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Rapid tour ringed spaces and schemes

There are two standard perspectives on “global geometry” today:

  1. Figure out a model for your space and build some global construct that locally looks like your model.
  2. Declare that all interesting spaces are cut out in some fixed global models.

These two approaches are not mutually exclusive. For example, a smooth manifold is locally modeled on open subsets of $\mathbb{R}^n$ and smooth maps between such. But, we are interested when such things embed in some $\mathbb{R}^N$ which is the content of the Whitney Embedding Theorem.

At this point, we feel pretty good about rings and modules over them, especially if the ring is commutative. We will take approach 1. above to build a geometry, called schemes, locally modeled on rings and then turn to 2. for most practical tasks.

The following will be the global setting.

Definition. A ringed space is a pair $(X,\mathcal O)$ where $X$ is a topological space and $\mathcal O$ is a sheaf of commutative
rings on $X$. We will often omit $\mathcal O$ from the notation or write $\mathcal O_X$.

We say $(X,\mathcal O)$ is a locally ringed space if for each $x \in X$, $\mathcal O_x$ is a local ring.

A map of ringed spaces \(f : (X, \mathcal O_X) \to (Y, \mathcal O_Y)\) consists of the data of a continuous map $f: X \to Y$ and a map of sheaves of rings \(f^{\#} : f^{-1} \mathcal O_Y \to \mathcal O_X\)

Note that by adjunction we get a map \(\mathcal O_Y \to f_\ast \mathcal O_X\) additionally.

A map of locally ringed spaces is a map of ringed spaces which is a local homomorphism on the stalks of $\mathcal O$.

Example. The central example where $\mathcal O_X$ is the sheaf of $\mathbb{R}$-valued functions. If $X$ admits more structure, say of a smooth or complex manifold, we can then talk about functions satisfying local properties like $C^\infty$ or holomorphic.

Given a continuous map $F : X \to Y$ and continuous function $g : V \to \mathbb{R}$, we get a continuous function \(g \circ F : F^{-1} V \to \mathbb{R}\) which furnishes \(F^{\#} : F^{-1} \mathcal O_Y \to \mathcal O_X\)

Definition. For a ringed space $(X,\mathcal O)$, we let $\operatorname{Mod} \mathcal O$ denote the (abelian) category of sheaves of $\mathcal O$-modules.

Note that map of ringed space $f: X \to Y$ furnishes a functor \(f_\ast : \operatorname{Mod} \mathcal O_X \to \operatorname{Mod} \mathcal O_Y\)

The inverse image of a module is, in general, not a module.

Definition. The pullback of an $\mathcal O_Y$ module $\mathcal N$ is \(f^\ast \mathcal N := f^{-1} \mathcal N \otimes_{f^{-1}\mathcal O_Y} \mathcal O_X\)

Proposition. We have an adjunction $f^\ast \vdash f_\ast$ on sheaves of modules.

Spectra of commutative rings

Now we need our local models built from commutative rings.

Definition. Let \(\operatorname{Spec} R := \lbrace \mathfrak p \lneq R \mid \mathfrak p \text{ is prime} \rbrace\) and declare subset closed if it is of the form \(V(J) := \lbrace \mathfrak p \mid J \leq \mathfrak p \rbrace\) for some ideal $J \leq R$. This topological space is the (prime) spectrum of $R$ with the Zariski topology.

Of course, one needs to check that we actually have a topology. But, we suppress that check.

Given a $a \in R$, we have a principal open \(U_a := V(a)^c\)

Lemma. The open sets $U_a$ for $a \in R$ form a basis for the Zariki topology.

Proof. (Expand to view)

Take any open set $V(J)^c$ and pick $a \in J \neq 0$. Then $$ U_a \subseteq V(J) $$

Lemma. We have an isomorphism of (locally) ringed spaces \(\operatorname{Spec} R_a \to U_a\)

Proof. (Expand to view)

The isomorphism is given by $$ \mathfrak{p} \mapsto i^{-1}\mathfrak{p} $$ for the canonical map $i: R \to R_a$. We suppress further details.

In general, not all open subsets of $\operatorname{Spec} R$ will be of the form $\operatorname{Spec} S$ for some ring $S$ but we have the following.

Lemma. We have an isomorphism \(V(J) \cong \operatorname{Spec} R/J\)

Proof. (Expand to view)

For the quotient map $\pi: R \to R/J$ we get $$ \pi^{-1} : \operatorname{Spec} R/J \to \operatorname{Spec} R $$ whose image is $V(J)$.

