The complement of an algebraic set in is path-connected.
Are there any decomposable quotients of (viewed as a -module)?
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.
Let be continuous functions. We call the -th moment of . Prove that if and have identical -th moments for all , then .
Show that any function that verifies
must be constant.
Let be an integral domain, ideals such that . If is principal, then and are projective.
Finite subgroups of the group of units of a field are cyclic.