24 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…

Yagita's conjecture. The restriction map $\Omega^*(E/P)\to\Omega^*(G/P)$ is injective. This conjecture is an algebraic-cobordism form of a question originally stated for Brown--Pet…

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 $k$ be a field, let $\Omega^{\ast}(k)$ denote algebraic cobordism of $k$, and let $\mathbb{L}^{\ast}$ denote the Lazard ring. In characteristic zero, there is an isomorphism…

Donaldson–Thomas degree-zero partition-function conjecture.

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 $\ell^{\nu}\geq 4$,…

Eventual constancy conjecture. The inverse system used to define $\Omega_{m,r}(X)$ is eventually constant for all $r$, not just for $r=0$.

Let $k$ be a field and let $X$ be a smooth scheme over $k$. Let ${\rm MGL}$ be the algebraic cobordism spectrum, let ${\rm H}_{\mathbb Z}$ be the motivic Eilenberg–Mac Lane spectru…

Let $k$ be the base field and let ${\rm MGL}$ denote the algebraic cobordism spectrum. Write … The complex cobordism ring is denoted by ${\rm MU}^{*}$. Algebraic cobordism subring…

Let $F$ be a field, and let $\mathrm{SH}_F$ denote the motivic stable homotopy category over $F$. Let $\mathrm{MGL}$ be the algebraic cobordism spectrum, let $a_i$ be generators of…

Let $S\in\mathrm{Rg}$ be a base ring, let $\mathrm{MS}_{\mathrm{Nis}}(S)$ denote the Nisnevich motivic stable category over $S$, and let $\mathbf{MGL}$ be the algebraic cobordism s…

Let $R$ be a regular local ring, and let $\mathrm{MGL}$ denote the motivic Thom spectrum. Consider the $\mathbb{P}^1$-diagonal of the coefficient ring of $\mathrm{MGL}$ over $R$. V…

Let $K$ be a compact connected Lie group, $T$ its maximal torus, and $\pi\colon K\rightarrow K/T$ the natural projection. Denote by $G=K_{\mathbb C}$ the corresponding split reduct…

Let $A$ be a local Noetherian ring. Write $d53f^*$ for the Lazard ring with cohomological grading. Lazard-ring computation conjecture. The natural map … is an isomorphism. This con…

Let $k$ be a field, let $\\T$ denote the Tate sphere, and let $\MGL$ be the algebraic cobordism spectrum. Write \Omega^\infty_\\T \MGL for its infinite loop space and let its mot…

Let $k$ be a field, let $\bMGL$ denote the motivic algebraic cobordism spectrum, let $\bT$ be the motivic suspension coordinate, and let $\bDM(k)$ be the category of motives over…

Let $X$ be a smooth variety of dimension $d$. A free ${\mathbb{L}}$-resolution of $\Omega^*(X)$ is one whose $j$-th term has generators concentrated in codimensions between $j$ and…

Voevodsky's slice conjecture. The map $\psi_S$ is an isomorphism. The conjecture was proved over fields of characteristic zero, while over a field of characteristic $p$ the map is…

Let $G$ be an algebraic group. Write $\Omega^*(G)$ for the $BP$-version of algebraic cobordism, and let … be the realization map. Realization conjecture. The map $t_{\mathbb C}$ is…

Syzygy conjecture. In particular, the cohomological dimension of the ${\Bbb L}$-module $\Omega^*(X)$ is at most $d$.

Generalized double point relation conjecture. The generalized double point relation is stronger than the double point relation.

Let $G$ be a reductive group, let $F_G$ be its flag variety, and let $\mathbb{L}$ denote the Lazard ring. A Borel presentation is a presentation of the algebraic cobordism ring as…

Rost's generalized degree formula conjecture. The Rost degree formula should follow from a generalized degree formula for some universal cohomology theory.

Levine's comparison conjecture. The natural transformation $\vartheta_{{\operatorname{MGL}}'}$ is an isomorphism.