a category requires knowledge of both its support and its cosupport.

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arXiv2023

Stratification and the comparison between homological and tensor triangular support

We compare the homological support and tensor triangular support for `big' objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support.

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arXiv2022

The Chromatic Fourier Transform

We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology of $π$-finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$. We use this theory to generalize said duality in three different directions. First, we extend it from $\mathbb{Z}$-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of $E_n$. Second, we lift it to the telescopic setting by replacing $E_n$ with $T(n)$-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg--Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal $\infty$-categories of local systems of $K(n)$-local $E_n$-modules, and relate it to (semiadditive) redshift phenomena.

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arXiv2022

Stratifying integral representations via equivariant homotopy theory

We prove that the derived category of $R$-linear representations of a finite group $G$ is stratified for any regular commutative ring $R$. As an application, we obtain a classification of localizing tensor ideals of ordinary $R$-linear $G$-representations whose underlying $R$-module is projective.

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arXiv2022

On conjectures of Hovey--Strickland and Chai

We prove the height two case of a conjecture of Hovey and Strickland that provides a $K(n)$-local analogue of the Hopkins--Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross--Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava $E$-theory is coherent, and that every finitely generated Morava module can be realized by a $K(n)$-local spectrum as long as $2p-2>n^2+n$. Finally, we deduce consequences of our results for descent of Balmer spectra.

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arXiv2021

Morava K-theory and Filtrations by Powers

We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin-Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabilizer group. As an application, we compute the zeroth limit at all primes and heights.

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arXiv2021

Stratifying integral representations of finite groups

We classify the localizing tensor ideals of the integral stable module category for any finite group $G$. This results in a generic classification of $\mathbb{Z}[G]$-lattices of finite and infinite rank and globalizes the modular case established in celebrated work of Benson, Iyengar, and Krause. Further consequences include a verification of the generalized telescope conjecture in this context, a tensor product formula for integral cohomological support, as well as a generalization of Quillen's stratification theorem for group cohomology. Our proof makes use of novel descent techniques for stratification in tensor-triangular geometry that are of independent interest.

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arXiv2021

Constructing the determinant sphere using a Tate twist

Following an idea of Hopkins, we construct a model of the determinant sphere $S\langle det \rangle$ in the category of $K(n)$-local spectra. To do this, we build a spectrum which we call the Tate sphere $S(1)$. This is a $p$-complete sphere with a natural continuous action of $\mathbb{Z}_p^\times$. The Tate sphere inherits an action of $\mathbb{G}_n$ via the determinant and smashing Morava $E$-theory with $S(1)$ has the effect of twisting the action of $\mathbb{G}_n$. A large part of this paper consists of analyzing continuous $\mathbb{G}_n$-actions and their homotopy fixed points in the setup of Devinatz and Hopkins.

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arXiv2021

On stratification for spaces with Noetherian mod $p$ cohomology

Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;\mathbb{F}_p)$ be the commutative ring spectrum of $\mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of module spectra on $C^*(X;\mathbb{F}_p)$ is stratified in the sense of Benson, Iyengar, Krause, providing a classification of all its localizing subcategories. We establish stratification in this sense for classifying spaces of a large class of topological groups including Kac--Moody groups as well as whenever $X$ admits an $H$-space structure. More generally, using Lannes' theory we prove that stratification for $X$ is equivalent to a condition that generalizes Chouinard's theorem for finite groups. In particular, this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory.

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arXiv2020

Local Gorenstein duality for cochains on spaces

We investigate when a commutative ring spectrum $R$ satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of $k$-algebras. Our main examples are of the form $R = C^*(X;k)$, the ring spectrum of cochains on a space $X$ for a field $k$. In particular, we establish local Gorenstein duality in characteristic $p$ for $p$-compact groups and $p$-local finite groups as well as for $k = \Q$ and $X$ a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.

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arXiv2020

Power operations in the Stolz--Teichner program

The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.

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arXiv2020

Gross-Hopkins Duals of Higher Real K-theory Spectra

We determine the Gross-Hopkins duals of certain higher real $K$-theory spectra. More specifically, let $p$ be an odd prime, and consider the Morava $E$-theory spectrum of height $n=p-1$. It is known, in the expert circles, that for certain finite subgroups $G$ of the Morava stabilizer group, the homotopy fixed point spectra $E_n^{hG}$ are Gross-Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups $G$ which contain $p$-torsion. This generalizes previous results for $n=2$ and $p=3$.

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arXiv2020

Transfer ideals and torsion in the Morava $E$-theory of abelian groups

Let $A$ be a finite abelian $p$ group of rank at least $2$. We show that $E^0(BA)/I_{tr}$, the quotient of the Morava $E$-cohomology of $A$ by the ideal generated by the image of the transfers along all proper subgroups, contains $p$-torsion. The proof makes use of transchromatic character theory.

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arXiv2020

Chromatic homotopy theory is asymptotically algebraic

Inspired by the Ax--Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the $E(n,p)$-local categories over any non-prinicipal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.

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arXiv2019

Completing perfect complexes

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a proper foundation for completing a triangulated category.

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