Math and stuff

Proposition Show that $\Spec(A)$ is irreducible if and only if the nilradical of $A$ is a prime ideal. Solution Let $\mathfrak{R}$ denote the nilradical of $A$. Let $a, b \in A$. Suppose that $ab \in \mathfrak{R}$. We will show that $a \in \mathfrak{R}$ or $b \in \mathfrak{R}$. Case 1: $V(\{...

Proposition Find a Mobius transformation that maps the unit disk to $\{ x + iy \in \mathbb{C} : x + y > 0 \}$. Solution First, $f(z) = 1 / (z - 1)$ maps the unit circle to the line $\Re z = -1/2$ because \[\begin{align*} w = \frac{1}{z -...

Proposition Show that if $f = u + iv$ is holomorphic, then the Jacobian equals $\abs{f’(z)}^2$. Solution $u_xv_y - u_yv_x = u_x^2 + u_y^2 = \abs{u_x + iv_y}^2$.

Proposition Find the fixed points in $\hat{\mathbb{C}}$ of $f(z) = \frac{z^2 - 1}{2z + 1}$. Solution $z = \infty$ is clearly a fixed point. Other than that, by solving $f(z) = z$, we obtain $z = (-1 + \sqrt{3}i) / 2$.

Proposition If $\omega \ne 0$, show that there is a chain $c$ such that $\int_{c} \omega \ne 0$. Use this fact, Stokes’ theorem and $\partial^2 = 0$ to prove $d^2 = 0$. Solution Let $\omega$ be a $k$-form on an open set $A \subset \mathbb{R}^n$. Then there exist $\omega_{i_1, \cdots,...