1. Affine schemes and schemes. We will take a deeper look at a commutative ring $R$ and its spectrum $Spec(R)$, by constructing the structure of ringed space for $Spec(R)$, and more than that, the stalk at any point $P\in Spec(R)$ is a local ring. This gives rise to define such structure as a locally ringed space. One needs to look at my previous post for the topology on $Spec$ before going further.
First, one can see that $Spec(R)$ equipped by closed sets of the form $V(I)=\{P\in Spec(R)|I\subset P\}$ forms a topological space. And we will give it the structure sheaf, which is very close to the structure sheaf of affine varieties. But the main difference things is that for affine varieties, we have "points" are exactly maximal ideals, and because $k$ is algebraically closed, the quotient fields are exactly $k$, and one has the "ground fields". For $Spec(R)$, we define the structure sheaf in a more general way, i.e. we accept "points" for prime ideals. That will lead to the fact that the quotient fields in this case is not unique, and we don't have the ground field at all. This is replaced by defining "regular functions" by "sections" on local rings at prime ideals, which also admits the local properties. Formally, we have
Definition 1.1. Let $X:=Spec(R)$ and $U\subset Spec(R)$ be any open subset, we define
$$\mathscr{O}_X(U)=\{\varphi=(\varphi_P)_{P\in U}|\varphi_P\in R_P, \text{ and }\varphi \text{ is locally represented by the form }\frac{f}{g}\in R_P \}$$
That means, for any $P\in U$, there exists a neighborhood $V_P\subset U$ of $P$ such that there exists $f,g\in R, g\notin Q, \forall Q\in V_P$, and $\varphi_Q=\frac{f}{g}$. And $\mathscr{O}_X(U)$ is called regular functions on $U$.
Via this definition, one can see that $\mathscr{O}_X$ is a structure sheaf, and hence, $(X,\mathscr{O}_X)$ is a ringed space. Recall that in a ring space $(X,\mathscr{O}_X)$, the stalk at a point is defined $\{(U,\varphi|U \text{ is neighborhood of } P \text{ and } \varphi \text{ is regular in } U)\}/\sim$, where $\sim$ is the equivalent relation $(U,\varphi)\sim (V,\psi)$ if there exists $O\subset U\cap V$ is an open neighborhood at $P$ and $\varphi_{|O}=\psi_{|O}$. We denote the stalk at a point $P$ as $\mathscr{O}_{X,P}$. And the important properties of $(Spec(R), \mathscr{O}_{Spec(R)})$ are
Proposition 1.2.
(i) $\mathscr{O}_{X, P}\cong R_P$.
(ii) Let $X_f:=\{P\in Spec(R)|f\notin P\}$ be the distinguished open subset, then $\mathscr{O}_{X}(X_f)\cong R_f$, and in particular $\mathscr{O}_X(X)=R$.
Proof.
(i) We consider the map $\psi$ from $\mathscr{O}_{X,P}$ to $R_P$ defined by sending $(U,\varphi)$ to $\varphi_P$. Due to the definition of regular functions, this map is well-defined. Furthermore, $\psi$ is surjective since for any $\frac{f}{g}\in R_P$, for $f,g\in R, g\notin P$, then $X_g$ is actually an open set containing $P$. Hence, $\psi(X_g,\frac{f}{g})=\frac{f}{g}$.
Assume there exists $\psi(U,\varphi) = \psi(V,\phi)$, for $U,V$ are neighborhoods of $P$, and $\varphi, \phi$ are regular functions on $U,V$, respectively. Then there exists $O\subset U\cap V$ a neighborhood of $P$ such that $\varphi_{|O}=\frac{f}{g}=\phi_{|O}=\frac{m}{n}$. That means, there exists $h\notin P$ such that $h(fn-gm)=0$. And then, $X_h\cap X_n\cap X_g$ is also an open neighborhood of $P$, and $\frac{f}{g}=\frac{m}{n}$ on this open set. And hence, $(U,
\varphi)\sim(V,\phi)$, and $\psi$ is injective.
