468 problems

Let $F$ be a field, and let $K^{(q)}_{2q}(F)$ and $\operatorname{CH}^q(\operatorname{Spec}F,2q)$ have the meanings used in the Beilinson–Soulé vanishing conjecture. Beilinson–Soulé…

Let $X$ be a smooth projective variety over a finite field, let $i,n$ be integers, and let $\ell$ be a prime invertible in the finite field. Parshin's conjecture. The motivic cohom…

Let $\mathbb{F}_q$ be a finite field and let $n\geq 1$. Write $\zeta(\mathbb{F}_q,s)$ for the zeta function of $\mathbb{F}_q$, and let $\mathrm{K}_i(\mathbb{F}_q)$ denote its algeb…

Ausoni–Rognes chromatic redshift conjecture. The algebraic $\mathrm{K}$-theory of an $\mathbb E_1$-ring spectrum of fp type $n$ is fp of type $n+1$.

Let $F$ be a field, and let $X$ be an affine finite-type $F$-scheme of dimension $d$. A scheme is regular if all of its local rings are regular, and it is $K_m$-regular when the ca…

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…

Morel's conjecture.

Let $R$ be a Euclidean or local ring, and let $\operatorname{CB}(R^n)$ be the simplicial complex whose vertices are the proper nonzero direct summands of $R^n$, with simplices give…

Let $X$ be a scheme of finite type over $\operatorname{Spec}\mathbf Z$, and let $n\geq 0$. The $K'$-theory (also called $G$-theory) group $K'_n(X)$ is the abelian group associated…

Let $X$ be a noetherian scheme of dimension at most $d$, and let $\text{K}_{-d}(X)$ and $\text{KH}_{-d}(X)$ denote algebraic and homotopy algebraic $K$-theory in degree $-d$. Weibe…

Rosenberg's conjecture. 1. For $n>0$, the map

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramifie…

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…

For a group $G$, let $\mathbb{Z}G$ be its integral group ring, let $K_{0}(\mathbb{Z}G)$ be its Grothendieck group of finitely generated projective $\mathbb{Z}G$-modules, and let ……

Antieau–Gepner–Heller's theorem of the heart conjecture. The map

Let $\mathcal A$ be an abelian category. Its negative $K$-theory is defined using an enhancement, with $K$-theory understood as Waldhausen $K$-theory of that enhancement. Schlichti…

Soient $F$ un corps de nombres et $n\geq 2$. Notons $d_n=r_1+r_2$ si $n$ est impair et $d_n=r_2$ si $n$ est pair, et soit…

Let $\ell$ be a prime number, let $K$ be a number field with $\zeta_\ell\in K$, and let $S$ be a finite set of places containing the infinite places and the places over $\ell$. The…

Let $\mathbb D$ be a triangulated derivator, let $\mathbb E$ be a Verdier quotient of $\mathbb D$, and let $K$ denote derivator K-theory. A Verdier quotient is a triangulated local…

Let $\mathcal W$ be a saturated Waldhausen category satisfying the cylinder axiom, and let $\mathbb D_\mathcal W$ be the associated left pointed derivator. The comparison map … is…

Hsiang's conjecture.

Let $R$ be a commutative Noetherian ring of Krull dimension $d$, and let $t_1,\ldots,t_n$ be indeterminates with $n\geq 0$. Weibel's K-dimension conjecture. … and … The conjecture…

Isomorphism Conjecture. For the chosen $K$- or $L$-theory, the assembly map $A_{X_G,G/H}$ should be a weak equivalence of spectra.

Let $A$ be a real C-algebra, let $K$ denote nonconnective algebraic $K$-theory, and let $\mathscr{C}([0,1];A)$ be the ring of continuous functions from $[0,1]$ to $A$. Rosenberg's…

Gersten's conjecture. The group $H^n(X)$ should be computable from the groups