Dependencies

Let \(k\) be an algebraically closed filed and \(A\) a finitely generated \(k\)-algebra, then \({A}\Big/{\mathfrak m}\) is isomorphic to \(k\).

The property “finite type” of ring homomorphisms is local in the sense of Definition 2.1.

If \(P\) is a property of ring homomorphisms then the property affine locally \(P\) of scheme morphism \((\phi , \phi ^*): \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) holds if and only if \(P\) holds for all ring homomorphism \(\Gamma (U, \mathcal{O}_X) \to \Gamma (V, \mathcal{O}_Y)\) for all affine subsets \(U \subseteq X\) and \(V \subseteq Y\) such that \(\phi (U) \le V\).

Let \(\Phi : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right) := (\phi , \phi ^*)\) be a morphism of schemes. We say

1. \(\Phi \) is locally of finite type if for any affine open \(V \subseteq Y\) and affine open \(U \subseteq X\) such that \(\phi (U) \subseteq V\), we have the induced map \(\Gamma (U, \mathcal{O}_X) \to \Gamma (V, \mathcal{O}_V)\) is a ring map of finite type. In another word, \(\Phi \) is affine locally a ring homomorphism of finite type.

2. \(\Phi \) is of finite type if it is locally of finite type and \(\phi \) is quasi-compact.

A scheme \(\left(X, \mathcal{O}_{X}\right)\) is of finite type over \(\mathbb {C}\) if \(\left(X, \mathcal{O}_{X}\right)\) is locally of finite type over \(\mathbb {C}\) and the structure morphism is quasicompact.

An affine open covering of spectra of finitely generated \(\mathbb {C}\)-algebra for a scheme \(\left(X, \mathcal{O}_{X}\right)\) over \(\mathbb {C}\) is the following data:

1. indexing set: \(I\);

2. a family of finitely generated algebras: \(R : I \to \mathsf{FGCAlg}_{\mathbb {C}}\);

3. a family of open immersions: for each \(i \in I\), \(\iota _i: \left(\operatorname *{Spec}{R_i}, \widetilde{R_i}\right) \to \left(X, \mathcal{O}_{X}\right)\);

4. covering: \(c : X \to I\) such that for each \(x \in X\), \(c_x \in \operatorname *{range}\left({\iota _i}\right)\).

A scheme \(\left(X, \mathcal{O}_{X}\right)\) is locally of finite type over \(\mathbb {C}\) if it is a scheme over \(\mathbb {C}\) and it admits an affine open cover of spectra of finitely generated \(\mathbb {C}\)-algebras.

If \(U \subseteq \operatorname {Spec}A\) is an open subset where \(A\) is a finite \(\mathbb {C}\)-algebra, then \(U\) admits a finite covering of \(D(f_{1}),\dots , D(f_{n})\) where \(D(x)\) is the basic open around \(x \in A\).

If \(\left(X, \mathcal{O}_{X}\right)\) is a scheme of finite type over \(\mathbb {C}\) and \(V \subseteq X\) is an open subset, then \(V\) is quasi-compact.

Let \(\left(X, \mathcal{O}_{X}\right)\) be a scheme of finite type over \(\mathbb {C}\) and \(U \subseteq X\) be open, then the restriction \((U, \mathcal{O}_{X}|_{U})\) is a scheme of finite type over \(\mathbb {C}\) as well.

Let \(\left(\operatorname *{Spec}{A}, \widetilde{A}\right)\) and \(\left(\operatorname *{Spec}{B}, \widetilde{B}\right)\) be two affine finite schemes over \(\mathbb {C}\). Then any morphism \(\Phi : \left(\operatorname *{Spec}{A}, \widetilde{A}\right) \to \left(\operatorname *{Spec}{B}, \widetilde{B}\right)\) is of the form \((\phi , \phi ^{\star })\) where \(\phi : B \to A\) is a \(\mathbb {C}\)-algebra homomorphism.

The category of affine scheme over \(\mathbb {C}\) is antiquivalent to commutative \(\mathbb {C}\)-algebras. The category of affine finite scheme over \(\mathbb {C}\) is antiequivalent to finitely generated commutative \(\mathbb {C}\)-algebras.