(ii) We consider the map $\psi$ from $R_f$ to $\mathscr{O}_X(X_f)$ sending $\frac{g}{f^r}$ to $\frac{g}{f^r}$. By definition of regular functions, it is a well-defined map. And $\psi$ is injective since if $\psi(g/f^r)=\psi(h/f^s)$, then $g/f^r=h/f^s$ holds in all $R_P, P\in X_f$. That means, for each $P$, there exists $h_P$ such that $h_P(gf^s-hf^r)=0$. Let $I$ be the annihilator of $(gf^s-hf^r)$, then $I\not\subset P$, for all $P\in X_f$. This yields $V(I)\cap X(f)=\emptyset$, and hence, $V(I)\subset V(f)$. By facts from commutative algebra, one can easily seen $f\in \sqrt{I}$, and there exists $n$ such that $f^n\in I$, which means $f^n(gf^s-hf^r)=0$. Hence, $g/f^r = h/f^s\in R_f$, and $f$ is injective.
For the subjectivity of $\psi$, we refer to the note of Gathmann on AG (version 2002), Chapter V.
(Q.E.D)
We are now ready to define affine schemes and locally ringed space.
Definition 1.3. Let $(X, \mathscr{O}_X)$ be a ringed space, we call $X$ an affine scheme, if $(X,\mathscr{O}_X)$ is isomorphic as ring spaces with $(Spec(R), \mathscr{O}_{Spec(R)})$ for some commutative ring $R$.
Definition 1.4. Let $(X,\mathscr{O}_X)$ be a ringed space, we call $X$ a locally ringed space if for any point $P\in X$, the stalk $\mathscr{O}_{X,P}$ is a local ring.
And via these definitions, one can see actually, an affine scheme is a locally ringed space. And we now come to the definition of schemes
Definition 1.5. Let $(X,\mathscr{O}_X)$ be a locally ringed space. Then $(X,\mathscr{O}_X)$ is called a scheme if $X$ is covered by open subset $\{U_i\}_{i\in I}$ and for each $i\in I$, $(U_i, \mathscr{O}_X(U_i))$ is isomorphic as ring spaces with $(Spec(R_i),\mathscr{O}_{Spec(R_i)})$, where $R_i$ is a commutative ring.
As we can see, all definitions above can be clearly seen from the case of spectrums of commutative rings. In the next section, we will understand about morphisms between schemes via a very familiar thing, the ring homomorphism.
2. Morphisms between schemes. The morphisms between schemes can be described exactly under the deeper look into the ring homomorphism. Let $\psi: S\to R$ be a ring homomorphism, $\psi$ will induce the map $f: Spec(R)\to Spec(S)$ defined by $f(P)=\psi^{-1}(P)$. The map $f$ is continuous, since if $F\in Spec(S)$ is closed, means $F=V(I)=\{Q\in Spec(S)|I\subset Q\}$, then $f^{-1}(F)={P\in Spec(R)|f(P)=\psi^{-1}(P)=Q\supset I}=\{P\in Spec(R)|\psi(I)\subset P\}=V(\psi(I))$ is closed in $Spec(R)$.
And these induce the map on local ring $\psi_P:S_{f(P)}\to R_P$ defined by $\psi_P(a/b)=\psi(a)/\psi(b)$. It can be checked directly that $\psi_P$ is a well-defined map, moreover, it is a ring homomorphism. Let us denote $X:=Spec(R), Y:=Spec(S)$, and $U\subset Y$ is open, we have $U=\{Q\in Spec(S)|...\}$, and $f^{-1}(U)=\{P\in Spec(R)|f(P)=\psi^{-1}(P)=Q, Q\in U\}$. And hence, $U$ consists of $f(P)$, for some $P\in Spec(R)$. Let $\varphi_{f(P)}\in S_{f(P)}$, then $\psi_P(\varphi_{f(P)})\in R_P$. These information give rise to define the map $f^*_U: \mathscr{O}_Y(U)\to \mathscr{O}_X(f^{-1}(U))$ by sending $(\varphi_{f(P)})$ to $(\psi_P(\varphi_{f(P)}))$. And this map is a ring homomorphism.
What we have constructed so far is the morphism between two ring spaces $(X,\mathscr{O}_X)$ and $(Y, \mathscr{O}_Y)$. But more than that, if $M_P, M_{f(P)}$ be the maximal ideas of $R_P, S_{f(P)}$, respectively, one can see $f^{-1}(M_P)=M_{f(P)}$. This leads us to the following
Definition 2.1. Let $(X,\mathscr{O}_X), (Y,\mathscr{O}_Y)$ be two locally ringed spaces. A morphism between them consists of the following data
(i) A continuous map: $f: X\to Y$
(ii) For any $U\subset Y$, a ring homomorphism $f^*: \mathscr{O}_Y(U)\to \mathscr{O}_X(f^{-1}(U))$, such that $f^*$ is compatible with restriction map, and the induced ring homomorphism $f^*_P$ between stalks $\mathscr{O}_{Y,f(P)}\to \mathscr{O}_{X,P}$ satisfies $(f^*_P)^{-1}(M_{X,P})=M_{Y,f(P)}$, where $M_{X,P}$ and $M_{Y,f(P)}$ are the maximal ideals of $\mathscr{O}_{X,P}$, and $\mathscr{O}_{Y,f(P)}$, respectively.
