一些走過的路: 代數幾何2 重點整理

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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