Consider the CFG of vector bundles on a scheme $S$, with a Zariski cover $\{U_i\}$ of $S$. A descent datum consists of:

• Vector bundles $\mathcal{E}_i$ on each $U_i$.
• Isomorphisms $\varphi_{ij} : \mathcal{E}_i|_{U_{ij}} \xrightarrow{\sim} \mathcal{E}_j|_{U_{ij}}$ on overlaps (transition functions).
• Cocycle condition $\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$ on triple overlaps.

This is exactly the classical transition-function description of a vector bundle! The descent datum glues to a global vector bundle $\mathcal{E}$ on $S$ with $\mathcal{E}|_{U_i} \cong \mathcal{E}_i$.

For line bundles on $S$ with cover $\{U_i\}$, the transition functions $\varphi_{ij} \in \mathcal{O}^*(U_{ij})$ satisfy the Cech cocycle condition $\varphi_{ik} = \varphi_{jk} \cdot \varphi_{ij}$. This is a Cech 1-cocycle with values in $\mathbb{G}_m$, and two descent data give isomorphic line bundles if and only if the cocycles differ by a coboundary. So $\operatorname{Pic}(S) \cong \check{H}^1(S, \mathbb{G}_m)$.

Consider finite etale covers of a scheme $S$ with the etale topology. A descent datum for the covering $T \to S$ consists of a finite etale $X \to T$ together with an isomorphism $\varphi : \operatorname{pr}_1^*X \xrightarrow{\sim} \operatorname{pr}_2^*X$ over $T \times_S T$ satisfying the cocycle condition over $T \times_S T \times_S T$. By faithfully flat descent (Grothendieck), such descent data are effective: they glue to a unique finite etale cover of $S$.

Any sheaf $F$ on $(\mathcal{C}, \tau)$ defines a stack: view $F(U)$ as a discrete groupoid (set). Axiom 1 holds because $\operatorname{Isom}(x, y) = \emptyset$ if $x \neq y$ and $\{*\}$ if $x = y$ (or rather, equality is detected locally). Axiom 2 is the gluing axiom for $F$.

In particular, any scheme $X$ (via the functor of points $h_X$) defines a stack on the etale (or fppf) site.

Let $G$ be a smooth group scheme over $S$. The CFG $BG$ (classifying $G$-torsors) is a stack on the fppf site of $\mathbf{Sch}/S$.

Axiom 1: For two $G$-torsors $P, Q$ over $T$, the presheaf $\operatorname{Isom}(P, Q)$ is a sheaf -- in fact, it is representable by the scheme $P \times^G Q^{\mathrm{op}}$ (a twisted form of $G$), so it is automatically a sheaf.

Axiom 2: Descent for torsors is effective by faithfully flat descent. Given a covering $\{T_i \to T\}$ and $G$-torsors $P_i$ on $T_i$ with compatible transition isomorphisms, we can glue to a $G$-torsor on $T$.

$BG$ is typically not a scheme: the fiber over $\operatorname{Spec} k$ ($k$ algebraically closed) has only one isomorphism class (the trivial torsor) but automorphism group $G(k)$. If $BG$ were a scheme, it would be $\operatorname{Spec} k$ with trivial automorphisms.

The CFG $\mathcal{M}_g$ of smooth genus-$g$ curves ($g \geq 2$) is a stack on the etale site.

Axiom 1: For two smooth curves $C, C'$ over $S$, the functor $\operatorname{Isom}(C, C')$ is representable by a scheme (a locally closed subscheme of the Hilbert scheme), hence a sheaf.

Axiom 2: By descent theory for morphisms of schemes, smooth proper curves can be glued from local data. This uses the fact that smooth proper morphisms satisfy effective descent for the etale (and fppf) topology.

The Deligne-Mumford theorem (1969) shows that $\mathcal{M}_g$ is a smooth, proper Deligne-Mumford stack of dimension $3g - 3$ (for $g \geq 2$).

Let $G$ be an algebraic group acting on a scheme $X$ over $S$. The quotient stack $[X/G]$ is the stack whose objects over $T$ are pairs $(P, \sigma)$ where $P$ is a $G$-torsor on $T$ and $\sigma : P \to X$ is a $G$-equivariant morphism.

Equivalently, $[X/G](T) = \{(P \to T, P \to X) : P \text{ is a } G\text{-torsor}, P \to X \text{ is } G\text{-equivariant}\}$.

