# Spooky geometry V

The complement of an algebraic set in $A^n(\Bbb{C})$ is path-connected.

# Splitting Q

Are there any decomposable quotients of $\Bbb{Q}$ (viewed as a $\Bbb{Z}$-module)?

# Tardily projective? You’re free then

Suppose $A$ is a $\mathbb Z$-graded connected algebra over a field $k$ concentrated in positive degrees, and suppose $P$ is a projective graded $A$-module concentrated in degrees above some $m\in \mathbb Z$ . Then $P$ is actually a free $A$-module.

Under the same hypothesis over $A$, show that if $M$ is a graded module and $M\otimes_A k=0$ where $k$ is the quotient of $A$ by the ideal of positively graded elements, then $M=0$.

Using the above, show that if $M$ is graded and $\mathrm{Tor}_A^1(M,k)=0$, then $M$ is free. To illustrate, it follows projective graded modules over polynomial algebras are free.

Let $f,g:[a,b]\to\Bbb{R}$ be continuous functions. We call $\int_a^b x^kf(x)\,\mathrm{d}x$ the $k$-th moment of $f$. Prove that if $f$ and $g$ have identical $k$-th moments for all $k\geq 0$, then $f=g$.

# Liouville is off the grid

Show that any function $f:\mathbb{Z}\times\mathbb{Z}\to (0,1)$ that verifies

$f(m,n)=\dfrac{f(m-1,n)+f(m+1,n) + f(m,n-1)+f(m,n+1)}{4}$

must be constant.

# Wish I had known this in 2012

Let $A$ be an integral domain, $I,J\subseteq A$ ideals such that $I+J=A$. If $I\cap J$ is principal, then $I$ and $J$ are projective.

# On primitive roots of unity

Finite subgroups of the group of units of a field are cyclic.