189-570A: Higher Algebra Assignment 2

Due: Monday, October 23

1. Let R = Z[\sqrt{−5}], and let M be the submodule of R generated by 3 and (1 + \sqrt{−5}). Show that M is not free over R, but is projective.

2. Let k be a field of characteristic ≠ 2, and let k[[x]] denote the ring of formal power series with coefficients in k. Show that k[[x]] is a complete local ring. Show that the polynomial (1 + x) has a square root in k[[x]].

3. Give an explicit formula for the square root of (1+x) whose existence you proved in the previous exercise. What happens if 2 = 0 in k?

4. Let \underline{C_{1}} denote the category of Z-modules and \underline{C_{2}} the category of F_{p}-vector spaces, where F_{p} denotes the finite field with p elements. Consider the rule F which to every object M of \underline{C}_{1} associates the object F (M) := M/pM of \underline{C}_{2}.

Show that F defines a covariant functor in a natural way.

5. Show that the functor F defined in the previous problem is right exact, i.e., that if

0⟶M′⟶M⟶M′′⟶0

is a short exact sequence in \underline{C}_{1}, then the induced sequence

F (M′)⟶F (M)⟶F M′′)⟶0

is also exact (in the category of F_{p}-vector spaces). Show that this functor is not left exact.

6. Let G be a group of p-power order, and let Z_{p} be the ring of p-adic integers. Show that the ring Z_{p}[G] is a complete local ring.

7. Let k be a (commutative) field, and let R be a k-algebra which is finitedimensional over k. Show that R is semisimple if and only if R contains no non-zero nilpotent elements.

8. Let D be the dihedral group of order 8, and let Q be the quaternion group of order 8, generated multiplicatively by the elements i, j, k in the ring of Hamilton quaternions. Compute the character tables for D and Q. What do you observe?

9. Keeping the notations of the previous exercise, show that the group rings C[D] and C[Q] are isomorphic, but that the group rings R[D] and R[Q] are not isomorphic. (Here, C and R denote the complex and real numbers respectively.)

10. Let G = GL_{3}(F_{2}) be the simple group of order 168 which you encountered in the previous assignment. Write down its character table. (The solution is worked out in a number of textbooks, but obviously you will learn more from this problem if you refrain from peeking, and try to work it out on your own!)