MathOverflow

Let $k$ be a number field. Is the Galois cohomology group $H^3(k,\mathbb{Z})$ trivial?
Number fields often behave like fields of cohomological dimension $2$. For example $H^3(k,\mathbb{G}_m)$ is trivial. However depending on the real properties of $k$, it can be that $H^3(k,\mathbb{Z}/2\mathbb{Z})$ is non-trivial....

Define the closed immersion $\mathbb{A}^n_k \hookrightarrow \mathbb{A}^{n+1}_k$ by $T_{n+1} \mapsto 0$ and $T_i \mapsto T_i$ for $1 \leq i \leq n$. I think that the sequence$$\mathbb{A}^1_k \hookrightarrow \mathbb{A}^2_k \hookrightarrow \mathbb{A}^3...

Assume that $R\to U$ is a ring epimorphism such that $U-\operatorname{Mod}$ is a (full) exact subcategory of $R-\operatorname{Mod}$. I am interested in the subcategory $T$ of $D(R)$ that consists of those complexes such that their cohomology $R$-modules...

Consider a commutative ring $R$ and its Picard group $\mathrm{Pic}(R)$. For two invertible $R$-modules $M,N$, we may define $[M]\le [N]$ to mean that there exists an injective $R$-linear map from $M$ to $N$, which makes $\mathrm{Pic}(R)$ a (pre)ordered...