Mathematisches Institut WS 2001/02 der Universität München Set 12 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(45) (a) Let A- Mod be equivalent to B- Mod. Show that Mod -A

and Mod -B are also equivalent.

(b) Let 𝕂 be an integral domain (commutative without zerodivisors). Show that 𝕂- Mod ≃ Mod -𝕂.

(46) Show that an equivalence of arbitrary categories preserves monomorphisms.

(47) Show that an equivalence of module categories preserves projective modules, but not free modules.

(48) Let D be a division algebra. Show that D- Mod ≃ Mod -D if and only if there is an isomorphism of algebras D ≅ D^{op}.

Due date: Tuesday, 22.1.2002, 16:15 in Lecture Hall 138