arXiv subjects

Haldun Özgür Bayındır

Publications and source records attributed to Haldun Özgür Bayındır.

9 recordsLinked to original sources

arXiv2026

On the integral algebraic K-theory of Morava K-theory

We compute the cardinalities of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in all degrees that are not congruent to $0$ or $1$ modulo $2p-2$. After base-changing the coefficients of $k(n)$ to the algebraic closure $\overline{\mathbb{F}}_p$, we determine the corresponding cardinalities in all degrees and show that the groups vanish in even degrees. Our approach uses what we call the orbit filtration on topological cyclic homology arising from the May filtration on topological Hochschild homology. Combining this with an analysis of the topological cyclic homology of formal DGAs of the form $\mathbb{F}[x_{2m}]$, where $\mathbb{F}$ is a perfect field of characteristic $p$, we prove strong cardinality results for the topological cyclic homology of $\mathbb{E}_1$-rings with homotopy $\mathbb{F}[x_{2m}]$. For Morava K-theories over finite fields, we further use the motivic filtration of Hahn--Raksit--Wilson. As an application of our methods, we compute the cardinalities of the algebraic K-theory groups of the truncation $W(\overline{\mathbb{F}}_p)/p^n$ of the $p$-typical Witt vectors of $\overline{\mathbb{F}}_p$; in particular, they vanish in even degrees.

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arXiv2025

Towards the classification of DGAs with polynomial homology

We study the classification of $\mathbb{Z}$-DGAs with polynomial homology $\mathbb{F}_p[x]$ with $\lvert x \rvert >0$, motivated by computations in algebraic $K$-theory. This classification problem was left open in work of Dwyer, Greenlees, and Iyengar. We prove that there are infinitely many such DGAs for even $\lvert x \rvert$ and that for $\lvert x \rvert \geq 2p-2$ any such DGA is formal as a ring spectrum. Through this, we obtain examples of triangulated categories with infinitely many DG-enhancements and a classification of prime DG-division rings. Combining our results with earlier work of the second author and Tamme, we obtain new (relative) algebraic $K$-theory computations for rings such as the mixed characteristic coordinate axes $\mathbb{Z}[x]/px$ and the group ring $\mathbb{Z}[C_{p^n}]$.

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arXiv2024

Algebraic $K$-theory of the two-periodic first Morava $K$-theory

Using the root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of $T(2)_*\text{K}(ku)$ for $p>3$. Through this, we also produce a new algebraic $K$-theory computation; namely we obtain $T(2)_*\text{K}(ku/p)$, where $ku/p$ is the $2$-periodic Morava $K$-theory spectrum of height $1$.

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arXiv2023

Adjunction of roots, algebraic $K$-theory and chromatic redshift

Given an $E_1$-ring $A$ and a class $a \in π_{mk}(A)$ satisfying a suitable hypothesis, we define a map of $E_1$-rings $A\to A(\sqrt[m]{a})$ realizing the adjunction of an $m$th root of $a$. We define a form of logarithmic THH for $E_1$-rings, and show that root adjunction is log-THH-étale for suitably tamely ramified extension, which provides a formula for THH$(A(\sqrt[m]{a}))$ in terms of THH and log-THH of $A$. If $A$ is connective, we prove that the induced map $K(A) \to K(A(\sqrt[m]{a}))$ in algebraic $K$-theory is the inclusion of a wedge summand. Using this, we obtain $V(1)_*K(ko_p)$ for $p>3$ and also, we deduce that if $K(A)$ exhibits chromatic redshift, so does $K(A(\sqrt[m]{a}))$. We interpret several extensions of ring spectra as examples of root adjunction, and use this to obtain a new proof of the fact that Lubin-Tate spectra satisfy the redshift conjecture.

