der Universitat Munchen Set 8
Prof. Dr. B. Pareigis
Problem set for
Advanced Algebra
(29) Let F : C ! Set be a covariant representable functor and let
f :M !N beamonomorphisminC. ShowthatF(f):F(M)
!F(N) isa monomorphism, too.
(30) (a) Let U : Ab ! Set be the underlying functor. Show that
U is arepresentable functor.
(b) Show that F : Q-Vec 3 V 7! U(Q n
Q
V) 2 Set is a
representable functor.
(31) Show that the functor F :Ab3M 7!U(Z=(n)
Z
M)2Set is
not representable (for n2).
(32) For a K-algebra A let A
denote the multiplicative group of
elementsthat are invertibleunder the multiplicationof A. Let
U :Gr !Setbetheunderlyingfunctor. Showthatthefunctor
F :K -Alg3A7!U(A
)2Set isa representable functor.
![0 K ( t,q ,q ) ∝ e . ′ im ( q − q ) / 2¯ ht 0 t 2 t cl cl 0 cl q ( s )=1 { sq +( t − s ) q } = ⇒ S [ q ]= dsL [ q ( s )]= ( q − q ) . ′ ′ 2 m t Z 0 q t q ′ mp 2˙ 0 0 H =12 L = q 2 2 m h ¯ cl q q q t ′ K ( t,q ,p ) ∝ e , ′ iS [ q ] / ¯ h K ( t,q ,q )= h q](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)