22 problems

The telescope conjecture. Every smashing localizing subcategory $\mathcal{L}$ is generated by compact objects of $\mathcal{T}$.

Let $\mathcal{T}$ be a compactly generated triangulated category, and let $\mathcal{S}$ be a smashing subcategory, meaning a subcategory whose inclusion admits a coproduct-preservi…

Telescope Conjecture. Then

Telescope conjecture for $fp$-spectra. The telescope conjecture holds for $fp$-spectra; equivalently, for every $fp$-spectrum $X$, the localization map

Krause's reformulated telescope conjecture. The ideal $I_{\mathbf{t}}$ is generated by identity morphisms of objects in $\mathcal{D}^c$.

An fp spectrum is a finite-periodic spectrum in the sense used in the paper. Mahowald–Rezk conjecture. All fp spectra satisfy the conditions of the telescope conjecture. The paper…

Let $X$ be a spectrum of fp-type $n$, and let $L_n^f$ and $L_n$ denote, respectively, finite and ordinary chromatic localization at height $n$. Mahowald-Rezk conjecture. $X$ satisf…

Finite-defect telescope conjecture. Any spectrum with finite chromatic defect satisfies the condition of the telescope conjecture at all heights and primes.

Let $X$ be a finite spectrum of type $h$, and let $\varpi_{\mathrm{T}(h)}(X,n)$ denote the size function measuring its telescopic homotopy groups through degree $n$. Telescopic gro…

Let $y(h)$ be the spectrum under consideration, and let $b_{h+i,j}$ denote the indicated classes in its localized Adams spectral sequence. MRS collapse conjecture. The classes…

Let $y(h)$ be the spectrum under consideration, and let $h_{2h+i-j,j}$, $b_{h+i,j}$, and $q_h$ denote the classes and coefficient appearing in its localized Adams spectral sequence…

Periodicity-one self-map conjecture. The spectrum $\mathcal{Z}_\mathbb{R}$ is of type $(2,2)$ and admits a $v_{(2,\mathrm{nil})}$-self-map

Let $\widehat Z$ be the telescope of the $v_2$-self-map on $Z$. Let $x_i=h_{i,0}^2$. Parabola conjecture, height 2. The differentials of the preceding height-$2$ differentials conj…

Let $\widehat{Z}$ be the telescope of the $v_2$-self-map on $Z$, and let $x_i:=h_{i,0}^2$. Parabola conjecture. The localized Adams spectral sequence for $Z$ collapses at $E_5$, an…

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated category. A localization $L:\mathcal{T}\to\mathcal{T}$ is smashing if it preserves coproducts, and it is fini…

Motivic Telescope Conjecture. The natural transformation

Let $X$ be a finite motivic spectrum of type $(m,n)$ with a non-nilpotent self-map $v:\Sigma^{-r,-s}X\to X$ of type $(m,n)$. Write $\langle E\rangle$ for the Bousfield class of a m…

Let $\mathscr{T}$ be a compactly generated $\otimes$-triangulated category and let $\mathscr{U}$ be a compactly generated triangulated category, both with all coproducts. Suppose ……

Let $\mathscr{T}$ be a compactly generated triangulated category with all coproducts, and let … be a smashing localization, meaning that its right adjoint preserves coproducts. Wri…

The telescope conjecture. Every smashing localising subcategory of $\mathsf T$ is generated by objects that are compact in $\mathsf T$.

Let $\Lambda$ be an artin algebra, and let $(\mathcal{A},\mathcal{B})$ be a hereditary cotorsion pair in the category of all left $\Lambda$-modules. A cotorsion pair is of finite t…

Let $Lambda$ be an artin algebra, and let $(\mathcal{A},\mathcal{B})$ be a hereditary cotorsion pair in the category of all left $Lambda$-modules. A cotorsion pair is hereditary wh…