HW due Friday Mar 2

1. In class we determined the Spec of C[x], using the fact that complex polynomials of degree > 1 all factor. It had a nice description as C union a weird extra point. (Don't confuse that point with the very different point at infinity in the Riemann sphere!)
Determine the Spec of R[x] (here R is the reals), and describe the map Spec C[x]-> Spec R[x].

2. We showed that a finitely generated torsion free module over an ED is free.
Show that the ring C[z,z^-1] of Laurent polynomials, considered as a module over the subring C[z], is torsion free but not free (i.e. it doesn't have a basis).

3. Let R = F[[z]], power series in one variable, with coefficients in some field F.

a. Show that r in R is invertible iff r not in < z >.

b. What are the elements of Spec R?

c. If F = C, so that we understand Spec C[z], describe the map Spec C[[z]] -> Spec C[z] induced by the inclusion C[z] -> C[[z]].

4. Let f:R->S be a homomorphism and r in R. Show that S @_R (R/< r >) is isomorphic to S/< f(r) > as an S-module.

5. Let V be a finite-dimensional complex vector space, and T:V->V an endomorphism. Consider it as a C[z]-module in the usual way.

a. Let C[z]->C take z |-> lambda, making C a C[z]-algebra.
Show that C @_C[z] V is finite-dimensional over C, and that its dimension is that of the lambda eigenspace of T.

b. Show that C[z,z^-1] @_C[z] V is finite-dimensional over C, and that its dimension is the number of nonzero eigenvalues (with multiplicity) of T.

c. Show that C[[z]] @_C[z] V is finite-dimensional over C, and that its dimension is the lowest degree occurring in T's characteristic polynomial.