If \(\left(X, \mathcal{O}_{X}\right)\) is a scheme locally of finite type over \(\mathbb {C}\), then \(\operatorname {Max}\left\{ X\right\} \) is in bijection with \(X(\mathbb {C}) := \operatorname {Hom}_{\mathsf{Sch}/\mathbb {C}}\left(\left(\operatorname *{Spec}{\mathbb {C}}, \widetilde{\mathbb {C}}\right), \left(X, \mathcal{O}_{X}\right)\right)\), such that every closed point \(p\), is the image of \(\star \) of a unique morphism \(\Phi _{p}\); and for each morphism \(\Phi : \left(\operatorname *{Spec}{\mathbb {C}}, \widetilde{\mathbb {C}}\right)\to \left(X, \mathcal{O}_{X}\right)\), \(\Phi (\star )\) is closed in \(X\) where \(\star \) is the unique point of \(\operatorname {Spec}\mathbb {C}\).

Let \(\Phi : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be an open immersion, then \(\operatorname {Max}\left\{ \phi \right\} \) is an embedding.

Let \(\theta : R \to S\) be a surjective \(\mathbb {C}\)-algebra homomorphism between finite \(\mathbb {C}\)-algebras. \(\operatorname {MaxSpec}\theta \) is an embedding.

The set of closed points in \(\operatorname {Spec}{\mathbb {C}[X_{1},\dots , X_{n}]}\) corresponds bijectively to \(\mathbb {C}^{n}\).

If \(S\), as \(\mathbb {C}\)-algebras, is generated by both \(a_{1},\dots , a_{n}\) and \(b_{1},\dots ,b_{m}\), we would have as topological spaces \({\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}_{a_{i}}\) and \({\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}_{b_{i}}\) are homeomophic.

Let \(a_{1},\dots , a_{n}\) be a set of generators of \(S\) as \(\mathbb {C}\)-algebra and \(R\) be \(\mathbb {C}[X_{1},\dots ,X_{n}]\) and \(\theta : R \to S\) be the surjective map defined by \(\theta (X_{i})=a_{i}\). The image of \(\operatorname {Spec}{\theta } : {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}} \to {\left\{ {\operatorname {Spec}{R}}\right\} }^{\mathsf{an}}\cong \mathbb {C}^{n}\) is

where \(f_{i}\) generates \(\ker \theta \).

If \(S \to S'\) is an isomorphism of finite \(\mathbb {C}\)-algebras, then \({\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\) and \({\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}}\) are homeomorphic.

If \(\phi : S \to S'\) is a surjective \(\mathbb {C}\)-algebra homomorphism between two finite \(\mathbb {C}\)-algebras, then \({\left\{ {\operatorname {Spec}{\phi }}\right\} }^{\mathsf{an}} : {\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}} \to {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\) is an embedding.

Let us write \(\mathbb {C}[X_{1},\dots ,X_{n}]\) as \(R\) and let \(f \in R\) be a polynomial, then the localization map \(\alpha : R \to R_{f}\) induces an embedding \({\left\{ {\operatorname {Spec}{R_{f}}}\right\} }^{\mathsf{an}} \hookrightarrow {\left\{ {\operatorname {Spec}{R}}\right\} }^{\mathsf{an}}\).

If \(S\) is a finite \(\mathbb {C}\)-algebra and \(f \in S\), then the localization map \(\alpha : S \to S_{f}\) induces an embedding \({\left\{ {\operatorname {Spec}{S_{f}}}\right\} }^{\mathsf{an}} \hookrightarrow {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\). In fact \({\left\{ {\operatorname {Spec}{S_{f}}}\right\} }^{\mathsf{an}}\) is identified as the subset \(D(f) \cap \operatorname {Spec}{S}\) where \(D(f)\) is the basic open set in \(\operatorname {Spec}{S}\).

Let \(\left(X, \mathcal{O}_{X}\right)\) be a scheme locally of finite type over \(\mathbb {C}\), let \(\mathcal{I}\) be the collection of open immersions \(\left(\operatorname *{Spec}{R}, \widetilde{R}\right) \to X\) where \(R\) is some finite \(\mathbb {C}\)-algebra. Then the complex topology on the set of closed points \(\max X\) is defined as the weak topology with respect to \(\left\{ {\left\{ {{\phi }}\right\} }^{\mathsf{an}} | (\phi , \phi ^{*}) \in \mathcal{I}\right\} \) where \({\left\{ {\phi }\right\} }^{\mathsf{an}}\) is the restriction of \(\phi \) to the subset of closed points. When we talk about complex toplogy, we write \(\max {X}\) as \({\left\{ {X}\right\} }^{\mathsf{an}}\).

