Transfinite Adams representability

Oriol Raventos
Thursday, July 25, 2013 - from 17:30 to 17:55
We consider the following problems in a well generated triangulated category $\mathscr{T}$. Let $\alpha$ be a regular cardinal and $\mathscr{T}^{\alpha}\subset \mathscr{T}$ the full subcategory of $\alpha$\nobreakdash-compact objects. Is every functor $H\colon (\mathscr{T}^{\alpha})^{\rm op}\rightarrow{\rm Ab}$ that preserves products of $<\alpha$ objects and takes exact triangles to exact sequences of the form $H\cong\mathscr{T}(-,X)_{|_{\mathscr{T}^{\alpha}}}$ for some $X$ in $\mathscr{T}$? Is every natural transformation $\tau\colon \mathscr{T}(-,X)_{|_{\mathscr{T}^{\alpha}}}\rightarrow \mathscr{T}(-,Y)_{|_{\mathscr{T}^{\alpha}}}$ of the form $\tau=\mathscr{T}(-,f)_{|_{\mathscr{T}^{\alpha}}}$ for some $f\colon X\rightarrow Y$ in $\mathscr{T}$? If the answer to both questions is positive we say that $\mathscr{T}$ satisfies $\alpha$\nobreakdash-Adams representability. A classical result going back to Brown and Adams shows that the stable homotopy category satisfies $\aleph_{0}$\nobreakdash-Adams representability. The case $\alpha=\aleph_{0}$ is well understood thanks to the work of Christensen, Keller and Neeman. We define an obstruction theory to decide when $\mathscr{T}$ satisfies $\alpha$\nobreakdash-Adams representability. We derive necessary and sufficient conditions of homological nature, and we exhibit several examples. In particular, we show that, for all $\alpha\geq\aleph_{0}$, there are rings whose derived category satisfies $\alpha$\nobreakdash-Adams representability and also rings for which the answer to the second question is~no. All the results are in joint work with Fernando Muro.