When $\mathcal{F}$ is a sheaf of sets $F$ (viewed as a CFG with discrete fibers), a descent datum is:

• Elements $s_i \in F(U_i)$ satisfying $s_i|_{U_{ij}} = s_j|_{U_{ij}}$ (the cocycle condition reduces to equality since the isomorphisms in discrete groupoids are identities).

A morphism of descent data requires $s_i = t_i$ for all $i$ (identity is the only isomorphism). So the category of descent data is a set (discrete groupoid), and the stack condition reduces to the usual sheaf condition: $F(U) \xrightarrow{\sim} \operatorname{eq}\left(\prod_i F(U_i) \rightrightarrows \prod_{i,j} F(U_{ij})\right)$

Let $S$ be a scheme with Zariski cover $\{U_i\}$. A descent datum for rank-$n$ vector bundles consists of:

• $\mathcal{E}_i$ on $U_i$: these are locally free sheaves of rank $n$.
• Transition isomorphisms $g_{ij} : \mathcal{E}_i|_{U_{ij}} \xrightarrow{\sim} \mathcal{E}_j|_{U_{ij}}$.
• Cocycle: $g_{jk} \circ g_{ij} = g_{ik}$ on $U_{ijk}$.

This glues to a vector bundle $\mathcal{E}$ on $S$: take $\mathcal{E}|_{U_i} = \mathcal{E}_i$ and use $g_{ij}$ to identify on overlaps. Effectiveness is guaranteed, so $\mathbf{Vect}_n$ is a stack for the Zariski topology.

In terms of matrices: after trivializing $\mathcal{E}_i \cong \mathcal{O}_{U_i}^n$, the transition maps become $g_{ij} \in GL_n(\mathcal{O}(U_{ij}))$ satisfying $g_{jk}g_{ij} = g_{ik}$. This is a Cech 1-cocycle with values in $GL_n$.

Let $G$ be a smooth group scheme over $S$ and let $T \to S$ be a faithfully flat etale cover. A descent datum for $G$-torsors relative to $T \to S$ consists of:

• A $G_T$-torsor $P$ over $T$.
• An isomorphism $\varphi : \operatorname{pr}_1^*P \xrightarrow{\sim} \operatorname{pr}_2^*P$ over $T \times_S T$.
• Cocycle: $\operatorname{pr}_{23}^*\varphi \circ \operatorname{pr}_{12}^*\varphi = \operatorname{pr}_{13}^*\varphi$ over $T \times_S T \times_S T$.

By Grothendieck's faithfully flat descent, this descent datum is effective: it glues to a $G$-torsor $Q$ over $S$ with $Q|_T \cong P$. This proves $BG$ is a stack for the fppf (and hence etale) topology.

The CFG $\mathbf{QCoh}$ over $\mathbf{Sch}$ (with the fppf topology) is a stack.

S1 (Isom is a sheaf): For quasi-coherent sheaves $\mathcal{E}, \mathcal{F}$ on $S$, the Hom sheaf $\mathcal{H}om(\mathcal{E}, \mathcal{F})$ is quasi-coherent, hence a sheaf for any topology finer than Zariski. In particular, $\operatorname{Isom}(\mathcal{E}, \mathcal{F}) \subseteq \mathcal{H}om(\mathcal{E}, \mathcal{F})$ is a sheaf.

S2 (Effective descent): This is Grothendieck's faithfully flat descent theorem (SGA1, Expose VIII): given a faithfully flat morphism $f : T \to S$, the functor $f^* : \mathbf{QCoh}(S) \to \mathbf{QCoh}(T)$ identifies $\mathbf{QCoh}(S)$ with the category of quasi-coherent sheaves on $T$ equipped with descent data relative to $f$.

The CFG $\mathcal{M}_g$ ($g \geq 2$) is a stack for the etale topology.

S1: For two families of curves $C, C' \to S$, the functor $\operatorname{Isom}(C, C')$ is representable by a quasi-projective $S$-scheme (an open subscheme of the Hilbert scheme). Representable functors on $\mathbf{Sch}/S$ are sheaves for the etale (and fppf) topology.

