課程名稱︰代數二
課程性質︰數學系選修，可抵必修代數導論二
課程教師︰林惠雯
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/6/19
考試時限（分鐘）：180分鐘
試題 :
1.(50%)
(a) Let A be the ring of integers in Q(√(d)). Show that A is a Euclidean
d 5
domain but A is not.
－5
(b) Show that if M is a finitely generated R-module (R：commutative with 1)
and IM = M (I：an ideal of R contained in the Jacobson radical of R),
then M = 0.
(c) Let R be a local ring and M be a finitely generated projective R-module.
Show that M is free.
(d) State and show the Going-up theorem.
(e) Let R be a Dedekind domain and I≠0 an ideal in R. Show that every ideal
in R/I is principal. Deduce that every ideal in R can be generated by at
most two elements. (You can use the properties of Dedekind domains
directly.)
2.(20%) Justify your answers.
n
(a) Let N be a Z/12Z-module. Compute Ext (Z/3Z, N), for all n≧0.
Z/12Z
(b) Compute Tor (Z/12Z, Z/15Z).
1
3.(25%) Let x ,..., x be a regular sequence in a commutative ring R (with 1)
1 n
and I be the ideal of R generated by x ,..., x .
1 n
(a) Construct a free resolution of R/I. (Justify your answer.)
(b) Show that R/I does not have any projective resolution of length shorter
than n.
n ～
(c) Let M be an R-module. Show that Ext (R/I, M) ＝ M/IM.
4.(25%) Let A, B be abelian categories, F：A → B be an additive left exact
functor and R be the class in A adapted to F.
(a) Show that the class S of quasi-isomorphisms in A is localizing and so
A
is S .
B
＋ －1 ＋
(b) Show that a triangle in K (R)[S ] which is distinguished in D (A) is
R
＋ －1
also distinguished in K (R)[S ]. (Hint： don't forget a map in
R
＋ －1
K (R)[S ] is represented by a left roof.)
R
＋
(c) Show that the derived functor RF defined in D (A) is exact.