極樂島

一個從小到大聽爸媽話認真讀書交大台大gpa 超過4 拿過一堆獎狀甚至拿獎學金出國去美國讀博士最後讀書輸沒車沒房沒錢沒妹。一堆獎狀成績分數考卷怎成空。只剩數學物理coding的愛好者。請君聽我唱: 聽說讀書無量苦千思萬算勞腸肚地水火風箏勝固和牢固到頭盡化為塵去; 聽說鬼島無量苦賺錢不比二代富 %妹難剩8+9 君須悟二十二k天天度; 聽說台男無量苦不婚不生不承諾沒車沒防沒未來嗟今古何人踏著無生路~~~~ 來美國學到最多東西就是知足，其他東西跟這東西相比，真的等於0，以後我如果有機會回到寶島生活打零工，沒車沒房，不婚不生，我也願意。就算要跟島一起沉沒....我真的是極樂島天龍人。寶島不需要我，但我需要寶島。神力女超人生在天堂島，我生在極樂島。 Youtube 頻道 (極樂島 Ecstasy Island): https://www.youtube.com/channel/UCcqaEy3cDCCUMvgg9IxggAg 科學頻道我一生領悟: https://www.youtube.com/channel/UCLSL9YGTU-BZjqtYwknv9fA

Monday, January 25, 2021

代數幾何2 重點整理

(Affine schemes) Given a ring $A$ and an $A$ -module $M$ :
We get an affine scheme $LaTeX: X=\operatorname{Spec}A$ $X$ = Spec ⁡ A and a quasi-coherent $LaTeX: \mathcal O_X$ $O$ X-module $LaTeX: \widetilde{M}$ $M$ ~. For any $LaTeX: p\in \operatorname{Spec} A$ $p$ ∈ Spec ⁡ A and any $LaTeX: f\in A$ $f$ ∈ A we have $LaTeX: \widetilde{M}_p\cong M_p$ $M$ ~ p ≅ M p and $LaTeX: \Gamma(D(f),\widetilde{M})\cong M_f$ $Γ$ ( D ( f ) , M ~ ) ≅ M f. In particular, $LaTeX: \Gamma(X,\widetilde{M})\cong M$ $Γ$ ( X , M ~ ) ≅ M. Conversely, for any quasi-coherent sheaf $LaTeX: \mathcal F$ $F$ on $LaTeX: X=\operatorname{Spec}A$ $X$ = Spec ⁡ A there is an $A$ -module $M$ such that $LaTeX: \mathcal F\cong \widetilde M$ $F$ ≅ M ~.
(Affine morphisms) Given a scheme $Y$ , a quasi-coherent $LaTeX: \mathcal O_Y$ $O$ Y-algebra $LaTeX: \mathcal A$ $A$ , and a quasi-coherent $LaTeX: \mathcal A$ $A$ -module $LaTeX: \mathcal M$ $M$ on $Y$ :
We get a scheme $LaTeX: X=\operatorname{Spec}\mathcal A$ $X$ = Spec ⁡ A together with an affine morphism $LaTeX: \pi:\operatorname{Spec}\mathcal A\to Y$ $π$ : Spec ⁡ A → Y and a quasi-coherent $LaTeX: \mathcal O_X$ $O$ X-module $LaTeX: \widetilde{\mathcal M}$ $M$ ~. Conversely, any quasi-coherent sheaf on $LaTeX: X=\operatorname{Spec}\mathcal A$ $X$ = Spec ⁡ A is of this form. This construction specializes to the previous item if we take $LaTeX: Y=\operatorname{Spec}\mathbb Z$ $Y$ = Spec ⁡ Z.
Given a graded ring $S$ and a graded $S$ -module $M$ :
We get a scheme $LaTeX: X=\operatorname{Proj}S$ $X$ = Proj ⁡ S and a quasi-coherent $LaTeX: \mathcal O_X$ $O$ X-module $LaTeX: \widetilde{M}$ $M$ ~. For any $LaTeX: f\in S_d$ $f$ ∈ S d with $d$ > 0 there is an open affine $LaTeX: D_+(f)\cong\operatorname{Spec} S_{(f)}\cong\operatorname{Spec} \frac{S^{(d)}}{(f-1)}$ $D$ + ( f ) ≅ Spec ⁡ S ( f ) ≅ Spec ⁡ S ( d ) ( f − 1 )in $X$ , and we have $LaTeX: \Gamma(D_+(f),\widetilde{M})\cong M_{(f)}\cong \frac{M^{(d)}}{(f-1)M^{(d)}}$ $Γ$ ( D + ( f ) , M ~ ) ≅ M ( f ) ≅ M ( d ) ( f − 1 ) M ( d ) (If $LaTeX: M=\oplus M_i$ $M$ = ⊕ M i then $LaTeX: M^{(d)}:=\oplus M_{id}$ $M$ ( d ) := ⊕ M i d ). For any $LaTeX: p\in \operatorname{Proj} S$ $p$ ∈ Proj ⁡ S we have $LaTeX: \widetilde{M}_p\cong M_{(p)}$ $M$ ~ p ≅ M ( p ). Conversely, if $S$ is generated by finitely many elements in $S$ 1 then any quasi-coherent $LaTeX: \mathcal O_X$ $O$ X-module $LaTeX: \mathcal F$ $F$ is of this form. An important example is the invertible $LaTeX: \mathcal O_X$ $O$ X -modules $LaTeX: \mathcal O_X(m)=\widetilde{S(m)}$ $O$ X ( m ) = S ( m ) ~ for any $LaTeX: m \in \mathbb Z$ $m$ ∈ Z (If S(m) is the grades S-module with $LaTeX: S(m)_d=S_{m+d}$ $S$ ( m ) d = S m + d for any $LaTeX: d \in \mathbb Z$ $d$ ∈ Z).
Given a scheme $Y$ , a quasi-coherent graded $LaTeX: \mathcal O_Y$ $O$ Y-algebra $LaTeX: \mathcal S$ $S$ , and a quasi-coherent graded $LaTeX: \mathcal S$ $S$ -module $LaTeX: \mathcal M$ $M$ on $Y$ :
We get a scheme $LaTeX: X=\operatorname{Proj}\mathcal S$ $X$ = Proj ⁡ S together with a morphism $LaTeX: \pi:\operatorname{Proj}\mathcal S\to Y$ $π$ : Proj ⁡ S → Y and a quasi-coherent $LaTeX: \mathcal O_X$ $O$ X-module $LaTeX: \widetilde{\mathcal M}$ $M$ ~. For any $LaTeX: f\in \Gamma(Y,\mathcal{S}_d)$ $f$ ∈ Γ ( Y , S d ) with $d$ > 0 there is an open $LaTeX: X_f\cong\operatorname{Spec} \frac{\mathcal S^{(d)}}{(f-1)}$ $X$ f ≅ Spec ⁡ S ( d ) ( f − 1 )in $X$ , and we have $LaTeX: \Gamma(X_f,\widetilde{M})\cong \frac{\mathcal M^{(d)}}{(f-1)\mathcal M^{(d)}}$ $Γ$ ( X f , M ~ ) ≅ M ( d ) ( f − 1 ) M ( d ). Conversely, if $LaTeX: \mathcal S_1$ $S$ 1is finite type and generates $LaTeX: \mathcal S$ $S$ then any quasi-coherent $LaTeX: \mathcal O_X$ $O$ X-module is of this form. This construction specializes to the previous item if we take $LaTeX: Y=\operatorname{Spec}\mathbb Z$ $Y$ = Spec ⁡ Z.

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