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 $k$ be an imaginary quadratic field, let $F/k$ be a finite abelian extension with Galois group $G$, let $r<0$, and let $onzerocharchiinwidehat{G}$ 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…

Let $F$ be a field of characteristic zero and $G$ a group. Let $\operatorname{class}_F(G)_f$ denote the indicated $F$-vector space of finite conjugacy classes, and let…

Let $R$ be a commutative integral domain and $G$ a group. For $g\in G$, write $\operatorname{HS}_{RG}(P)(g)$ for the value at the conjugacy class $(g)$ of the Hattori–Stallings ran…

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…

Let $F$ be an algebraically closed field. Here $\mathfrak{p}_n(F)$ denotes the primitive part of the homology group associated with $\operatorname{GL}_n(F)$, and $l$ is a prime. Th…

Rognes' localization conjecture. These maps should fit into a cofiber sequence in the stable category

Exactness conjecture. This sequence is exact.

Let $R$ be the ring of integers in an unramified extension $K$ of $\mathbb Q_p$, and let $\pi_R\colon X_R\to C_R$ be an elliptic surface over $R$ satisfying condition (Rat). Let…

Let $K$ be the number field, ${\cal P}$ a prime ideal in its ring of integers, and let $\gamma {\Bbb B}_d$ be a polyhedron in the Bianchi decomposition of the orbifold…

Let $\pi$ be a group, and let $A^L_\pi$ and $A^K_\pi$ denote the assembly maps in $L$-theory and $K$-theory, respectively. Their rationalizations are obtained by tensoring with the…

Let $p$ be an odd prime, let $\mathcal C_2$ be the thick subcategory of finite $p$-local spectra of type at least $2$, and let $V(1)$ denote the Smith–Toda complex. Adams's conject…

Let $G$ be a group and let $F$ be a field of characteristic zero. Define … where the colimit uses induction along inclusions of finite subgroups. Covariant representation-ring conj…

Let $X$ be a Noetherian scheme of dimension $d$. For a contravariant functor $F$ on schemes, call $X$ $F$-regular if, for every $r\geq 0$, the map $F(X)\to F(X\times\mathbb{A}^r)$…

Let $R$ be a commutative ring and let $P$ be a Col-divisible polytope of arbitrary dimension. Write $K_i(R,P)$ for its associated $K$-groups, and let ${\mathfrak c}(P)\leq\dim P$ b…

Let $G$ be a group, and let $\overline{{\mathcal H}}$ be the class of possibly infinite hyperelementary groups. Say that a group satisfies induction for a class of groups when the…

Let $K$ be an algebraically closed field of characteristic zero, and let $A$ and $B$ be $K$-algebras essentially of finite type over $K$. Assume that $B$ is a finite-dimensional pr…