Let \(\Phi = (\phi , \phi ^{*}) : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be a morphism of schemes locally of finite type over \(\mathbb {C}\), then \({\left\{ {\phi }\right\} }^{\mathsf{an}} : {\left\{ {X}\right\} }^{\mathsf{an}} \to {\left\{ {Y}\right\} }^{\mathsf{an}}\) is continuous.

Let \(\Phi : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be an open immersion of schemes locally of finite type over \(\mathbb {C}\), then \({\left\{ {\phi }\right\} }^{\mathsf{an}}\) is an embedding of \({\left\{ {X}\right\} }^{\mathsf{an}}\) into \({\left\{ {Y}\right\} }^{\mathsf{an}}\).

Given two morphisms among schemes locally of finite type over \(\mathbb {C}\)

we have \({\left\{ {\psi \circ \phi }\right\} }^{\mathsf{an}} = {\left\{ {\psi }\right\} }^{\mathsf{an}} \circ {\left\{ {\phi }\right\} }^{\mathsf{an}}\)

Let \(\Phi = (\phi , \phi ^{*}) : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be a morphism of schemes locally of finite type over \(\mathbb {C}\), we have

All generalized polydiscs form a topological basis of \(\mathbb {C}^{n}\).

On generalized polydiscs \(\Delta _{1} \subseteq \Delta _{2}\), \({\Gamma \left(\mathcal{O}^{\mathsf{pre}},\Delta _{1}\right)}\) is isomorphic to

and the restriction map \({\Gamma \left({\mathcal{O}^{\mathsf{pre}}},\Delta _{2}\right)} \to {\Gamma \left({\mathcal{O}^{\mathsf{pre}}},\Delta _{1}\right)}\) is the restriction map

defined by \([f] \mapsto [f\mid _{U}]\).

Let \(V\subseteq \mathbb {C}^{n}\) be an open subset then we have an isomorphism

Let \(\left(\operatorname *{Spec}{A}, \widetilde{A}\right)\) be an affine finite scheme over \(\mathbb {C}\). We have that the \(\operatorname {MaxSpec}A\) is in bijection with \(\operatorname {Hom}_{\mathsf{Sch}/\mathbb {C}}\left(\left(\operatorname *{Spec}{\mathbb {C}}, \widetilde{\mathbb {C}}\right), \left(\operatorname *{Spec}{A}, \widetilde{A}\right)\right)\)

Let \(k\) be an algebraically closed field and \(A\) a finitely generated \(k\)-algebras, then each maximal ideals \(\mathfrak m \subseteqq A\) is the kernel of a unique algebra homomorphism \(\phi _{\mathfrak m} : A \to k\)

The analytification \({\left\{ {\widetilde{S}}\right\} }^{\mathsf{an}}\) of \(\widetilde{S}\) is a presheaf on \({\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\) defined by the following:

• For any open set \(U \subseteq {\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\), we define the sections \({\Gamma \left({\left\{ {\widetilde{S}}\right\} }^{\mathsf{an}},U\right)}\) to be

  \[ {\Gamma \left(\mathcal{O}^{\mathsf{pre}},U \cup \left(\mathbb {C}^{n}-{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\right)\right)}. \]

• For any open sets \(U \subseteq V \subseteq {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\), we define the restriction map of \({\left\{ {\widetilde{S}}\right\} }^{\mathsf{an}}\) from \(V\) to \(U\) is the restriction map of \(\mathcal{O}^{\mathsf{pre}}\) from \(V \cup \left(\mathbb {C}^{n}-{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\right)\) to \(U \cup \left(\mathbb {C}^{n}-{\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\right)\).

This defines a presheaf, the satisfaction of sheaf axioms are essentially from that of \(\mathcal{O}^{\mathsf{pre}}\).

The complex topology of \(\operatorname {Spec}S\) is the subspace topology of \(\mathbb {C}^{n}\) via the injective map \(\operatorname {MaxSpec}\theta \). With the complex topology, we denote \(\operatorname {MaxSpec}S\) as \({\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}\) and, if \(\phi : S \to S'\) is a \(\mathbb {C}\)-algebra homorphism, we the induced map between \({\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}}\) and \({\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\) as \({\left\{ {\operatorname {Spec}\phi }\right\} }^{\mathsf{an}}\).

