Textbook Sections

Problems

due

Part III, Amodules.
Section 15: Path algebras 
15.7/ 1, 3 
Friday, 1/20 
Section 15: Path algebras 
15.7/ 2, 4, 5 
Friday, 1/27 
Projective representations 
Suppose Q is a quiver with no oriented cycles.
Show that the representations P(i)
introduced in class are all projective,
and specify them for the example quiver shown. 
Friday, 2/3 
Projective covers 
For the example quiver in the previous problem,
specify the projective covers of the simple modules
S(i) and compute the kernels.
Are the kernels projective modules?
Make a conjecture about the kernels of the projective
covers of the simple modules over a path algebra
of a quiver with no oriented cycles!
 Friday, 2/17 
Section 25: Pushout and pullback 
Exercise on p.150 (Uniqueness of pushout), exercise on p.151 (Universal property of the pullback) 
Monday, 3/26 
Section 25: Induced short exact sequence 
Exercise 25.8/ 2.
 Friday, 4/ 6 
Section 27: Projective resolutions 
Proof of Lemma 27.1 on page 173
 Friday, 4/ 20 