242 problems

Let $\mathcal{H}$ be a complex Hilbert space and let $U,V\in\mathcal{B}(\mathcal{H})$ be invertible. If $(U-\mathbf{1})(V-\mathbf{1})$ and $(V-\mathbf{1})(U-\mathbf{1})$ belong to…

For every noetherian scheme $X$ of finite Krull dimension $d$, one has $K_i(X)=0$ for every integer $i<-d$.

Let $F$ be a global field and let $l\ge 2$. If $l=2$, assume that $\mu_4\subset F$. For $n>1$ and $l\ne\operatorname{char} F$, let $K_n^{\mathrm{et}}$ denote étale $K$-theory. Quil…

Let $G$ be a torsion-free group and let $R$ be a regular ring. Farrell–Jones conjecture for $K_0$. The map induced by the inclusion $R\to RG$ is an isomorphism … This is described…

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 $k$ be a finite field of characteristic $p>0$, and let $X$ be a smooth projective variety over $k$. For each integer $n\geq 1$, let $K_n(X)$ denote the algebraic $K$-group of…

Vorst's conjecture. If $R$ is $K_{d+1}$-regular, then $R$ is regular.

Let $G$ be a group, let $r \colon K_0\mathbb{Z}G\rightarrow HH_0(\mathbb{Z}G)$ be the Hattori–Stallings trace map, and write the image of a finitely generated projective module $P$…

Isomorphism Conjecture in algebraic $K$-theory. The map from the colimit over finite subgroups

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 $X_{{\mathbb Q}}$ be a smooth projective variety over ${{\mathbb Q}}$, and let $X_{{\mathbb R}}:=X_{{\mathbb Q}}\times_{{\mathbb Q}}{{\mathbb R}}$. Let…

A-theoretic Farrell--Jones conjecture. The assembly map

Suslin's conjecture. The image of Suslin's Hurewicz map coincides with

Rationalized Farrell-Jones conjecture. For every group $G$ and every $n\in\mathbb{Z}$, this rationalized Farrell-Jones assembly map is an isomorphism.

Let $X$ be a Noetherian scheme of dimension $d$, and write $K_m(X)$ for its algebraic $K$-groups. A scheme is $K_n$-regular when homotopy invariance holds in degree $n$, equivalent…

Let $F/K$ be an abelian CM extension of a totally real number field $K$, let ${\bf f}$ be the conductor of $F/K$, and let $\Theta_n({\bf b},{\bf f})$ be the higher Stickelberger el…

Let ${\cal Z}(N)$ be the algebra of multiple polylogarithm values at $N$-th roots of unity, filtered by weight, and let $UC_{\bullet}(N)^{\vee}$ denote the graded dual of the unive…

Let $k$ be an imaginary quadratic field, let $F/k$ be a finite abelian extension with Galois group $G$, let $r<0$, and let $onzerochar\chi\in\widehat{G}$ be a character. Write…

Let $k$ be a field and $R$ a coefficient ring. Write $\operatorname{DM}_{\operatorname{gm}}^{\operatorname{eff}}(k;R)$ for geometric effective motives and…

Quotient conjecture. This composition is an isomorphism.

Let $S$ be the open arithmetic scheme and let $A(n)$ denote the real Tate-twist object used in the source. Theorem 12.1(a) gives a comparison isomorphism for positive twists betwee…

Let $\mathcal B(\mathbb C)$ be the Bloch group, defined as the kernel of the complex Dehn invariant, and let … be the Bloch regulator map. Bloch regulator injectivity conjecture. T…

Let $F$ be a subfield of $\mathbb{C}$ with embedding $\sigma:F\hookrightarrow\mathbb{C}$. Let $\mathcal B(F)$ and $\mathcal B(\mathbb{C})$ denote the corresponding Bloch groups, an…

For an integer $N\geq 3$, let $D_2$ denote the Bloch–Wigner dilogarithm, and consider the real numbers $D_2(e^{2\pi\sqrt{-1}j/N})$ indexed by integers $j$ relatively prime to $N$ w…

Let $G$ be a group, let $\operatorname{Wh}(G)$ be its Whitehead group, and let $A\in GL_n(\mathbb ZG)$. Let $r_A^{(2)}$ be the $G$-equivariant bounded operator on $\ell^2(G)^n$ giv…