S2: A smooth proper morphism of relative dimension 1 satisfies descent. Given a covering $\{S_i \to S\}$ and smooth curves $C_i \to S_i$ with compatible transition isomorphisms, the curves glue to a smooth curve $C \to S$ by descent for proper morphisms (EGA IV, or Grothendieck's general descent for morphisms of schemes).

For a smooth affine group scheme $G/S$, the CFG $BG$ is a stack for the fppf topology.

S1: $\operatorname{Isom}(P, Q)$ for $G$-torsors is representable by $P \times^G Q^{\mathrm{op}}$ (a twisted form of $G$), hence a sheaf.

S2: Faithfully flat descent for affine morphisms. A $G$-torsor $P \to T$ is an affine morphism (since $G$ is affine), and affine morphisms descend effectively along faithfully flat morphisms.

For $G$ acting on $X$ over $S$, the quotient $[X/G]$ is a stack for the fppf topology.

S1: $\operatorname{Isom}((P_1, \sigma_1), (P_2, \sigma_2))$ parametrizes $G$-equivariant isomorphisms $P_1 \to P_2$ compatible with $\sigma_1, \sigma_2$. This is a closed subscheme of $\operatorname{Isom}(P_1, P_2)$, hence representable.

S2: A descent datum for $[X/G]$ consists of torsors $P_i$ on $U_i$ with equivariant maps to $X$ and compatible transition data. Since torsors descend (by the $BG$ argument) and equivariant maps descend (morphisms of schemes descend), the full descent datum is effective.

Let $S = \operatorname{Spec} \mathbb{R}$ and $T = \operatorname{Spec} \mathbb{C}$ with $\pi : T \to S$ the natural map. The Cech nerve is:

• $T = \operatorname{Spec} \mathbb{C}$
• $T \times_S T = \operatorname{Spec}(\mathbb{C} \otimes_\mathbb{R} \mathbb{C}) \cong \operatorname{Spec}(\mathbb{C} \times \mathbb{C})$ (two points, corresponding to $\operatorname{id}$ and complex conjugation)
• $T \times_S T \times_S T = \operatorname{Spec}(\mathbb{C}^4)$ (four points)

A descent datum for this covering is: an object $x$ over $\operatorname{Spec} \mathbb{C}$ with an isomorphism $\varphi : x \xrightarrow{\sim} \bar{x}$ (where $\bar{x}$ is the conjugate) satisfying $\bar{\varphi} \circ \varphi = \operatorname{id}$. This is precisely a real form of the complex object $x$.

For the fppf topology, any covering family $\{U_i \to U\}$ can be refined to a single covering $V = \coprod U_i \to U$. So it suffices to check the stack condition for single covering morphisms $V \to U$. This simplifies computations.

For algebraic applications, the most useful criterion is:

A CFG $\mathcal{F}$ over $\mathbf{Sch}$ (with the fppf topology) is a stack if:

1. For all $S$ and $x, y \in \mathcal{F}(S)$, $\operatorname{Isom}(x, y)$ is representable by an algebraic space (or scheme) over $S$. [This gives S1.]
2. The objects parametrized by $\mathcal{F}$ satisfy effective fppf descent. [This gives S2.]

Condition (1) is typically checked using the theory of Hilbert schemes or Artin's criteria. Condition (2) follows from descent theorems for specific types of objects (quasi-coherent sheaves, morphisms of schemes, etc.).

To verify that a CFG $\mathcal{F}$ is a prestack (condition S1), it often suffices to show that $\operatorname{Isom}(x, y)$ is representable. In the fppf topology, representable functors are automatically sheaves.

For $\mathcal{M}_g$: $\operatorname{Isom}(C, C')$ is a locally closed subscheme of $\operatorname{Hom}(C, C')$, which itself is a locally closed subscheme of the Hilbert scheme $\operatorname{Hilb}_{C \times C'/S}$. Since Hilbert schemes are projective (when the ambient space is projective), $\operatorname{Isom}(C, C')$ is a quasi-projective scheme.