DGAs with polynomial homology

Bayındır, Haldun Özgür

doi:10.1016/j.aim.2021.107907

Cited by 3 publications

(3 citation statements)

References 20 publications

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“…\end{equation*}$$Precomposing with this automorphism, the action of the second factor of

\Lambda _{\mathbb {F}_p}({z_{-1}}) \otimes \Lambda _{\mathbb {F}_p}({z_{-1}})

\Lambda _{\mathbb {F}_p}({z_{-1}})

becomes trivial, that is, this action is the one obtained by the augmentation map of

\Lambda _{\mathbb {F}_p}({z_{-1}})

. Using the Künneth formula [3, eq. (2)], we obtain the following:

\begin{matrix} E^{2} ≅ & {sans-serifTor}^{{normalΛ}_{F_{p}} false(z_{- 1} false) \otimes {normalΛ}_{F_{p}} {(z_{- 1})}^{o p}} false(Λ_{{double-struckF}_{p}} (z_{- 1}), Λ_{{double-struckF}_{p}} (z_{- 1}) false) \\ newline≅ & {sans-serifTor}^{{normalΛ}_{F_{p}} false(z_{- 1} false)} false(Λ_{{double-struckF}_{p}} (z_{- 1}), Λ_{{double-struckF}_{p}} (z_{- 1}) false) \otimes {sans-serifTor}^{{normalΛ}_{}} \end{matrix}

…”

Section: Computationsmentioning

confidence: 99%

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Spanier–Whitehead duality for topological coHochschild homology

Bayındır

Péroux

2023

Journal of London Math Soc

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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 topological Hochschild homology (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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“…\end{equation*}$$Precomposing with this automorphism, the action of the second factor of

\Lambda _{\mathbb {F}_p}({z_{-1}}) \otimes \Lambda _{\mathbb {F}_p}({z_{-1}})

\Lambda _{\mathbb {F}_p}({z_{-1}})

becomes trivial, that is, this action is the one obtained by the augmentation map of

\Lambda _{\mathbb {F}_p}({z_{-1}})

. Using the Künneth formula [3, eq. (2)], we obtain the following:

\begin{matrix} E^{2} ≅ & {sans-serifTor}^{{normalΛ}_{F_{p}} false(z_{- 1} false) \otimes {normalΛ}_{F_{p}} {(z_{- 1})}^{o p}} false(Λ_{{double-struckF}_{p}} (z_{- 1}), Λ_{{double-struckF}_{p}} (z_{- 1}) false) \\ newline≅ & {sans-serifTor}^{{normalΛ}_{F_{p}} false(z_{- 1} false)} false(Λ_{{double-struckF}_{p}} (z_{- 1}), Λ_{{double-struckF}_{p}} (z_{- 1}) false) \otimes {sans-serifTor}^{{normalΛ}_{}} \end{matrix}

…”

Section: Computationsmentioning

confidence: 99%

“…Remark In [3, proof of Theorem 1.6], the first author shows that every

\mathbb {F}_p

‐DGA with homology

\Lambda _{\mathbb {F}_p}(z_{-1})

is formal as a DGA. However, we need a slightly stronger result.…”

Section: Computationsmentioning

confidence: 99%

Spanier–Whitehead duality for topological coHochschild homology

Bayındır

Péroux

2023

Journal of London Math Soc

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“…1 / 2 D 2 1 ˝F2 1. However, there is no element in H F 2 H Z˝F 2 H F 2 Y that squares to 2 1 ˝F2 1. Since this does not use Dyer-Lashof operations, this argument at p D 2 also works for DGAs and H Z-algebras and provides a proof of Theorem 1.8.…”

Section: E -Infinity F P -Dgas Are Not Extensionmentioning

confidence: 99%

Extension DGAs and topological Hochschild homology

Bayındır

2023

Algebr. Geom. Topol.

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Algebraic 𝐾-theory of 𝑇𝐻𝐻(𝔽_{𝕡})

Bayındır¹,

Moulinos

2022

Trans. Amer. Math. Soc.

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In this work we study the E ∞ E_{\infty } -ring THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) as a graded spectrum. Following an identification at the level of E 2 E_2 -algebras with F p [ Ω S 3 ] \mathbb {F}_p[\Omega S^3] , the group ring of the E 1 E_1 -group Ω S 3 \Omega S^3 over F p \mathbb {F}_p , we show that the grading on THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) arises from decomposition on the cyclic bar construction of the pointed monoid Ω S 3 \Omega S^3 . This allows us to use trace methods to compute the algebraic K K -theory of THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) . We also show that as an E 2 E_2 H F p H\mathbb {F}_p -ring, THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) is uniquely determined by its homotopy groups. These results hold in fact for THH ⁡ ( k ) \operatorname {THH}(k) , where k k is any perfect field of characteristic p 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 K -theory of formal DGAs.

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DGAs with polynomial homology

Cited by 3 publications

References 20 publications

Spanier–Whitehead duality for topological coHochschild homology

Spanier–Whitehead duality for topological coHochschild homology

Extension DGAs and topological Hochschild homology

Algebraic 𝐾-theory of 𝑇𝐻𝐻(𝔽_{𝕡})

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