Fix an abelian group $A$. Define $\mathcal{F}^{\mathrm{pre}}(U) = A$ for every nonempty open $U$, with all restriction maps $\operatorname{id}_A$. This is a presheaf but not a sheaf in general (it fails gluing when $U$ is disconnected).

Let $X$ be a topological space. Define $\mathcal{C}(U) = \{f : U \to \mathbb{R} \mid f \text{ continuous}\}$, with restriction maps being literal restriction of functions. This is a presheaf (in fact a sheaf, as we'll verify shortly).

Let $X = \mathbb{R}$. Define $\mathcal{F}(U) = \{f : U \to \mathbb{R} \mid f \text{ continuous and bounded}\}$. This is a presheaf under restriction. But it is not a sheaf: the function $f(x) = x$ is locally bounded on every bounded open interval, but not globally bounded on $\mathbb{R}$. So gluing fails.

$\mathcal{C}(U) = C^0(U, \mathbb{R})$ is a sheaf on any topological space $X$:

• Identity: If $f|_{U_i} = g|_{U_i}$ for all $i$, then $f(x) = g(x)$ for all $x \in U$ since the $U_i$ cover $U$.
• Gluing: Given compatible $f_i \in C^0(U_i)$, define $f(x) = f_i(x)$ for $x \in U_i$. Compatibility ensures this is well-defined, and the pasting lemma gives continuity.

On a smooth manifold $M$, define $C^\infty(U) = \{f : U \to \mathbb{R} \mid f \text{ smooth}\}$. This is a sheaf of $\mathbb{R}$-algebras. Similarly for $C^\omega$ (real-analytic), holomorphic functions $\mathcal{O}$ on a complex manifold, etc.

Let $X$ be an algebraic variety. The structure sheaf $\mathcal{O}_X$ assigns to each open $U \subseteq X$ the ring of regular functions:

$\mathcal{O}_X(U) = \{f : U \to k \mid f \text{ is regular}\}.$

For an affine variety $X = V(I) \subseteq \mathbb{A}^n$, a function $f$ is regular on $U$ if it is locally a ratio $g/h$ of polynomials with $h \neq 0$. This is the central sheaf of classical algebraic geometry.

The constant sheaf $\underline{A}$ on $X$ assigns to each open $U$:

$\underline{A}(U) = \{f : U \to A \mid f \text{ locally constant}\} = A^{\pi_0(U)}$

where $\pi_0(U)$ is the set of connected components of $U$. Unlike the constant presheaf, this is a genuine sheaf: if $U = U_1 \sqcup U_2$ is disconnected, then $\underline{A}(U) = A \times A$, not $A$.

Over an irreducible variety (where every open is connected), $\underline{A}(U) = A$ for all nonempty $U$, so the constant presheaf and constant sheaf coincide.

Fix a point $p \in X$ and an abelian group $A$. The skyscraper sheaf $i_{p*}A$ is defined by:

$i_{p*}A(U) = \begin{cases} A & \text{if } p \in U, \\ 0 & \text{if } p \notin U. \end{cases}$

This is a sheaf (check!). It is supported at the single point $p$. In the derived category, skyscraper sheaves generate a useful class of objects.

For the sheaf $\mathcal{C}$ of continuous functions on $\mathbb{R}$, the stalk $\mathcal{C}_p$ is the ring of germs of continuous functions at $p$. Two functions $f$ and $g$ represent the same germ at $p$ if they agree on some neighborhood of $p$.

The germ of $|x|$ at $0$ is different from the germ of $x$ at $0$ (they differ on every punctured neighborhood), but the germ of $|x|$ at $1$ equals the germ of $x$ at $1$.

For an affine variety $X$ with coordinate ring $A = k[X]$, the stalk of $\mathcal{O}_X$ at a point $p$ (corresponding to the maximal ideal $\mathfrak{m}_p$) is the local ring:

$\mathcal{O}_{X,p} = A_{\mathfrak{m}_p} = \left\{\frac{f}{g} \mid f, g \in A, \ g(p) \neq 0\right\}.$

This is a local ring with maximal ideal $\mathfrak{m}_p A_{\mathfrak{m}_p}$ and residue field $k$. For $X = \mathbb{A}^1$, the stalk at $p = a$ is $k[x]_{(x-a)}$, the ring of rational functions regular at $a$.

For the constant sheaf $\underline{A}$, the stalk at every point is $\underline{A}_p = A$. For the skyscraper sheaf $i_{p*}A$:

$(i_{p*}A)_q = \begin{cases} A & \text{if } q \in \overline{\{p\}}, \\ 0 & \text{otherwise}. \end{cases}$

In a $T_1$ space (e.g., a Hausdorff space), $\overline{\{p\}} = \{p\}$, so the stalk is $A$ only at $p$ itself. But in the Zariski topology, $\overline{\{p\}}$ can be much larger (the whole space if $p$ is the generic point!).

