The complement of an algebraic set in is path-connected.

# Month: October 2016

# Splitting Q

Are there any decomposable quotients of (viewed as a -module)?

# Tardily projective? You’re free then

Suppose is a -graded connected algebra over a field concentrated in positive degrees, and suppose is a projective graded -module concentrated in degrees above some . Then is actually a free -module.

Under the same hypothesis over , show that if is a graded module and where is the quotient of by the ideal of positively graded elements, then .

Using the above, show that if is graded and , then is free. To illustrate, it follows projective graded modules over polynomial algebras are free.

# One moment please

Let be continuous functions. We call theĀ *-th moment* of . Prove that if and have identical -th moments for all , then .

# Liouville is off the grid

Show that any function that verifies

must be constant.

# Wish I had known this in 2012

Let be an integral domain, ideals such that . If is principal, then and are projective.

# On primitive roots of unity

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