39 problems

Let $F$ be an infinite field and let … be the Suslin–Hurewicz map, whose image is the subgroup of Milnor $K$-theory obtained from Quillen $K$-theory. Suslin's conjecture. The image…

Let $i$ and $q$ be non-negative integers, let $k$ be a field, and let $C_i^q$ denote the Kato–Kuzumaki property defined in terms of Milnor $K$-theory of degree $q$. Kato–Kuzumaki's…

Let $R$ be a regular Noetherian local ring, let $K_j^M(R)$ denote Milnor $K$-theory, and write $\widehat K_j^M(R)$ for the improved Milnor $K$-group. For each codimension $i$, writ…

Let $k$ be a field of characteristic different from $2$. The fundamental ideal $\mathbf{I}(k)$ is the ideal in the Witt group $\mathbf{W}(k)$ generated by even-dimensional inner pr…

Let $A$ be a henselian discrete valuation ring of characteristic zero with residue field of characteristic $p$ and function field $K$. Let $X$ be smooth and projective of relative…

Graph isomorphism conjecture. For every such $A$, the graph homomorphism is an isomorphism.

Let $k$ be a field of positive characteristic, and let $K_n^M(k)$ and $K_n(k)$ denote its Milnor and Quillen $K$-groups, respectively. Write…

Let $F$ be an infinite field, let $k$ be either $\mathbb{Q}$ or $\mathbb{Z}/l$ for a prime $l$, and assume $K_2^M(F)\otimes k=0$. Let $\mathfrak{t}_r^{(k)}(F)$ denote the homologic…

Milnor–Kato conjecture. This homomorphism from Milnor K-theory modulo $m$ to Galois cohomology with cyclotomic coefficients is an isomorphism for every $n$. This is the central com…

Let $K$ be a field, let $p$ be a prime distinct from the characteristic of $K$, assume that $K$ contains a primitive $p$th root of unity $\zeta$, and let $r\ge0$. For $w\in H^r(K)$…

Let $A$ be a local ring, let $i\geq 0$ be an integer, let $\widehat{\text{K}}{}^{\text{M}}_i(A)$ denote the improved Milnor $K$-group, and let…

Let $k$ be a flexible field and let $F/k$ be a finitely generated field extension. The isotropic Milnor K-ring of $F/k$, denoted $\operatorname{\mathrm{k}}^M_*(F/k)$, is the quotie…

Tate's conjecture. The comparison morphism is an isomorphism. Equivalently, for every smooth $X$ and every $i$, the induced homomorphism

Conjectural description of invariant function-field cohomology. There should be an isomorphism

Let $K$ be a field, let $W(K)$ be its Witt ring, and let $I(K)$ be the fundamental ideal. For each $n>0$, let $I^n(K)$ be the ideal generated by $n$-fold Pfister forms, and conside…

Let $q$ be a non-negative integer and let $K$ be a field. Say that $K$ has the $C^q_{\mathrm{sc}}$ property if, for every finite extension $L/K$ and every principal homogeneous spa…

Let $\sgf$ be a field, let $m$ be a positive integer, and let $n$ be a positive integer. For $a\in\sgf^\times$, write $(a)\in H^1(\sgf,\mu_m)$ for its Kummer class. Bloch–Kato conj…

Let $A$ be an excellent henselian discrete valuation ring with uniformising parameter $\pi$ and residue field $k$, let $X$ be a smooth projective scheme over…

Bloch–Quillen formula. The cycle class map $\operatorname{cyc}_X^M$ is an isomorphism. Together with the known surjectivity, this would identify the Levine–Weibel Chow group of zer…

Weak Lefschetz conjecture. This map is an isomorphism when $2p<\dim(L)$ and is injective modulo torsion when $2p=\dim(L)$. This is the Chow-group analogue of the classical Weak Lef…

Let $\mathbf{k}$ be a field of characteristic $0$. For a field extension $F/\mathbf{k}$, write … A dominant rational map induces pullback on Milnor $K$-groups. The logarithmic Miln…

Let $F$ be a field of characteristic different from $l$, set … and let $E$ be a finite extension of $F$. Denote by $K^{\mathrm M}_2(E)/l$ and $K^{\mathrm M}_2(KE)/l$ the degree-two…

Let $l$ be a prime and let $F$ satisfy the conditions under which the Milnor K-theory algebra modulo $l$ is a quotient of the exterior algebra on $F^*/F^{*l}$. Set … and let…

Let $F$ be a field, let $G_F=\operatorname{Gal}(\overline{F}/F)$ be its absolute Galois group, and let $l\ne\operatorname{char} F$ be prime. Write $\mu_l$ for the group of $l$th ro…

Let $k$ be a field. Write $K^M_i(k)$ for Milnor $K$-theory, let $\mathbb{S}_k$ be the motivic sphere spectrum, and let $\mathrm{KO}$ denote Hermitian $K$-theory. Let $\nu$ be the m…