EGA Section 3: Sheaves with Values in a Category
Before diving into schemes, Grothendieck sets up machinery to handle sheaves of rings, modules, and algebras—not just sheaves of sets.
Presheaves on a Base of Opens
The practical motivation: defining the structure sheaf $\mathcal{O}$ on $\text{Spec}(A)$.
The annoying way: define $\mathcal{O}(U)$ for every open $U$. But arbitrary opens in $\text{Spec}(A)$ are complicated unions.
The good way: only define $\mathcal{O}$ on distinguished opens $D(f)$—the complement of the algebraic set $V(f)$ where $f$ vanishes. Set:
\[\mathcal{O}(D(f)) = A_f\]Check the sheaf condition only for covers of $D(f)$ by other distinguished opens. Then extend automatically.
This is why Section 3.2 exists. It’s the theorem that lets us define $\mathcal{O}_{\text{Spec}(A)}$ without pain.
Sheaves of Topological Rings
For ordinary sheaves, the condition is: sections that agree on overlaps glue uniquely.
For sheaves of topological rings (3.1.4), there’s an additional requirement: the topology on $\mathcal{F}(U)$ must be the coarsest making all restrictions $\mathcal{F}(U) \to \mathcal{F}(U_\alpha)$ continuous.
Why “coarsest”? If we just required restrictions to be continuous, we could take the discrete topology on everything—every map is continuous when the domain is discrete. But that destroys all topological structure.
By requiring the coarsest such topology, Grothendieck pins down a unique topology on global sections.
Example: Continuous Functions on $[0,1]$
Consider continuous functions on $[0,1]$ with two candidate topologies:
- Pointwise convergence: $f_n \to f$ iff $f_n(x) \to f(x)$ for each $x$
-
Uniform convergence: $f_n \to f$ iff $\sup_x f_n(x) - f(x) \to 0$
Pointwise is coarser (fewer open sets, more sequences converge).
Define $f_n(x) = nx(1-x)^n$.
Pointwise: $f_n \to 0$. At any fixed $x \in (0,1]$, the exponential decay $(1-x)^n$ beats the linear growth $n$.
Uniformly: $f_n \not\to 0$. The maximum of $f_n$ occurs at $x = \frac{1}{n+1}$, and:
\[f_n\left(\frac{1}{n+1}\right) \to e^{-1} \neq 0\]Each $f_n$ is a bump that gets narrower and moves toward $0$, but maintains height $\approx 0.37$.
If we tried to use pointwise convergence on $\mathcal{F}([0,1])$, restrictions wouldn’t be continuous—this fails to be a sheaf of topological rings. The coarsest topology making restrictions continuous turns out to be uniform convergence.
Gluing Sheaves
Given an open cover $(U_\lambda)$ of $X$, sheaves $\mathcal{F}\lambda$ on each piece, and isomorphisms $\theta{\lambda\mu}$ on overlaps satisfying the cocycle condition:
\[\theta_{\lambda\nu} = \theta_{\lambda\mu} \circ \theta_{\mu\nu}\]on triple overlaps, there exists a unique global sheaf $\mathcal{F}$ restricting to each $\mathcal{F}_\lambda$.
This is how we build $\mathbb{P}^n$ from affine pieces, or any scheme by gluing.
Direct and Inverse Images
For a morphism $f : X \to Y$:
Direct image: $(f_*\mathcal{F})(V) = \mathcal{F}(f^{-1}(V))$
Inverse image: $(f^*\mathcal{G})x = \mathcal{G}{f(x)}$ on stalks.
These are adjoint: $\text{Hom}X(f^*\mathcal{G}, \mathcal{F}) \cong \text{Hom}_Y(\mathcal{G}, f*\mathcal{F})$.
A morphism of schemes $f : X \to Y$ includes a map $f^*\mathcal{O}Y \to \mathcal{O}_X$, giving local ring maps $\mathcal{O}{Y,f(x)} \to \mathcal{O}_{X,x}$ at each point.