Transfinite Adams representability

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• 作者：Fernando Muro^a ; ^{fmuro@us.es" class="auth_mail" title="E-mail the corresponding author} ; Oriol Raventó ; s^b ; ^{oriol.raventos-morera@ur.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：18E30 ; 55S35
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：9 April 2016
• 年：2016
• 卷：292
• 期：Complete
• 页码：111-180
• 全文大小：995 K

文摘

We consider the following problems in a well generated triangulated category T$\mathcal{T}$. Let α   be a regular cardinal and T^α⊂T${\mathcal{T}}^{\alpha }\subset \mathcal{T}$ the full subcategory of α  -compact objects. Is every functor H:(T^α)^op→Ab$H:{({\mathcal{T}}^{\alpha })}^{\mathrm{op}}\to \mathrm{Ab}$ that preserves products of <α   objects and takes exact triangles to exact sequences of the form H≅T(−,X)_|T^α$H\cong \mathcal{T}{(-,X)}_{{|}_{{\mathcal{T}}^{\alpha }}}$ for some X   in T$\mathcal{T}$? Is every natural transformation τ:T(−,X)_|T^α→T(−,Y)_|T^α$\tau :\mathcal{T}{(-,X)}_{{|}_{{\mathcal{T}}^{\alpha }}}\to \mathcal{T}{(-,Y)}_{{|}_{{\mathcal{T}}^{\alpha }}}$ of the form τ=T(−,f)_|T^α$\tau =\mathcal{T}{(-,f)}_{{|}_{{\mathcal{T}}^{\alpha }}}$ for some f:X→Y$f:X\to Y$ in T$\mathcal{T}$? If the answer to both questions is positive we say that T$\mathcal{T}$ satisfies α  -Adams representability. A classical result going back to Brown and Adams shows that the stable homotopy category satisfies ℵ₀${\mathrm{\aleph }}_0$-Adams representability. The case α=ℵ₀$\alpha ={\mathrm{\aleph }}_0$ is well understood thanks to the work of Christensen, Keller, and Neeman. In this paper we develop an obstruction theory to decide whether T$\mathcal{T}$ satisfies α-Adams representability. We derive necessary and sufficient conditions of homological nature, and we compute several examples. In particular, we show that there are rings satisfying α  -Adams representability for all α≥ℵ₀$\alpha \ge {\mathrm{\aleph }}_0$ and rings which do not satisfy α  -Adams representability for any α≥ℵ₀$\alpha \ge {\mathrm{\aleph }}_0$. Moreover, we exhibit rings for which the answer to both questions is no for all ℵ_ω>α≥ℵ₂${\mathrm{\aleph }}_{\omega }>\alpha \ge {\mathrm{\aleph }}_2$.