Modern Algebra II, Spring 2014

• Northcott's chapter on Koszul complex is available here.

• (Due 2/10) VI: 1(cefgkl), 2(dfg), 3(def), 4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, Properties 1-5 on p.280.
• (Due 3/17) XVI: 6, 7, 9. XX: 1, 2, 3, 5, 19, 21, 22, 26-29 and the following:
  (1) Give details of proof of Theorem 2.1.
  (2) Show enough injectives implies existence of injective resolutions.
  (3) Show the category of bounded-below complexes of A modules have enough injectives.
  (4) Show the category of SES's of A modules have enough injectives.
  (Bonus 1) Prove (the isomorphism in) Theorem 6.2 via spectral sequence technique.
  (Bonus 2) No. 20.
• (Due 4/23) XXI: 1, 2(a,b,c,d), 3, and the following 9 problems.
  (1) Show the 2 sequences given in class have the same cohomologies but are not quasi-isomorphic.
  (2) Show that a similar property holds for the sufficient family of injectives in Proposition 1.1' (explained in class).
  (3) Show the short exact sequence for the mapping cone (explained in class) holds.
  (4) Prove Proposition 4.3.
  (5) Give details to proof of Theorem 4.10.
  (6) Give details to proof of Theorem 4.15.
  (7) Let A be a local ring, M and N finite A-modules. Prove that if $M \otimes N =0$, then either M=0 or N=0. (Hint: Use Nakayama's lemma.)
  (8) Let A be a commutative ring and I an ideal. Show that $$(A/I) \otimes_A M \cong M/IM.$$
  (9) Let A be a regular local ring of dimension n with residue field k. Show that $$ Tor^A_p (k,k) \cong Ext_A^p (k,k) \cong \wedge^p k^n.$$