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 an integral domain, ideals such that . If is principal, then and are projective.
If is a projective -module, there exists a free -module such that .
Consider the group ring . How many module structures, up to isomorphism, does admit? Do the same for . With this information, calculate in such cases.
Let be a polynomial ring over a commutative ring (!) in variables. Show that for any -module and , there are natural isomorphisms
Consider the canonical embedding of a sphere of radius , say inside as the set of pairs with , and the curve . Calculate the homology of the intersection for as many values of as possible where , with of your liking.