A morphism between two schemes is a morphism between two locally ringed spaces.
By this definition, one obtains, by our earlier example about ring homomorphism.
Proposition 2.2. Let $R, S$ be two commutative rings, then there exists one-to-one correspondence between ring homomorphisms from $S$ to $R$ and morphisms from $Spec(R)$ to $Spec(S)$.
Proof. As we have considered, a ring homomorphism from $S$ to $R$ will induce a morphism from $Spec(R)$ to $Spec(S)$. Conversely, a morphism from $X:=Spec(R)$ to $Y:=Spec(S)$ will induce a map $f^*: S=\mathscr{O}_Y(Y)$ to $R=\mathscr{O}_X(X)$, by Proposition 1.2 (ii).
(Q.E.D)
Due to the gluing properties of sheaves, given a morphism $f$ between two schemes $X$ and $Y$ is the same as given morphism from $f_i: U_i\to Y$ for $U_i\subset X$ open and $\cup_{i}U_i=X$, and $f_{i_{U_i\cap U_j}}=f_{j_{U_i\cap U_j}}$. This leads us to an important
Proposition 2.3. Let $X$ be any scheme, and $Y$ is affine, then there exists one-to-one correspondence between morphisms from $X$ to $Y$ and ring homomorphism from $\mathscr{O}_Y(Y)$ to $\mathscr{O}_X(X)$.
Proof. Let $\{U_i\}_{i\in I}$ be affine covers of $X$, then given a morphism from $X$ to $Y$ is the same as given morphisms from $U_i$ to $Y$, which agrees on the intersections. By Proposition 2.2, given morphisms between $U_i$ to $Y$ is the same as given ring homomorphisms from $\mathscr{O}_{Y}(Y)$ to $\mathscr{O}_{U_i}(U_i)=\mathscr{O}_X(U_i)$. However, given $\varphi_i\in U_i$ such that $\varphi_{i_{U_i\cap U_j}}=\varphi_{j_{U_i\cap U_j}}$ is the same as given $\varphi\in \mathscr{O}_{X}(X)$. And hence, given a morphism from $X$ to $Y$ is the same as given a ring homomorphism from $\mathscr{O}_Y(Y)$ to $\mathscr{O}_X(X)$.
(Q.E.D)
We know that $\mathbb{Z}$ is the initial object in the category of commutative rings. Hence, there exists only one homomorphism from $\mathbb{Z}$ to $\mathscr{O}_X(X)$ for any scheme $X$. By Proposition 2.3, it will induce a unique morphism from $X$ to $Spec(\mathbb{Z})$. We have the following
Corollary 2.4. Let $X$ be any scheme, then there exists a unique morphism from $X$ to $Spec(\mathbb{Z})$, that sends a point $P$ to the characteristic of $k(P)$, where $k(P)=\mathscr{O}_{X,P}/M_{X,P}$.
Proof. The existence and uniqueness have been proved by our earlier comment. Because stalks are the local objects, we can deal with this on affine open cover. And the problem is reduced to the case $X=Spec(R)$. Let $f$ be the morphism from $X:=Spec(R)$ to $Y:=Spec(\mathbb{Z})$. It then induce the map $f^*: \mathbb{Z}\to R$. And also the ring homomorphism between stalk $f^*_P: \mathscr{O}_{Y,f(P)}\to \mathscr{O}_{X,P}$ satisfying $(f^*_P)^{-1}(M_{X,P})=M_{Y,f(P)}$ (*). And $\mathscr{O}_{Y,f(P)}/M_{Y,f(P)}\cong \mathbb{Z}, \text{ or } \mathbb{F}_p$, for some prime $p$, and $p$ or $0$ are exactly the image of $f(P)$. Recall that if $\psi: R\to S$ is a ring homomorphism, then we can consider $S$ as $R$-module. And hence, due to (*), one can consider $k(P)=\mathscr{O}_{X,P}/M_{X,P}$ as $\mathbb{Z}$-module or $\mathbb{F}_p$-vector space, i.e. $k(P)$ is an extension field of $\mathbb{Q}$ or $\mathbb{F}_p$, and its characteristic follows from this.
