For any topological space $X$ and ring $R$, the constant sheaf $\underline{R}$ (or $R_X$) forms a ringed space $(X, \underline{R})$.

For a connected open set $U$, $\underline{R}(U) = R$. For disconnected $U = \bigsqcup_i U_i$, we have $\underline{R}(U) = \prod_i R$.

Specific case: $(X, \underline{\mathbb{Z}})$ where $X$ is any topological space. This is the most basic ringed space structure.

Let $X$ be a topological space. Define $\mathcal{C}_X$ by: $\mathcal{C}_X(U) = \{\text{continuous functions } f: U \to \mathbb{R}\}$

Then $(X, \mathcal{C}_X)$ is a ringed space, where ring operations are pointwise addition and multiplication.

Key observation: This shows that ringed spaces generalize the notion of spaces with continuous real-valued functions.

Let $M$ be a smooth manifold. Define the sheaf $\mathcal{C}^\infty_M$ by: $\mathcal{C}^\infty_M(U) = C^\infty(U, \mathbb{R}) = \{\text{smooth functions } f: U \to \mathbb{R}\}$

Then $(M, \mathcal{C}^\infty_M)$ is a ringed space. This is the standard structure for differential geometry.

Stalks: The stalk $\mathcal{C}^\infty_{M,p}$ consists of germs of smooth functions at $p$.

Let $X \subseteq \mathbb{C}^n$ be a domain. Define $\mathcal{O}_X$ by: $\mathcal{O}_X(U) = \{\text{holomorphic functions } f: U \to \mathbb{C}\}$

Then $(X, \mathcal{O}_X)$ is a ringed space. This is the basic structure of complex analytic geometry.

Note: The ring $\mathcal{O}_X(U)$ is actually a $\mathbb{C}$-algebra, reflecting the richer structure of holomorphic functions.

Let $X \subseteq \mathbb{A}^n_k$ be an affine variety over an algebraically closed field $k$. For open $U \subseteq X$, define: $\mathcal{O}_X(U) = \{f: U \to k \mid f \text{ is regular}\}$

where $f$ is regular if locally it can be written as $p/q$ with $p, q$ polynomials and $q$ non-vanishing.

Then $(X, \mathcal{O}_X)$ is a ringed space. This is the classical structure sheaf from variety theory.

Let $(X, \mathcal{O}_X)$ be a ringed space and $U \subseteq X$ open. Define $(U, \mathcal{O}_X|_U)$ with the restriction sheaf.

The inclusion $i: U \hookrightarrow X$ extends to a morphism of ringed spaces $(i, i^\sharp)$ where: $i^\sharp: \mathcal{O}_X \to i_* \mathcal{O}_X|_U$

is given by restriction: for open $V \subseteq X$, $i^\sharp_V: \mathcal{O}_X(V) \to \mathcal{O}_X(V \cap U)$ is the natural restriction map.

Let $(X, \mathcal{O}_X)$ be any ringed space and consider the one-point space $\{\text{pt}\}$ with structure sheaf $\mathcal{O}_{\text{pt}}(\{\text{pt}\}) = R$ for some ring $R$.

A morphism $(f, f^\sharp): (X, \mathcal{O}_X) \to (\{\text{pt}\}, R)$ consists of:

• The unique continuous map $f: X \to \{\text{pt}\}$
• A ring homomorphism $f^\sharp: R \to \mathcal{O}_X(X)$

Interpretation: This is the same as giving $\mathcal{O}_X(X)$ an $R$-algebra structure.

Let $M, N$ be smooth manifolds with structure sheaves $\mathcal{C}^\infty_M, \mathcal{C}^\infty_N$.

A smooth map $f: M \to N$ induces a morphism of ringed spaces $(f, f^\sharp)$ where: $f^\sharp: \mathcal{C}^\infty_N(V) \to \mathcal{C}^\infty_M(f^{-1}(V))$ $\phi \mapsto \phi \circ f$

This is the standard pullback of smooth functions.

Key point: Every smooth map of manifolds is naturally a morphism of ringed spaces.

Let $X \subseteq \mathbb{A}^n$ and $Y \subseteq \mathbb{A}^m$ be affine varieties with structure sheaves.

A regular map $f: X \to Y$ (given by polynomials) induces a morphism of ringed spaces where $f^\sharp$ pulls back regular functions via composition.

This shows that classical regular maps of varieties are morphisms of ringed spaces.

Let $M$ be a smooth manifold with $\mathcal{C}^\infty_M$ the sheaf of smooth functions.

Claim: $(M, \mathcal{C}^\infty_M)$ is a locally ringed space.

