der Universitat Munchen Set 2
Prof. Dr. B. Pareigis
Problem set for
Quantum Groups and Noncommutative Geometry
(5) LetX denotethe planecurvey=x 2
. ThenX isisomorphictothe aÆneline.
(6)
LetK beanalgebraicallyclosedeld. Letpbeanirreduciblesquarepolyno-
mialinK[ x;y]. LetZ bethe conicsectiondened bypwiththeaÆnealgebra
K[x ;y]=(p). Show that Z is naturally isomorphic either to X or to U from
problems (3)resp. (5).
(7) LetX bean aÆnescheme with aÆnealgebra
A=K[x
1
;:::;x
n ]=(p
1
;:::;p
m ):
Dene \coordinatefunctions"q
i
:X(B) !B which describethe coordinates
of B-points and identify these coordinatefunctions withelementsof A.
(8) Let S
3
be the symmetric group and A := K[S
3
] be the group algebra on S
3 .
DescribethepointsofX(B)=K-Alg(A;B)asasubspace ofA 2
(B). Whatis
the commutative part X
c
(B) of X and what is the aÆne algebraof X
c
?