$\mathcal{O}_X$ itself is quasi-coherent (and coherent): on $U = \operatorname{Spec} A$, it corresponds to $\widetilde{A}$, the module $A$ over itself.

For a closed subscheme $Z \hookrightarrow X$ defined by an ideal sheaf $\mathcal{I}_Z \subseteq \mathcal{O}_X$, both $\mathcal{I}_Z$ and the quotient $\mathcal{O}_Z = \mathcal{O}_X / \mathcal{I}_Z$ are quasi-coherent. On an affine open $\operatorname{Spec} A$, $\mathcal{I}_Z$ corresponds to an ideal $I \subseteq A$ and $\mathcal{O}_Z$ to $A/I$.

On $\mathbb{A}^1 = \operatorname{Spec} k[x]$, the sheaf $\mathcal{F}$ with $\mathcal{F}(\mathbb{A}^1) = k[x, 1/(x-1)]$ is quasi-coherent — it corresponds to the $k[x]$-module $k[x]_{(x-1)}$... but wait, that's a localization, not a finitely generated module. Quasi-coherent sheaves allow arbitrary modules, not just finitely generated ones.

On a Noetherian scheme $X$:

• $\mathcal{O}_X$ is coherent.
• Every ideal sheaf $\mathcal{I} \subseteq \mathcal{O}_X$ defining a closed subscheme is coherent (since $A$ is Noetherian, every ideal is finitely generated).
• $\mathcal{O}_Z$ for a closed subscheme $Z$ is coherent.

A locally free sheaf of rank $r$ (= vector bundle of rank $r$) is coherent. On each affine open, it corresponds to a free module $A^r$.

Key examples on $\mathbb{P}^n$:

• $\mathcal{O}(d)$ for $d \in \mathbb{Z}$: the twisting sheaf (rank 1, invertible).
• $\Omega^1_{\mathbb{P}^n}$: the cotangent sheaf (rank $n$).
• $\mathcal{T}_{\mathbb{P}^n}$: the tangent sheaf (rank $n$).

For a closed point $p \in X$ with residue field $k(p)$, the skyscraper sheaf $k(p)_p$ (= pushforward of $k(p)$ from $\{p\}$ to $X$) is coherent. On $\operatorname{Spec} A$ with $p = \mathfrak{m}$, it corresponds to $A/\mathfrak{m}$.

More generally, any sheaf supported on a finite set of closed points is coherent.

On $\mathbb{A}^1 = \operatorname{Spec} k[x]$:

• $\widetilde{k(x)}$ (the constant sheaf of the function field) is quasi-coherent but not coherent: $k(x)$ is not finitely generated over $k[x]$.
• $\widetilde{\bigoplus_{n \geq 0} k[x]}$ (infinite direct sum) is quasi-coherent but not coherent.

On a Noetherian scheme, infinite direct sums of coherent sheaves are quasi-coherent but typically not coherent.

On $\mathbb{P}^n$, for a coherent sheaf $\mathcal{F}$ and $d \in \mathbb{Z}$:

$\mathcal{F}(d) = \mathcal{F} \otimes \mathcal{O}(d).$

This is the twist of $\mathcal{F}$ by $d$. Key fact: for $d \gg 0$, $\mathcal{F}(d)$ is generated by global sections (Serre's theorem A). The global sections:

• $\Gamma(\mathbb{P}^n, \mathcal{O}(d)) = k[x_0, \ldots, x_n]_d$ for $d \geq 0$
• $\Gamma(\mathbb{P}^n, \mathcal{O}(d)) = 0$ for $d < 0$

On $\mathbb{P}^n$, there is a fundamental short exact sequence of locally free sheaves:

$0 \to \Omega^1_{\mathbb{P}^n} \to \mathcal{O}(-1)^{n+1} \to \mathcal{O} \to 0.$

Dualizing: $0 \to \mathcal{O} \to \mathcal{O}(1)^{n+1} \to \mathcal{T}_{\mathbb{P}^n} \to 0$.