Proof: For $p \in M$, the stalk $\mathcal{C}^\infty_{M,p}$ consists of germs of smooth functions at $p$. The maximal ideal is: $\mathfrak{m}_p = \{[f] \in \mathcal{C}^\infty_{M,p} : f(p) = 0\}$

A germ $[f]$ is a unit if and only if $f(p) \neq 0$ (by the inverse function theorem in a suitable sense). Thus $\mathcal{C}^\infty_{M,p}$ has a unique maximal ideal.

The residue field is $\mathcal{C}^\infty_{M,p}/\mathfrak{m}_p \cong \mathbb{R}$.

Let $X$ be a complex analytic space with structure sheaf $\mathcal{O}_X$ of holomorphic functions.

Then $(X, \mathcal{O}_X)$ is a locally ringed space. For $p \in X$:

• The stalk $\mathcal{O}_{X,p}$ consists of germs of holomorphic functions
• The maximal ideal is $\mathfrak{m}_p = \{f: f(p) = 0\}$
• A germ is a unit iff it doesn't vanish at $p$
• The residue field is $\kappa(p) \cong \mathbb{C}$

Let $X$ be an algebraic variety over $k$ with structure sheaf $\mathcal{O}_X$ of regular functions.

Then $(X, \mathcal{O}_X)$ is a locally ringed space. For $x \in X$:

• The stalk $\mathcal{O}_{X,x}$ consists of germs of regular functions (locally quotients of polynomials)
• The maximal ideal is $\mathfrak{m}_x = \{f: f(x) = 0\}$
• A regular function is invertible near $x$ iff $f(x) \neq 0$
• The residue field is $\kappa(x) = k$ when $k$ is algebraically closed

This is the classical structure that makes varieties into locally ringed spaces.

Let $A$ be a ring. The spectrum $\text{Spec } A$ with structure sheaf $\mathcal{O}_{\text{Spec } A}$ is a locally ringed space.

Construction: For a prime ideal $\mathfrak{p} \in \text{Spec } A$, the stalk is: $\mathcal{O}_{\text{Spec } A, \mathfrak{p}} = A_\mathfrak{p}$

the localization of $A$ at $\mathfrak{p}$.

Local ring structure: The ring $A_\mathfrak{p}$ is local with unique maximal ideal: $\mathfrak{m}_\mathfrak{p} = \mathfrak{p} A_\mathfrak{p} = \left\{\frac{a}{s} : a \in \mathfrak{p}, s \notin \mathfrak{p}\right\}$

The residue field is $\kappa(\mathfrak{p}) = A_\mathfrak{p}/\mathfrak{p} A_\mathfrak{p} = \text{Frac}(A/\mathfrak{p})$.

Why this is crucial: This is the fundamental example that motivates the entire theory of schemes. The locally ringed space structure on $\text{Spec } A$ is what makes schemes work.

Let $X$ be a topological space with at least two points. The ringed space $(X, \underline{\mathbb{Z}})$ with constant sheaf is not locally ringed.

Reason: For any $x \in X$, the stalk is: $\underline{\mathbb{Z}}_x = \mathbb{Z}$

But $\mathbb{Z}$ is not a local ring (it has distinct maximal ideals $(2), (3), (5), \ldots$).

This shows that not every ringed space is locally ringed.

Let $f: M \to N$ be a smooth map of manifolds. We showed earlier that $f$ induces a morphism of ringed spaces.

Claim: This is actually a morphism of locally ringed spaces.

Proof: For $p \in M$, we need to show $f^\sharp_p: \mathcal{C}^\infty_{N,f(p)} \to \mathcal{C}^\infty_{M,p}$ is local.

If $\phi \in \mathfrak{m}_{f(p)}$, meaning $\phi(f(p)) = 0$, then: $f^\sharp_p(\phi) = \phi \circ f$ and $(\phi \circ f)(p) = \phi(f(p)) = 0$, so $f^\sharp_p(\phi) \in \mathfrak{m}_p$.

Thus every smooth map is a morphism of locally ringed spaces.

Let $\phi: A \to B$ be a ring homomorphism. This induces a continuous map: $f = {}^a\phi: \text{Spec } B \to \text{Spec } A$ $\mathfrak{q} \mapsto \phi^{-1}(\mathfrak{q})$

and a comorphism $f^\sharp: \mathcal{O}_{\text{Spec } A} \to f_* \mathcal{O}_{\text{Spec } B}$.

Claim: This is a morphism of locally ringed spaces.

Proof: For $\mathfrak{q} \in \text{Spec } B$ with $\mathfrak{p} = \phi^{-1}(\mathfrak{q})$, the stalk map is: $f^\sharp_\mathfrak{q}: A_\mathfrak{p} \to B_\mathfrak{q}$

This is local because $\mathfrak{p} A_\mathfrak{p}$ maps into $\mathfrak{q} B_\mathfrak{q}$ (preimage of the maximal ideal is the maximal ideal).

