Blog

Taken from: (?). Exercise Let $I$ be an ideal on $R$ and $M$ an $R$-module. Prove that \[M\otimes_R \frac{R}{I}\cong \frac{M}{IM}.\] In particular: $M\otimes_R R\cong R\otimes_R M\cong M$. Proof Define the function \[\begin{align*} \varphi: M \times \frac{R}{I} &\to \frac{M}{IM}\\[3mm] (m, \overline{r}) &\mapsto \overline{...

Taken from: [1, Exercise 3.63]. Exercise 3.63 Let $R$ be a commutative ring which contains an infinite field $K$ as a subring. Let $I$ and $J_1, \ldots, J_n$, where $n \geq 2$, be ideals of $R$ such that \[I \subseteq \bigcup_{i=1}^n J_i.\] Prove that $I \subseteq J_j$ for some $j$ with $1 \leq j \leq n$. Proof It is easy to verify that $ ...

Taken from: [1, Exercise 12.26]. Exercise 12.26 Let $\Gamma$ and $\Delta$ be simplicial complexes on disjoint vertex sets $V$ and $W$, respectively. Define the join $\Gamma * \Delta$ to be the simplicial complex on the vertex set $V \cup W$ with faces $F \cup G$, where $F \in \Gamma$ and $G \in \Delta$. (If $\Gamma$ consists of a single point,...