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arXiv2023

Spanier-Whitehead duality for topological coHochschild homology

In this work, we compute the topological coHochschild homology (coTHH) of interesting coalgebras such as the Steenrod algebra spectrum. For this, we start by extending the Hess-Shipley definition of coTHH to $\infty$-categories, following the Nikolaus-Scholze approach to THH. Furthermore, we prove that coTHH of what we call quasi-proper coalgebras can be obtained from THH via Spanier-Whitehead duality which provides further insight into coTHH and its relationship to THH.

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arXiv2022

Algebraic $K$-theory of $\text{THH}(\mathbb{F}_p)$

In this work we study the $E_{\infty}$-ring $\text{THH}(\mathbb{F}_p)$ as a graded spectrum. Following an identification at the level of $E_2$-algebras with $\mathbb{F}_p[ΩS^3]$, the group ring of the $E_1$-group $ΩS^3$ over $\mathbb{F}_p$, we show that the grading on $\text{THH}(\mathbb{F}_p)$ arises from decomposition on the cyclic bar construction of the pointed monoid $ΩS^3$. This allows us to use trace methods to compute the algebraic $K$-theory of $\text{THH}(\mathbb{F}_p)$. We also show that as an $E_2$ $H\mathbb{F}_p$-ring, $\text{THH}(\mathbb{F}_p)$ is uniquely determined by its homotopy groups. These results hold in fact for $\text{THH}(k)$, where $k$ is any perfect field of characteristic $p$. Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic $K$-theory of formal DGAs.

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arXiv2021

DGAs with polynomial homology

In this work, we study the classification of differential graded algebras over $\mathbb{Z}$ (DGAs) whose homology is $\mathbb{F}_p[x]$, i.e. the polynomial algebra over $\mathbb{F}_p$ on a single generator. This classification problem was left open in work of Dwyer, Greenlees and Iyengar. For $\lvert y_{2p-2} \rvert = 2p-2$, we show that there is a unique non-formal DGA with homology $\mathbb{F}_p[y_{2p-2}]$ and a non-formal $2p-2$ Postnikov section. Among a classification result, this provides the first example of a non-formal DGA with homology $\mathbb{F}_p[x]$. By duality, this also shows that there is a non-formal DGA whose homology is an exterior algebra over $\mathbb{F}_p$ with a generator in degree $-(2p-1)$. Considering the classification of the ring spectra corresponding to these DGAs, we show that every $E_2$ DGA with homology $\mathbb{F}_p[x]$ (with no restrictions on $\lvert x \rvert$) is topologically equivalent to the formal DGA with homology $\mathbb{F}_p[x]$, i.e. they are topologically formal. This follows by a theorem of Hopkins and Mahowald.

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arXiv2021

Admissible replacements for simplicial monoidal model categories

Using Dugger's construction of universal model categories, we produce replacements for simplicial and combinatorial symmetric monoidal model categories with better operadic properties. Namely, these replacements admit a model structure on algebras over any given colored operad. As an application, we show that in the stable case, such symmetric monoidal model categories are classified by commutative ring spectra when the monoidal unit is a compact generator. In other words, they are strong monoidally Quillen equivalent to modules over a uniquely determined commutative ring spectrum.

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arXiv2021

Extension DGAs and topological Hochschild homology

In this work, we study those differential graded algebras (DGAs) that arise from ring spectra through the extension of scalars functor. Namely, we study DGAs whose corresponding Eilenberg-Mac Lane ring spectrum is equivalent to $H\mathbb{Z} \wedge E$ for some ring spectrum $E$. We call these DGAs extension DGAs. We also define and study this notion for $E_\infty$ DGAs. The topological Hochschild homology (THH) spectrum of an extension DGA splits in a convenient way. We show that formal DGAs with nice homology rings are extension and therefore their THH groups can be obtained from their Hochschild homology groups in many cases of interest. We also provide interesting examples of DGAs that are not extension. In the second part, we study properties of extension DGAs. We show that in various cases, topological equivalences and quasi-isomorphisms agree for extension DGAs. From this, we obtain that dg Morita equivalences and Morita equivalences also agree in these cases.

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