This is the key construction that makes schemes into locally ringed spaces.

Consider $X = \{x\}$ with $\mathcal{O}_X(\{x\}) = \mathbb{R}$, and $Y = \{y\}$ with $\mathcal{O}_Y(\{y\}) = \mathbb{C}$.

As ringed spaces with constant sheaves, both are locally ringed spaces (with maximal ideals $(0)$ in the stalks... wait, actually $\mathbb{R}$ and $\mathbb{C}$ are fields, so they are local rings with maximal ideal $(0)$).

A morphism of ringed spaces is given by the unique map $f: X \to Y$ and a ring homomorphism $\mathbb{C} \to \mathbb{R}$.

But there is no ring homomorphism $\mathbb{C} \to \mathbb{R}$! (Because $i^2 = -1$ would need to map to an element whose square is $-1$, but no such real exists.)

So while both are locally ringed spaces, there is no morphism between them as locally ringed spaces.

The locally ringed space condition makes evaluation well-defined. Let $(X, \mathcal{O}_X)$ be a locally ringed space and $x \in X$.

For $f \in \mathcal{O}_{X,x}$ (a germ at $x$), we can "evaluate" $f$ at $x$ via the residue field map: $\text{ev}_x: \mathcal{O}_{X,x} \to \mathcal{O}_{X,x}/\mathfrak{m}_x = \kappa(x)$

Geometric interpretation:

• If $f \in \mathfrak{m}_x$, then $f(x) = 0$ in $\kappa(x)$ (the function vanishes)
• If $f \notin \mathfrak{m}_x$, then $f$ is a unit, meaning $f(x) \neq 0$ (the function is non-zero)

This is why we can talk about "the value of a function at a point" in the generalized setting.

The category of affine schemes AffSch is (equivalent to) the opposite category of rings: $\text{AffSch} \simeq \text{Ring}^\text{op}$

via the correspondence: $A \mapsto (\text{Spec } A, \mathcal{O}_{\text{Spec } A})$ $\phi: A \to B \mapsto ({}^a\phi, {}^a\phi^\sharp): \text{Spec } B \to \text{Spec } A$

This embeds fully and faithfully into LRSp. Every morphism of affine schemes is a morphism of locally ringed spaces.

The category of smooth manifolds Man embeds into LRSp via: $M \mapsto (M, \mathcal{C}^\infty_M)$

with smooth maps becoming morphisms of locally ringed spaces.

However, this embedding is not full: there exist morphisms of locally ringed spaces between manifolds that are not smooth maps (they might only be continuous, for instance).

A scheme is a locally ringed space $(X, \mathcal{O}_X)$ such that every point has an open neighborhood $U$ with $(U, \mathcal{O}_X|_U)$ isomorphic (as a locally ringed space) to $(\text{Spec } A, \mathcal{O}_{\text{Spec } A})$ for some ring $A$.

The locally ringed space structure is essential to this definition:

• It ensures that schemes can be glued from affine pieces
• It makes morphisms of schemes behave geometrically
• It allows us to define closed subschemes, fiber products, etc.

Without the locally ringed condition, the theory would not work.

For a scheme $X$ and a point $x \in X$, the residue field $\kappa(x) = \mathcal{O}_{X,x}/\mathfrak{m}_x$ encodes crucial geometric information:

• For $X = \text{Spec } k[t]$ over a field $k$:

  • At a closed point corresponding to $(t-a)$, we have $\kappa(a) = k$
  • At the generic point $(0)$, we have $\kappa(\eta) = k(t)$ (rational functions)

• For $X = \text{Spec } \mathbb{Z}$:

  • At $(p)$, we have $\kappa(p) = \mathbb{F}_p$
  • At $(0)$, we have $\kappa((0)) = \mathbb{Q}$

The locally ringed space structure makes these residue fields well-defined and functorial.

In a locally ringed space $(X, \mathcal{O}_X)$, a section $f \in \mathcal{O}_X(U)$ is a unit in $\mathcal{O}_X(U)$ if and only if its image in every stalk $\mathcal{O}_{X,x}$ for $x \in U$ is a unit.

By the local ring structure, $f_x \in \mathcal{O}_{X,x}$ is a unit if and only if $f_x \notin \mathfrak{m}_x$, i.e., $f(x) \neq 0$.

Conclusion: $f \in \mathcal{O}_X(U)$ is a unit if and only if $f$ is "non-vanishing on $U$" in the sense that $f(x) \neq 0$ for all $x \in U$.

This is a fundamental property used constantly in scheme theory.