A generalized polydisc

is the set \(\left\{ x \in \mathbb {C}^{n}| \left|g_{i}(x) - w_{i}\right| {\lt} r_{i}~ \text{for all}~ i = 1,\dots ,l \right\} \) where each \(g_{i} \in \mathbb {C}[X_{1}\dots ,X_{n}]\) is a polynomial of \(n\) variables, \(w := (w_{1},\dots , w_{n}) \in \mathbb {C}^{l}\) is a point and \(r := (r_{1},\dots ,r_{l})\in \mathbb {R}^{l}_{\ge 0}\) are all non-negative. We call \(w\) the center of the polydisc and \(r\) the polyradius ⁶ .

We have a sheaf \(\mathcal{O}^{\mathsf{pre}}\) on \(\mathbb {C}^{n}\) is defined to be the cokernel sheaf of the following exact sequence:

where the first arrow, on \(U\), is given by \((a_{1},\dots ,a_{m}) \mapsto f_{1}|_{U}\symbf {\cdot }a_{1}+\cdots +f_{m}|_{U}\symbf {\cdot }a_{m}\) where \(a_{i} \in {\Gamma \left(\operatorname {\mathcal{Hol}},U\right)}\) and \(f_{i}\) generates \(\ker \theta \)

Let \(P\) be a property of ring homomorphisms: we say the property \(P\) is local if

1. if \(P\) holds for \(\phi : A \to B\), then \(P\) holds for \(\phi _S: S^{-1}A \to \left\langle f(S)\right\rangle ^{-1} B\) for any submonoid \(S \subseteq A\).

2. Let \(\phi : A \to B\) be a ring homomorphism, if \(P\) holds for \(A \stackrel{\phi }{\to } B \to B_{f_i}\) for some \(\{ f_i\} \subseteq B\) such that \(\langle f_i\rangle = B\), then \(P\) holds for \(\phi \).

3. Let \(\phi : A \to B\) and \(\psi : B \to C\) be two ring homomorphisms, if \(P\) holds for \(\phi \) and \(\psi \), then \(P\) holds for \(\psi \circ \phi \).

4. \(P\) holds for \(A \to A_f\) for all \(f \in A\).

A scheme \(\left(X, \mathcal{O}_{X}\right)\) is locally of finite type over \(\mathbb {C}\) if \(\left(X, \mathcal{O}_{X}\right)\) is a scheme over \(\mathbb {C}\) and the structure morphism \(\left(X, \mathcal{O}_{X}\right) \to \left(\operatorname *{Spec}{\mathbb {C}}, \widetilde{\mathbb {C}}\right)\) is a morphism locally of finite type.

Let \(\Phi : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be a morphism of schemes and \(\mathcal{U}=\{ U_{i}\} \) be an affine open covering of \(Y\). Consider the pullback cover \(\mathcal{V} = \{ V_{i}\} \) of \(X\), and if for each \(i\), there is an affine open cover \(\mathcal{W}_{i} = \{ W_{i, j}\} \) for \(V_{i} \subseteq X\). Then \(\Phi \) is locally of finite type if and only if, for each \(i\) and \(j\), the ring map \(\Gamma (W_{i, j}, \mathcal{O}_{X}) \to \Gamma (U_{i}, \mathcal{O}_{Y})\) is of finite type.

Let \(\Phi : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be an open immersion between schemes, then \(\Phi \) is locally of finite type.

Let \(\Phi : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be a morphism of schemes and \(\mathcal{U} = \{ U_{i}\} \) be an open covering of \(X\), then affine locally \(\Phi \) is locally of finite type if and only if \(\Phi |_{U_i}\) is a morphism locally of finite type.

If \(\left(X, \mathcal{O}_{X}\right)\) is a scheme locally of finite type over \(\mathbb {C}\), and \(U = \left(\operatorname *{Spec}{A}, \widetilde{A}\right)\) is an open affine subscheme of \(\left(X, \mathcal{O}_{X}\right)\), then \(A \cong \Gamma (U, \mathcal{O}_{X})\) is a finitely generated \(\mathbb {C}\)-algebra as well.

