112 problems

Let $S\in\mathrm{Rg}$ be a base ring, and let $\mathrm{SH}(S)$ denote the stable motivic homotopy category over $S$. Let $\mathbf{MGL}\in\mathrm{SH}(S)$ be the algebraic cobordism…

Morel's conjecture.

Let $\mathcal{Y}$ be a space over a field, and let $\pi_i^{\mathbb{A}^1}(\mathcal{Y})$ denote its $i$-th $\mathbb{A}^1$-homotopy sheaf. Morel proved that…

Over $\mathbb{C}$, let $\eta^{-1}\mathbb{S}$ be the η-periodic motivic sphere, let $P$ and $v_n$ denote the classes on the $E_2$-page of its motivic Adams spectral sequence, and le…

Morel's stable $\mathbb{A}^1$-connectivity conjecture. Under these hypotheses, $L_{\operatorname{mot}}E$ is $n$-connective.

Let $\mathbb S$ denote the sphere spectrum over $\mathbb Z$. A spectrum $\mathbb E$ over $\mathbb Z$ is absolutely pure if, for every smoothable morphism $f:X\to S$ between regular…

Let $S$ be a noetherian excellent finite dimensional base scheme of characteristic $0$. The Ayoub conjecture. The heart of the perverse homotopy t-structure on $\mathrm{DM}(S)$ is…

For a non-negative integer $k$, let $e_0\mathbf{g}^k$ denote the corresponding subset of Ext. Non-emptiness conjecture. The set $e_0\mathbf{g}^k$ is non-empty if and only if … for…

Let $k$ be a field of characteristic zero. Write $K^M_2(k)$ for the second Milnor $K$-theory group and $\pi_{1,0}^k$ for the motivic stable homotopy group in weight zero. Morel's…

Let ${\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k)$ be the category of symmetric $\mathbb{P}^1$-spectra over $k$, and let ${\operatorname{{\mathcal S}{\mathcal H}}}(k)$ be its homotopy…

Let $k$ be the base field, let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the homotopy category of $S^1$-spectra, and let $\Sigma^\infty_{{\mathbb P}^1}$ and…

Let $k$ be the base field and let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the Morel–Voevodsky homotopy category of $S^1$-spectra. For an integer $d$, let…

Let $X$ be a smooth scheme of finite type over a separably closed field $k$ of characteristic different from a prime $\ell$. Suppose that $\ell$ is odd or that $\ell^{\nu}\geq 4$,…

Let $DMT(\mathbb Z)$ be the category of mixed Tate motives over $\mathbb Z$, let ${\rm Gal}_{\rm mot}(\overline{\mathbb Q}/\mathbb Q)$ be its motivic Galois groupscheme, and let…

Over $\mathbb{C}$, let $\sigma\oldin\pi_{7,4}^{\mathbb{C}}(\mathbb{S}^{\mathrm{top}})$ and $\overline{\sigma}\oldin\pi_{19,11}^{\mathbb{C}}(\mathbb{S})$ be the motivic classes name…

Over $\mathbb{C}$, let $\overline{\kappa}_2\oldin\pi_{44,24}^{\mathbb{C}}(\mathbb{S})$ be the motivic class obtained from the class detecting…

Over $\mathbb{C}$, let $\mathbb{S}$ be the motivic sphere, let $\eta$ be the first Hopf map, and let $w_1^4$ and $w_2^{32}$ denote the indicated motivic periodicity self-maps. The…

Let $F$ be a field, let $\mathrm{KGL}\oldin\mathrm{SH}(F)$ be algebraic $K$-theory, and let $\mathrm{K}(F)\oldin\mathrm{Sp}$ denote its topological realization. Let the slice filtr…

Let … Over the base field $\mathbb{C}$, let $M_n(\mathbb{C}):= H^{*,*}(BV_n/\mathbb{C};\mathbb{F}_2)$ and … where $A^{>0,*}$ consists of motivic Steenrod algebra elements of positi…

Let $S=\operatorname{Spec}\mathbf Z$. For an $R$-linear MW-module $M$ over $S$, write $\big(\mathrm{FT}^{R}\big)$ for the property that, for every $S$-scheme $X$ of finite type, al…

Uniqueness conjecture for relative motivic spheres. If $X$ is $\mathbb{A}^1$-homotopic to $\mathbb{A}^2_S\backslash\{0\}$, then $X$ is isomorphic to $\mathbb{A}^2_S\backslash\{0\}$…

Voevodsky's base-change conjecture. The formation of $\mathbf{H}\mathbb{Q}$ should be stable under base change between any Noetherian semi-normal schemes.

Let $k$ be an algebraically closed field, and let … be the realization map from the motivic stable homotopy group of the sphere spectrum to the classical stable homotopy group. Ker…

Let $\mathrm{S}\mathscr{H}_\mathrm{idh}$ denote the idh-local version of the motivic stable homotopy category over a base scheme. Six-functor formalism conjecture. The category…

The idh-topology is defined over arbitrary bases, and perfect schemes in characteristic $p$ are related to absolutely weakly normal schemes by inverting universal homeomorphisms. E…