Noether環上の有限生成加群の有限表示
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有限生成加群の有限表示を解説しました. 加群の有限表示を理解すると，単因子論による行列のジョルダン標準形の理論が理解できるようになりますよ！
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対称群上のフーリエ解析のメモです.
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- 1. Noether環上の 有限生成加群の有限表示 2017 March 23 Thursday M1 Tamura Takumi Age 23
- 3. 𝑅:Noether環 𝑀: 𝑅上の有限生成加群 𝑀 = 𝑖=1 𝑛 𝑅𝑚𝑖 , 𝑚1, 𝑚2, … , 𝑚 𝑛 : 𝑀の生成系 𝜑: 𝑅 𝑛 → 𝑀 (全射) ∈ ∈ 𝕖𝑖 ↦ 𝑚𝑖 𝜑 𝕖𝑖 = 𝑚𝑖, 𝕖1 = 1 0 ⋮ 0 , 𝕖2 = 0 1 ⋮ 0 , … , 𝕖 𝑛 = 0 0 ⋮ 1 𝑛 𝕖𝑖が𝑛次元列ベクトルである事に注意.
- 4. 全射準同型写像𝜑の核ker 𝜑 に注目する. 𝑅 𝑛 ⊃ ker 𝜑 = 𝑟1 ⋮ 𝑟𝑛 ∈ 𝑅 𝑛 𝑖=1 𝑛 𝑟𝑖 𝑚𝑖 = 𝑖=1 𝑚 𝑅 𝑎1𝑖 𝑎2𝑖 ⋮ 𝑎 𝑛𝑖 ∵ 𝑅 𝑛 の部分加群ker 𝜑 も有限生成に なることが知られているから． 𝑟𝑖 ∈ 𝑅
- 5. 𝜓: 𝑅 𝑚 → ker 𝜑 = 𝑖=1 𝑚 𝑅 𝑎1𝑖 ⋮ 𝑎 𝑛𝑖 (全射) ∈ ∈ 𝕖′𝑖 ↦ 𝑎1𝑖 ⋮ 𝑎 𝑛𝑖 𝜓 𝕖′ 𝑖 = 𝕧𝑖 = 𝑎1𝑖 ⋮ 𝑎 𝑛𝑖 𝕖′1 = 1 0 ⋮ 0 , 𝕖′2 = 0 1 ⋮ 0 , … , 𝕖′ 𝑚 = 0 0 ⋮ 1 𝑚 𝕖′𝑖は𝑚次元列ベクトル! 𝜓の表現行列は𝑛 × 𝑚行列 𝔸 = 𝕧1 … 𝕧 𝑚 となる． 𝕧𝑖は ker 𝜑 の生成元 また，明らかにI𝑚 𝜓 = ker 𝜑
- 6. ゆえに，以下の完全列(有限表示)が得られる. 𝑅 𝑚 → 𝜓 𝑅 𝑛 → 𝜑 𝑀 → 0 I𝑚 𝜓 = ker 𝜑 ∴ 𝑀 ≅ 𝑅 𝑛 /Im 𝜓