Let \(\left(\operatorname *{Spec}{A}, \widetilde{A}\right)\) be an affine finite scheme over \(\mathbb {C}\). We have that the set of closed points \(\operatorname {MaxSpec}A\) are in bijection to \(\operatorname {Hom}_{\mathsf{CAlg}_{\mathbb {C}}}\left(A, \mathbb {C}\right)\)

Let \(S\) and \(S'\) be two finitely generated \(\mathbb {C}\)-algebras and \(\phi : S \to S'\) be a \(\mathbb {C}\)-algebra homomorphism, the natural map \({\left\{ {\operatorname {Spec}{\phi }}\right\} }^{\mathsf{an}} : {\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}} \to {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\) is continuous (in the complex topology) and compatible with the inclusion map, i.e. the following diagram is commutative:

Let \(\lambda _{X} : {\left\{ {X}\right\} }^{\mathsf{an}} \hookrightarrow X\) be the inclusion map, then \(\lambda _{X}\) is continuous.

Since \({\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}\), as a set, is just the set of closed points of \(\operatorname {Spec}S\), we have a function \(\lambda : {\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}} \hookrightarrow \operatorname {Spec}S\). \(\lambda \) is continuous where \({\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}\) is with complex topology while \(\operatorname {Spec}S\) is with the Zaraski toplogy.

Let \(k\) be a field and \(A\) be a finitely generated \(k\)-algebra. Then for every prime ideal \(\mathfrak {p}\) of \(A\),

If \(\mathfrak m\) is a maximal ideal of \(A\), then \(k \hookrightarrow {A}\Big/{\mathfrak m}\) is a finite field extension.

Composition of morphisms locally of finite type is locally of finite type.

Let \(\theta : R \to S\) be a surjective \(\mathbb {C}\)-algebra homomorphism between finite \(\mathbb {C}\)-algebras. Then the image of \(\operatorname {MaxSpec}\theta : \operatorname {MaxSpec}{S} \to \operatorname {MaxSpec}{R}\) is identified via Proposition 2.21 with the set of \(\mathbb {C}\)-algebra homomorphisms \(\psi : R \to \mathbb {C}\) such that \(\psi (\ker \theta ) = 0\).

Let \(\Phi = (\phi , \phi ^{*}) : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)\) be a morphism of schemes locally of finite type over \(\mathbb {C}\), then \(\phi \) maps closed points of \(X\) to closed points of \(Y\). Thus, we have a well defined map \(\operatorname {Max}\left\{ \phi \right\} : \operatorname {Max}\left\{ X\right\} \to \operatorname {Max}\left\{ Y\right\} \)

Let \(\left(X, \mathcal{O}_{X}\right)\) be a scheme locally of finite type over \(\mathbb {C}\), \(R\) be a finite \(\mathbb {C}\)-algebra and \(\Psi = (\psi , \psi ^{*}) : \left(\operatorname *{Spec}{R}, \widetilde{R}\right) \to \left(X, \mathcal{O}_{X}\right)\) be an open immersion. Then \({\left\{ {\psi }\right\} }^{\mathsf{an}}\) is an embedding.

Let \(\left(X, \mathcal{O}_{X}\right)\) be a scheme locally of finite type over \(\mathbb {C}\), let \(U \subseteq X\) be an open subset, then \((U, \mathcal{O}_{X}\mid _{U})\) is a scheme locally of finite type over \(\mathbb {C}\).

Let \(\phi : R \to S\) be a surjective \(\mathbb {C}\)-algebra homomorphisms between finite \(\mathbb {C}\)-algebras, then \(\operatorname {Spec}\phi : \operatorname {Spec}{S} \to \operatorname {Spec}{R}\) is an embedding.

Let \(x\) be a point not in \({\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\), in another word, \(x \in \mathbb {C}^{n} - {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\). There is an open subset \(x \in U \subseteq \mathbb {C}^{n}\) such that for any generalized polydiscs \(\Delta \subseteq U\), we have \({\Gamma \left(\mathcal{O}^{\mathsf{pre}},\Delta \right)} = 0\).

Let \(V \subseteq \mathbb {C}^{n}-{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\) be an open set in \(\mathbb {C}^{n}\), then the sections of \(\mathcal{O}^{\mathsf{pre}}\) on \(V\) is trivial.