The sheafification of the constant presheaf $U \mapsto A$ is the constant sheaf $\underline{A}$ (locally constant functions to $A$). On a connected space, they agree on sections over connected opens, but differ on disconnected opens: $A^{\mathrm{pre}}(U_1 \sqcup U_2) = A$ while $\underline{A}(U_1 \sqcup U_2) = A \times A$.

The sheafification of the presheaf of bounded continuous functions on $\mathbb{R}$ is the sheaf of all continuous functions, since every continuous function is locally bounded.

On a smooth manifold $M$, differentiation $d : C^\infty \to \Omega^1$ is a morphism of sheaves: for each open $U$, $d_U : C^\infty(U) \to \Omega^1(U)$ sends $f \mapsto df$, and this commutes with restriction (differentiation is a local operation).

The kernel sheaf is $\ker(d) = \underline{\mathbb{R}}$ (locally constant functions). The image presheaf (exact $1$-forms) is already a sheaf. This is the beginning of the de Rham complex $0 \to \underline{\mathbb{R}} \to C^\infty \xrightarrow{d} \Omega^1 \xrightarrow{d} \Omega^2 \to \cdots$.

On a complex manifold $X$, the exponential sequence is an exact sequence of sheaves:

$0 \to \underline{2\pi i\mathbb{Z}} \hookrightarrow \mathcal{O}_X \xrightarrow{\exp} \mathcal{O}_X^* \to 0$

where $\mathcal{O}_X^*$ is the sheaf of nowhere-vanishing holomorphic functions. This is exact on stalks (every nonzero germ has a local logarithm). The long exact sequence in cohomology gives:

$\cdots \to H^1(X, \mathcal{O}_X) \to H^1(X, \mathcal{O}_X^*) \xrightarrow{c_1} H^2(X, \mathbb{Z}) \to \cdots$

Here $H^1(X, \mathcal{O}_X^*) \cong \operatorname{Pic}(X)$ (the Picard group of line bundles) and $c_1$ is the first Chern class. For $X = \mathbb{P}^1(\mathbb{C})$: $\operatorname{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$, generated by $\mathcal{O}(1)$.

On $X = \mathbb{C}^*$, consider $\exp : \mathcal{O} \to \mathcal{O}^*$. On small discs $U$, the exponential map is surjective onto $\mathcal{O}^*(U)$ (every nonvanishing holomorphic function on a disc has a logarithm). But $\exp : \mathcal{O}(\mathbb{C}^*) \to \mathcal{O}^*(\mathbb{C}^*)$ is not surjective: $z \in \mathcal{O}^*(\mathbb{C}^*)$ has no global logarithm. The image presheaf assigns $\mathcal{O}^*(U)$ to small discs but not to $\mathbb{C}^*$ itself — this fails the gluing axiom.

Let $j : U \hookrightarrow X$ be an open inclusion and $\mathcal{F}$ a sheaf on $U$. Then $j_*\mathcal{F}$ is the extension by zero outside $U$... almost. Actually:

$(j_*\mathcal{F})(V) = \mathcal{F}(V \cap U).$

This extends $\mathcal{F}$ to all of $X$, but the stalks at points outside $U$ may be nonzero: $(j_*\mathcal{F})_p = \mathcal{F}_p$ if $p \in U$, but for $p \in \overline{U} \setminus U$, the stalk picks up "boundary behavior."

The true extension by zero $j_!\mathcal{F}$ has stalks $\mathcal{F}_p$ for $p \in U$ and $0$ otherwise.

Let $i : \{p\} \hookrightarrow X$ be the inclusion of a point. Then $i_*A =$ the skyscraper sheaf at $p$ with stalk $A$. This gives a conceptual explanation of skyscraper sheaves.

Let $f : X \to Y$ and $y \in Y$. The inclusion $i : \{y\} \hookrightarrow Y$ gives $i^{-1}\mathcal{G} = \mathcal{G}_y$ (the stalk). More generally, for the fiber $X_y = f^{-1}(y)$ with inclusion $i_y : X_y \hookrightarrow X$:

$i_y^{-1}\mathcal{F} = \mathcal{F}|_{X_y}$

is the restriction of $\mathcal{F}$ to the fiber. This is how sheaf cohomology varies in families.