We need a sheaf of rings on $\operatorname{Spec} R$.

Definition. The sheaf of regular functions on $\operatorname{Spec} R$ is the unique sheaf such that \(\mathcal O(U_a) = R_a\) and for $U_a \subset U_b$ the restriction \(R_b \to R_a\) is the localization map.

Note that if $U_a \subseteq U_b$ then $V(b) \subset V(a)$. So for any prime $\mathfrak{p}$ with $(b) \leq \mathfrak{p}$ we have $(a) \leq \mathfrak{p}$. One can see this implies that $(a) \leq (b)$ so $a = bc$ giving the map \(R_b \to R_a\)

We can boost this definition to apply to $R$-modules. Given an $R$-module $M$, we let $\widetilde{M}$ be the sheaf with \(\widetilde{M}(U_a) = M_a\) with localization as the restriction maps as before.

We say locally ringed space $X$ is affine if \(X \cong \operatorname{Spec} R\) Let $\operatorname{Aff}$ denote the full category of affine locally ringed spaces.

We will not prove the following result but it is necessary for the utility of the next definition.

Proposition. The functor \(\begin{aligned} \Gamma : \operatorname{Aff}^{op} & \to \operatorname{CRing} \\ X & \mapsto \Gamma(X,\mathcal O_X) \end{aligned}\) is an equivalence whose inverse is taking the spectrum.

Definition. A locally ringed space is a scheme if it is locally affine.

For a scheme $X$, an $\mathcal O_X$-module $\mathcal F$ is quasi-coherent if for each $x \in X$ there is affine open $x \in U = \operatorname{Spec} R$ with $\mathcal F|_U \cong \widetilde{M}$ for an $R$-module $M$.

A quasi-coherent sheaf is coherent if it is locally finitely-presented.

We denote the categories by $\operatorname{Qcoh} X$ and $\operatorname{coh} X$.

Lemma. Let $X=\operatorname{Spec} R$ be affine. The functor \(\begin{aligned} \Gamma : \operatorname{Qcoh} X & \to \operatorname{Mod} R \\ \mathcal F & \mapsto \Gamma(X,\mathcal F) \end{aligned}\) is an equivalence whose inverse is \(M \mapsto \widetilde{M}\)

Lemma. The category $\operatorname{Qcoh} X$ is an abelian subcategory of $\operatorname{Mod} \mathcal O_X$.

Under some conditions which we will discuss later, $\operatorname{coh} X$ is an abelian subcategory.

The main actors for the remainder of the term are the following categories.

Definition. Let $X$ be a scheme. The derived category of $X$ is $D(\operatorname{Qcoh} X)$. In the case that $\operatorname{coh} X$ is abelian, the bounded derived category of $X$ $D^b(\operatorname{coh} X)$ is the full subcategory of $D(\operatorname{Qcoh} X)$ consisting of complexes $\mathcal F$ with bounded and coherent cohomology sheaves.

Example. Let $k$ be a field. The simplest affine schemes are \(\mathbb{A}^n_k := \operatorname{Spec} k[x_1,\ldots,x_n]\)

For each point $p \in k^n$, we have a maximal ideal \((x_1 - p_1,\ldots,x_n - p_n)\) so we have an inclusion \(k^n \subset \mathbb{A}^n_k\) The induced topology on $k^n$ the one where the closed sets are the zero loci of polynomials.

In the case $k= \mathbb{R}$ or $k = \mathbb{C}$ we can compare to the Euclidean (metric) topology. For the Zariski topology, any two open subsets intersect and any open subset is dense. This is not the case for Euclidean topology.

In general, the points from $k^n$ are not all the points of $\mathbb{A}^n_k$. We also have points corresponding to each prime ideal $\mathfrak{p}$ and if $\mathfrak{p} \subset \mathfrak{p}^\prime$ we have \(\mathfrak{p} \in \overline{\lbrace \mathfrak{p}^\prime \rbrace}\) and vice-versa. Thus, points in the Zariski topology are, in general, not closed and are closed only when they are maximal ideals.

Even in the case of $\mathbb{R}[x]$ we can have maximal ideals not generated by linear polynomials, $(x^2+1)$.

Given a finitely-generated $k$-algebra $R$, we can write \(R \cong k[x_1,\ldots,x_n]/I\) for some ideal $I$. Then, we can use the zero locus of $I$ in $k^n$ to think about $\operatorname{Spec} R$.

Next, we will talk about central global examples: projective spaces.