T6=q[LpsKN
中国科学技术大学 8>y!=+9_
2016年秋季博士资格考试试卷 J SOgq/ \
P-c<[DSM'I
%juR6zB%8
代数学 07: N)y,
}3&~YBx;:
1." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">1.1.(40分) 考虑形式幂级数环 C[[x]]={a0+a1x+a2x2+⋯∣ai∈C}" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]={a0+a1x+a2x2+⋯∣ai∈C}C[[x]]={a0+a1x+a2x2+⋯∣ai∈C} 考虑 2" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">22 阶全矩阵环 R=M2(C[[x]])" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">R=M2(C[[x]])R=M2(C[[x]])． :
?.RZKXQF
(1) 证明 C[[x]]" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]C[[x]] 为 Noether 整环； v1Wz#oP
(2) 描述 C[[x]]" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]C[[x]] 全部的有限生成不可分解模，并给出论证； nC> 'kgRt
(3) 给出环 R" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RR 全部的双边理想，并给出论证； &lp5W)D
(4) 描述 R" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RR 上全部的有限生成不可分解左模，以及这些模的自同态环． %Z#s9QC
"\l O1D
2." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">2.2.(40分) 将 Abel 群与 Z" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">ZZ-模等同起来，考虑 Abel 群 G=Z3⊕Z" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">G=Z3⊕ZG=Z3⊕Z． %Qc5 _ of
(1) 列出群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的全部子群，并给出论证； ao(lj
(2) 将 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的每个商群都分解成不可分解群的直和，并给出论证； D ^~G(m;-
(3) 列出群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的全部直和项，并给出论证； [c=T)]E1
(4) 描述 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的自同构群． ^
KeJ=VT
回顾：Abel 群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的子群 A" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">AA 称为直和项，若存在另一子群 B" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">BB 满足 G=A+B" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">G=A+BG=A+B 以及 A∩B={0}" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">A∩B={0}A∩B={0}． zu@5
,AH
w6
x{ <d
3." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">3.3.(20分) 具体给出代数同构 X>`5YdT~+
CS3→∼C×C×M2(C)，" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; text-align: left; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">CS3−→～C×C×M2(C)，CS3→～C×C×M2(C)， 其中 CS3" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">CS3CS3 为 S3" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">S3S3 的群代数；并给出相应的论证． :jc
?T
提示：利用不可约复表示． ,GH;jw)P
|
