Math and stuff

Proposition Show that the set $\{ x \}$ is closed in $\Spec(A)$ if and only if $p_x$ is maximal. $\overline{\{ x \}} = V(p_x)$. $y \in \overline{\{ x \}} \iff p_x \subset p_y$. $X$ is a $T_0$-space. Solution 1 Suppose $x$ is maximal. Clearly, $x \in V(x)$. Let $y \in...

Proposition If $\omega \in \Lambda^n(V)$ is a volume element, define a “cross product” $v_1 \times \cdots \times v_{n - 1}$ in terms of $\omega$. Solution A cross product is defined in the textbook on P.83 and P.84. However, the textbook only defines a cross product for $\mathbb{R}^n$. This problem asks...

Proposition Let $M$ be a Noetherian $A$-module and let $a$ be the annihilator of $M$ in $A$. Prove that $A / a$ is a Noetherian ring. If we replace “Noetherian” by “Artinian” in this result, is it still true? Solution Since $M$ is Noetherian, it is finitely generated by Proposition...

Proposition if $M$ is a $k$-dimensional manifold in $\mathbb{R}^n$ and $k < n$, show that $M$ has measure 0. If $M$ is a closed $n$-dimensional manifold-with-boundary in $\mathbb{R}^n$, show that the boundary of $M$ is $\partial M$. Give a counterexample if $M$ is not closed. If $M$ is a compact...

Proposition Let $M$ be an $A$-module and let $N_1, N_2$ be submodules of $M$. If $M / N_1$ and $M / N_2$ are Noetherian, so is $M / (N_1 \cap N_2)$. Similarly with Artinian in place of Noetherian. Solution We have two exact sequences induced by the canonical inclusion and...