(Q.E.D)
3. Projective schemes. We are now turning to another interesting construction, the projective scheme. By generalizing projective varieties, one want to construct such schemes on graded rings. Recall that if $R$ is a commutative ring, $R$ is called graded ring if $(R,+)=\oplus_{d\ge 0}R_d$, and for all $a\in R_d, b\in R_e$, we have $ab\in R_{d+e}$. If $a\in R_d$, then $a$ is called homogeneous element of degree $d$, and denote $\deg a = d$. An ideal $I$ of $R$ is called homogeneous ideal if $I$ is generated by homogeneous elements.
It can be seen from the definition that $R_0$ is a subring of $R$. And if $I$ is an homogeneous ideal of $R$, then $R/I$ is also a graded ring. We next define $Proj(R)$ the sets of all homogeneous prime ideal $P$ of $R$, such that $\oplus_{d>0}R_d\not\subset P$. One can see that $Proj(R)$ inherits the subspace topology on $Spec(R)$. For any open subset $U$ of $Proj(R)$, we define the set
$$\mathscr{O}_{Proj(R)}(U)=\{\varphi=(\varphi_P)_{P\in U}|\varphi_P\in R_P\text{ is locally represented of the form }\frac{f}{g} \text{ with }f, g\in R, \deg f=\deg g\}$$
By this definition, $\mathscr{O}_{Proj(R)}$ becomes a structure sheaf. Denote $X:=Proj(R)$, we have, $(X, \mathscr{O}_X)$ is a ringed space. We will prove that it is in fact a locally ringed space, and a scheme, respectively. It is not hard to see, actually, we repeat the proof of Proposition 1.2 to obtain $\mathscr{O}_{X, P}$ is a local ring, for all $P\in Proj(R)$.
The difficult part is to prove $(X, \mathscr{O}_X)$ is covered by affine schemes. In fact, we will prove that the distinguished open sets $X_f\subset X$ is affine, for any $f$ is a homogeneous element of positive degree. It can be seen from the definition of $Proj(R)$ that $\{X_f\}$ is an open cover of $X$. And hence, it is sufficient to prove $X_f$ is affine.
First, it can be seen that $X_f=\{P\in Proj(R)| f\notin P\}$ corresponds to $\{P\in Proj(R_f)\}$. And it is actually the isomorphism between two ringed spaces. We now prove that $Proj(R_f)$ is actually affine.
Lemma 3.1. Let $R$ be a graded ring and $f$ a homogeneous element of positive degree $d$. There exists a one-to-one correspondence between $Proj(R_f)$ and $Spec((R_f)_0)$, the homogeneous part of $R_f$ (which is a ring) of degree 0. This correspondence is a homeomorphism between $Spec(R_f)$ and $Spec((R_f)_0)$. Furthermore, it defines the isomorphism between two ringed spaces.
Proof sketch. Let $Q\in R_f$ be a prime ideal, then $Q\cap (R_f)_0\in Spec((R_f)_0)$. Now, take any $P\in Spec((R_f)_0)$, we construct $Q=\oplus_{i\in \mathbb{N}}Q_i$, for $Q_i\subset (R_f)_d$ and for all $a\in R_f$, $a\in Q_i$ iff $\frac{a^d}{f^i}\in P$. It can be seen from this that $Q_0=P=Q\cap (R_f)_0$. Furthermore, $Q$ is an ideal of $R_f$, and $Q$ is a prime ideal of $R_f$. We then obtain the bijection between $Proj(R_f)$ and $Spec((R_f)_0)$. Also, this bijection give a continuous map from $Proj(R_f)$ to $Spec((R_f)_0)$, and its inverse. That means $Proj(R_f)\cong Spec((R_f)_0)$ as topological spaces. Now, via the definition of structure sheaf for graded ring $R_f$, one can see that it is the same as structure sheaf of the ring $(R_f)_0$. Hence, the homeomorphism above is the isomorphism between the two ringed spaces.
(Q.E.D)
Via Lemma 3.1, one can see $Proj(R)$ is a scheme. These are basic constructions of schemes, and it generalizes what we have considered for affine and projective varieties.
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