For $A = k[x, y]$ and $X = \operatorname{Spec} A = \mathbb{A}^2$, the distinguished opens $D(f) = \{\mathfrak{p} \mid f \notin \mathfrak{p}\}$ form a basis. We define:

$\mathcal{O}_X(D(f)) = A_f = k[x, y, 1/f].$

For example:

• $\mathcal{O}_X(D(x)) = k[x, y, 1/x]$ — functions that can have poles along $\{x = 0\}$.
• $\mathcal{O}_X(D(xy)) = k[x, y, 1/(xy)]$ — functions with poles along $\{x = 0\} \cup \{y = 0\}$.
• $\mathcal{O}_X(X) = \mathcal{O}_X(D(1)) = A$ — global regular functions.

The restriction $\mathcal{O}_X(D(f)) \to \mathcal{O}_X(D(fg))$ is the natural localization map $A_f \to A_{fg}$.

On $\mathbb{P}^n$, the twisting sheaf $\mathcal{O}(1)$ is the tautological line bundle dual. Its sections over the standard open $U_i = \{x_i \neq 0\}$ are fractions of degree $1$. The global sections are:

$\Gamma(\mathbb{P}^n, \mathcal{O}(d)) = \begin{cases} k[x_0, \ldots, x_n]_d & \text{if } d \geq 0, \\ 0 & \text{if } d < 0, \end{cases}$

where $k[x_0,\ldots,x_n]_d$ is the space of homogeneous polynomials of degree $d$. In particular:

• $\dim \Gamma(\mathbb{P}^n, \mathcal{O}(d)) = \binom{n+d}{d}$ for $d \geq 0$.
• $\Gamma(\mathbb{P}^n, \mathcal{O}(0)) = k$ (global regular functions are constant).
• $\operatorname{Pic}(\mathbb{P}^n) \cong \mathbb{Z}$, generated by $\mathcal{O}(1)$.

On a smooth variety $X$ of dimension $n$, the cotangent sheaf (or sheaf of Kähler differentials) $\Omega^1_{X/k}$ is a locally free sheaf of rank $n$. Its top exterior power is the canonical sheaf:

$\omega_X = \bigwedge^n \Omega^1_{X/k}.$

For $\mathbb{P}^n$, the Euler sequence $0 \to \Omega^1_{\mathbb{P}^n} \to \mathcal{O}(-1)^{n+1} \to \mathcal{O} \to 0$ gives $\omega_{\mathbb{P}^n} = \mathcal{O}(-(n+1))$.

For a smooth hypersurface $X = V(f) \subseteq \mathbb{P}^n$ of degree $d$: $\omega_X = \mathcal{O}_X(d - n - 1)$ (adjunction formula).

On $\mathbb{P}^1$, the tangent sheaf $\mathcal{T}_{\mathbb{P}^1} = (\Omega^1)^\vee \cong \mathcal{O}(2)$. So $\dim \Gamma(\mathbb{P}^1, \mathcal{T}) = 3$, corresponding to the $3$-dimensional Lie algebra $\mathfrak{sl}_2$ (infinitesimal automorphisms). Indeed $\operatorname{Aut}(\mathbb{P}^1) = PGL(2)$ has dimension $3$.

For genus $g \geq 2$ curves: $\deg \mathcal{T}_C = 2 - 2g < 0$, so $\Gamma(C, \mathcal{T}_C) = 0$ — the automorphism group is finite!

Grothendieck generalized sheaves to work with sites (categories with a Grothendieck topology). Key examples:

• Étale site $X_{\mathrm{ét}}$: covers are surjective families of étale morphisms. Sheaves on this site give étale cohomology $H^i_{\mathrm{ét}}(X, \mathcal{F})$, which works in characteristic $p$ (unlike singular cohomology). Deligne's proof of the Weil conjectures uses étale cohomology.

• fppf site: covers are faithfully flat, finitely presented morphisms. Needed for non-smooth group schemes.

• Zariski site: the usual Zariski topology. Coarsest but most accessible.

The relationship: Zariski $\subseteq$ Nisnevich $\subseteq$ étale $\subseteq$ fppf $\subseteq$ fpqc. Finer topologies see more information but are harder to compute with.

On a smooth manifold $M$:

| Sheaf | Sections over $U$ | Type | |---|---|---| | $C^\infty$ | smooth functions $U \to \mathbb{R}$ | sheaf of $\mathbb{R}$-algebras | | $\Omega^p$ | smooth $p$-forms on $U$ | sheaf of $C^\infty$-modules | | $\mathcal{E}$ (vector bundle) | smooth sections of $E|_U$ | locally free $C^\infty$-module | | $\underline{\mathbb{R}}$ | locally constant functions | constant sheaf |

The de Rham theorem: $H^i_{\mathrm{dR}}(M) \cong H^i(M, \underline{\mathbb{R}})$ — de Rham cohomology equals sheaf cohomology of the constant sheaf. This is a consequence of the Poincaré lemma (the de Rham complex is a resolution of $\underline{\mathbb{R}}$).