Key properties:

• $[*/G] = BG$ (the classifying stack).
• $[X/\{e\}] = X$ (trivial group gives the scheme back).
• If $G$ acts freely and the quotient $X/G$ is a scheme, then $[X/G]$ is equivalent to $X/G$.
• If $G$ acts with finite stabilizers, $[X/G]$ is a Deligne-Mumford stack.
• For general $G$, $[X/G]$ is an Artin stack (algebraic stack).

Let $X$ be a scheme, $D \subset X$ an effective Cartier divisor, and $r \geq 1$ an integer. The $r$-th root stack $\sqrt[r]{D/X}$ parametrizes "$r$-th roots of $D$": objects over $T \to X$ are pairs $(\mathcal{L}, \phi)$ where $\mathcal{L}$ is a line bundle on $T$ and $\phi : \mathcal{L}^{\otimes r} \xrightarrow{\sim} \mathcal{O}_T(D|_T)$ is an isomorphism.

This is a Deligne-Mumford stack. Away from $D$, it is isomorphic to $X \setminus D$. Along $D$, it has $\mu_r$-stabilizers (automorphisms are $r$-th roots of unity acting on $\mathcal{L}$).

Root stacks appear in logarithmic geometry and orbifold theory.

For a smooth proper morphism $f : X \to S$, the Picard stack $\mathcal{P}ic_{X/S}$ parametrizes line bundles on fibers: objects over $T \to S$ are line bundles $\mathcal{L}$ on $X_T = X \times_S T$.

This is a stack (descent for line bundles is effective). The associated coarse moduli is the Picard scheme $\operatorname{Pic}_{X/S}$ (when it exists). Every object has automorphism group $\mathbb{G}_m$ (scalar multiplication on the line bundle), so $\mathcal{P}ic_{X/S}$ is a $\mathbb{G}_m$-gerbe over $\operatorname{Pic}_{X/S}$.

The functor $\operatorname{Hilb}_{X/S}$ parametrizing closed subschemes of $X \to S$ is representable by a scheme (the Hilbert scheme, when $X/S$ is projective). This is a "scheme" stack (discrete fibers, no nontrivial automorphisms).

However, if we instead parametrize closed subschemes up to some equivalence (e.g., flat families of 0-dimensional subschemes of length $n$ up to permutation), we get a stack with nontrivial automorphisms, the symmetric product stack $[\operatorname{Hilb}^n / S_n]$.

Let $G = \mu_2 = \operatorname{Spec} k[t]/(t^2 - 1)$ over a field $k$ with $\operatorname{char} k \neq 2$. The $\mu_2$-torsor $\operatorname{Spec} k[x]/(x^2 - a) \to \operatorname{Spec} k$ (for $a \in k^*$, not a square) is trivial Zariski-locally (it splits over the extension $k(\sqrt{a})$) but this extension is etale, not a Zariski open. So $B\mu_2$ is not a stack for the Zariski topology.

It is, however, a stack for the etale and fppf topologies, since $\mu_2$-torsors satisfy effective descent for these finer topologies.

For a smooth affine group scheme $G$ over $S$:

• $H^1_{\mathrm{Zar}}(S, G) \subseteq H^1_{\mathrm{et}}(S, G) \subseteq H^1_{\mathrm{fppf}}(S, G)$

These inclusions can be strict. For example, $H^1_{\mathrm{Zar}}(\operatorname{Spec} \mathbb{R}, \mu_2) = 0$ (every $\mu_2$-torsor over $\operatorname{Spec} \mathbb{R}$ is Zariski-trivial? No -- there are none nontrivial Zariski-locally, but $\operatorname{Spec} \mathbb{C} \to \operatorname{Spec} \mathbb{R}$ is a nontrivial etale $\mu_2$-torsor).

Affine morphisms satisfy effective descent: if $T \to S$ is faithfully flat and $X \to T$ is affine, and if we have descent data $(X, \varphi)$ where $\varphi : \operatorname{pr}_1^*X \cong \operatorname{pr}_2^*X$ over $T \times_S T$ satisfies the cocycle condition, then there exists a unique (up to canonical isomorphism) affine morphism $Y \to S$ with $Y \times_S T \cong X$.

This follows from the equivalence between affine morphisms $X \to T$ and quasi-coherent $\mathcal{O}_T$-algebras, combined with faithfully flat descent for quasi-coherent sheaves.

Descent is not always effective for the Zariski topology on non-separated schemes. Consider gluing two copies of $\mathbb{A}^1$ along $\mathbb{A}^1 \setminus \{0\}$ via the identity -- this gives the non-separated "line with doubled origin." But this scheme does not arise from a separated scheme over a point.

More precisely, there exist descent data for the Zariski topology that are not effective when the objects are not quasi-projective. The fppf and etale topologies have much stronger effectivity results.