Question

10 . I tidl M M '. (7.3. Assume that the complex - O- L-M M ' N 0 is exact. Show that, up to natural identifications, L=kery and N = coker 7.4 Construct short exact sequences of Z-modules 0- ZƏN ZƏN —+2– 0 and 0- ZeN — ZON — ZeN - 0. (Hint: David Hilbert's Grand Hotel.) 7.5. Assume that the complex . L M N is exact and that L and N are Noetherian. Prove that M is Noetherian. 197.1 111 -

7.3 7.5

Answer 1

(7.3 ... is. Since the complex ... o M l H' N o is exact thayfone L sinjective and MiN surgutive exactress, Ker(49) - ingl →M) a mL (:L M is injective) As s' is surgutive therefore cro M' ~N ker(w' N) Ing(4) .. . ~ Cokery (7.5) ... & M B N is exact and Lenare Noethean Loin: Mis Noethenian let (Minne, be an ascending cham of submodules of it, then 6 (M) is a chain in a and (Blower, is a chain in N . For largen as and so are Noethenion therefore (a (Main, and (Bronin ax stationary. clains (selon is stationary. let xe Mote then Bla) E B (Mrs) and let's suppose that B (M) = B(Mnr.) we have B(x) E B (mm) so there is ye Mr so that B(x) = Bly) = x-y Eken (B) =ing (2) = y = x (2) for somezel Since x-yeMnt, we have REX (nt.) = x (MM) X-y= 2(2) EM XEMn (M) is stationary Mis Noetherion.