Taking determinants: $\omega_{\mathbb{P}^n} = \det \Omega^1 = \mathcal{O}(-(n+1))$.

This single sequence encodes the geometry of projective space: its tangent bundle, canonical class, and Chern classes.

For a closed subscheme $Z \hookrightarrow X$ with ideal sheaf $\mathcal{I}_Z$:

$0 \to \mathcal{I}_Z \to \mathcal{O}_X \to \mathcal{O}_Z \to 0.$

This is exact as a sequence of coherent sheaves. Taking cohomology gives the long exact sequence:

$0 \to H^0(X, \mathcal{I}_Z) \to H^0(X, \mathcal{O}_X) \to H^0(Z, \mathcal{O}_Z) \to H^1(X, \mathcal{I}_Z) \to \cdots$

which is the primary tool for computing cohomology by induction on dimension.

On $\mathbb{P}^1$, every coherent sheaf $\mathcal{F}$ splits as a direct sum:

$\mathcal{F} \cong \mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_r) \oplus \mathcal{T}$

where $a_1 \geq \cdots \geq a_r$ and $\mathcal{T}$ is a torsion sheaf (supported on finitely many points). This is Grothendieck's theorem (1957).

For vector bundles (locally free sheaves), every vector bundle on $\mathbb{P}^1$ splits as $\bigoplus \mathcal{O}(a_i)$. The integers $a_1 \geq \cdots \geq a_r$ are uniquely determined.

This fails spectacularly on $\mathbb{P}^n$ for $n \geq 2$: there exist indecomposable vector bundles of every rank $\geq 2$.

The tangent bundle $\mathcal{T}_{\mathbb{P}^2}$ is an indecomposable rank-2 vector bundle. From the Euler sequence:

$0 \to \mathcal{O} \to \mathcal{O}(1)^3 \to \mathcal{T}_{\mathbb{P}^2} \to 0.$

Since $\mathcal{T}_{\mathbb{P}^2}$ has rank 2 and $c_1 = 3$, if it split as $\mathcal{O}(a) \oplus \mathcal{O}(b)$ with $a + b = 3$, then $h^0(\mathcal{T}) = h^0(\mathcal{O}(a)) + h^0(\mathcal{O}(b))$. From the Euler sequence, $h^0(\mathcal{T}_{\mathbb{P}^2}) = 8$ (the dimension of $\operatorname{Aut}(\mathbb{P}^2) = PGL(3)$). No splitting $\mathcal{O}(a) \oplus \mathcal{O}(b)$ gives $h^0 = 8$.

Let $C = V(F) \subseteq \mathbb{P}^2$ be a smooth curve of degree $d$. The ideal sheaf sequence:

$0 \to \mathcal{O}_{\mathbb{P}^2}(-d) \xrightarrow{\cdot F} \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_C \to 0.$

Taking global sections: $H^0(\mathbb{P}^2, \mathcal{O}_C) = k$ (the curve is connected). The long exact sequence gives:

$h^0(\mathcal{O}_C) = 1, \quad h^1(\mathcal{O}_C) = \binom{d-1}{2} = \frac{(d-1)(d-2)}{2} = g(C).$

So the arithmetic genus of a degree-$d$ plane curve is $g = \binom{d-1}{2}$. For $d = 3$: $g = 1$ (elliptic curve).

The bounded derived category $D^b(\mathbf{Coh}(X))$ is one of the most powerful invariants of $X$. Two smooth projective varieties $X$ and $Y$ are derived equivalent if $D^b(\mathbf{Coh}(X)) \cong D^b(\mathbf{Coh}(Y))$.

Bondal–Orlov theorem: if $X$ is smooth projective with $\omega_X$ ample or anti-ample, then $X$ is determined by $D^b(\mathbf{Coh}(X))$. But derived equivalence can hold for non-isomorphic varieties:

• An abelian variety $A$ and its dual $A^\vee$ are derived equivalent (Mukai).
• Certain K3 surfaces can be derived equivalent without being isomorphic.